Introduction to Quantum Computing for Business

Groenland, Koen Book Introduction to Quantum Computing for Business Provided in Cooperation with: Amsterdam University Press (AUP) Suggested Citation: Groenland, Koen (2025) : Introduction to Quantum Computing for Business, ISBN 978-90-4856-899-4, Amsterdam University Press, Amsterdam, https://doi.org/10.5117/9789048568987 This Version is available at: https://hdl.handle.net/10419/321932 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ koen groenland INTRODUCTION TO QUANTUM COMPUTING FOR BUSINESS ow will businesses use quantum technology in the future? What problems will a quantum computer solve? How long will it take before these devices become commercially relevant? With the first generation of quantum computers on the horizon, understanding their impact is more relevant than ever. Luckily, you don’t need a physics degree to understand the functionality of these computers – just like you don’t need to know how a transistor works to excel in conventional it. This book is the perfect introduction to the opportunities and threats of quantum technologies. It equips you with the necessary knowledge to join cutting-edge discussions and make strategic decisions. koen groenland is a theoretical physicist with a PhD in the near-term applications of quantum computers. He works as an innovation officer at the University of Amsterdam, where he is responsible for setting up research collaborations and developing lifelong learning education for professionals. He is one of the driving forces behind Quantum. Amsterdam, the innovation hub that drives the commer cialisation of quantum technologies around the Dutch capital. “Easy to read and full of insights, a must-read for anyone looking to understand the real-world impact of quantum computing.” – Diederick Croese, Director of Center for Quantum and Society “This book offers a well-rounded, scientifically accurate overview of quantum technology, highlighting its significant potential for innovation.” – Christian Schaffner, Professor in Theoretical Computer Science, Director of QuSoft H koen groenland INTRODUCTION TO QUANTUM COMPUTING FOR BUSINESS    Introduction to Quantum Computing for Business.indd 1Introduction to Quantum Computing for Business.indd 1 11-02-2025 12:0511-02-2025 12:05 Introduction to Quantum Computing for Business Introduction to Quantum Computing for Business Koen Groenland Amsterdam University Press Cover illustration: © Dadara Cover design: Mijke Wondergem Lay-out: Crius Group, Hulshout Illustrations: © Dadara isbn 978 90 4856 898 7 e-isbn 978 90 4856 899 4 (pdf) doi 10.5117/9789048568987 nur 120 Creative Commons License CC-BY NC ND (http://creativecommons.org/licenses/by-nc-nd/4.0) K. Groenland / Amsterdam University Press B.V., Amsterdam 2025 Some rights reserved. Without limiting the rights under copyright reserved above, any part of this book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise). Table of Contents Part1 The essentials Preface: Why this book? 11 1 An introduction to the quantum world 15 1.1 What is quantum? 15 1.2 Four surprising phenomena 16 1.3 What does a quantum computer look like? 22 1.4 Further reading 26 2 The background: Why are we so enthusiastic about quantum technology? 27 2.1 What is quantum technology? 27 2.2 The importance of high-performance computing 28 2.3 Why can quantum computers have an advantage? 29 2.4 From algorithm to software 34 2.5 Further reading 35 2.6 Notes 35 3 The applications: What problems will we solve with quantum computers? 37 3.1 What applications offer a quantum speedup? 38 3.2 How can we compare different types of speedups? 44 3.3 Where is the killer application? 47 3.4 Further reading 51 3.5 Notes 52 4 Timelines: When can we expect a useful quantum computer? 55 4.1 What parameters are relevant? 55 4.2 How many qubits are needed? 58 4.3 How long until we have million-qubit machines? 63 4.4 Putting it all together 67 4.5 Further reading 69 4.6 Notes 69 5 Four myths about quantum computing 71 5.1 Myth 1: Quantum computers find all solutions at once 71 5.2 Myth 2: Qubits can store much more data than the same number of classical bits 72 5.3 Myth 3: Entanglement allows you to send information faster than light or to influence objects at a distance 73 5.4 Myth 4: Quantum computers are always ten years away. 75 5.5 Further reading 76 5.6 Notes 77 Part2 More about the applications 6 Applications in chemistry and material science 81 6.1 What problems in chemistry and material science will we solve? 81 6.2 Algorithms for quantum chemistry 83 6.3 A hype around quantum computing for climate change 85 6.4 A case study of a potential killer application: FeMoco 86 6.5 Further reading 88 6.6 Notes 89 7 The impact on cybersecurity 91 7.1 Cryptography is much more than just secrecy 91 7.2 The quantum threat is mainly to public key cryptography 93 7.3 What solutions exist? 97 7.4 Conclusion 100 7.5 Further reading 100 7.6 Note 101 8 Applications of quantum networks 103 8.1 The promises of the quantum internet 103 8.2 How useful is the quantum internet in practice? 104 8.3 The case for QKD 105 8.4 Conclusion 107 8.5 Further reading 107 9 Optimisation and AI: What are companies doing today? 109 9.1 Comparing Algorithms and Oranges 109 9.2 Where should we look for a new killer application? 113 9.3 Examples of results in different sectors 114 9.4 Further reading 122 9.5 Notes 123 Part3 The hardware and strategic actions 10 Quantum hardware 127 10.1 Different functionalities 127 10.2 Different building blocks 131 10.3 Further reading 132 10.4 Note 132 11 Error correction 133 11.1 What is error correction? 134 11.2 Longer computations need more qubits 138 11.3 What is the current state-of-the-art? 141 11.4 Conclusion 143 11.5 Further reading 144 12 What steps should your organisation take? 145 12.1 Common first steps 145 12.2 Prepare to use quantum applications 146 12.3 Migrating to post-quantum cryptography 149 12.4 Further reading 153 12.5 Note 153 Part4 The final bits 13 Further reading 157 13.1 I want to learn the technical details 157 13.2 I want to learn to program a quantum computer 159 13.3 I want to stay up to date with the latest developments 160 13.4 I want to learn more about business implications 161 14 Overview of quantum computers available today 163 15 Quantum Hype Bingo 165 16 Acknowledgements 167 17 Bibliography 169 18 Index 173 1 An introduction to the quantum world At a glance you don’t need to understand quantum mechanics to understand the functionality of quantum computers. But if you insist, quantum mechanics describes the behaviour of the smallest particles. It leads to many counterintuitive phenomena: computer memory can store multiple pieces of data simultaneously, but, when measured, nature selects just a single piece and throws away all the others. If you want to drive a car, do you need to understand how its engine works? Of course, you don’t! In a similar vein, you don’t need to know the details of quantum physics to read the rest of this book. So, feel free to skip this chapter. Nevertheless, we know that most people want to have some conceptual intuition about what quantum mechanics really is. It is not natural to leave one of the most used words in this book as an abstract concept, and it might be hard for the human brain to proceed without at least seeing some examples. Here is my best attempt to explain quantum mechanics in accessible terms. Proceed with caution, as things will almost certainly get confusing from here. 1.1 What is quantum? Quantum physics or quantum mechanicsis the theory that describes the tiniest particles, such as electrons, atoms, and small molecules. The theory is meant to describe the fundamental laws of nature using a set of mathematical equations, allowing us to predict cause and effect at the scale of nanometres. It answers questions like ‘What happens when I bring two electrons close together?’ or ‘Will these two substances undergo a chemical reaction?’. You can contrast quantum mechanics to Newton’s classical physics, which we learned in high school. The classical theory works great for objects the size of a building or a football but becomes inaccurate at much smaller scales. Quantum is, in a sense, arefinementof classical physics: the theories are effectively identical when applied to a coffee mug, but the more difficult quantum theory is needed to describe very small things. Some examples of systems where quantum could play a role are: – Atoms and the electrons that orbit around them. – Flows of electricity in microscopic (nano-scale) wires and chips. – Photons, the particles out of which light is made. 16 IntroduCtIon to Quantum ComputIng for BusIness We are going to need some physics jargon to proceed. We like to use the word ‘state’, which is a complete description of all the physical properties of the world at one instance: the locations of all the different particles, their velocities, how much they rotate, etc. Usually, the entire universe is too big to study, so we often simplify our world to a single, isolated particle or to a limited piece of computer memory. Let’s imagine a bare particle in an otherwise empty world. We may be interested in its location, which we’ll call x . For example, the world might look something likethe image below, which can be described by a very simple state: x = 5 (the ruler is just virtual). In the spirit of computing, we might look at a ‘bit’ that stores information. Think of it as a tiny magnet that can either point ‘up’ (1) or ‘down’ (0). The state of a piece of memory is easy to describe, simply by expressing the bit values one by one. For example: 11010. Importantly, the state of the world can change over time. We will often care about the state of the world at a certain moment, for example, at the beginning of a computation or at the end of it. 1.2 Four surprising phenomena The most iconic quantum phenomenon is superposition. Think about any property that we can (classically) measure, such as the position of a particle or the value of a bit on a hard drive (0 or 1). In quantum mechanics, many different measurement outcomes can be somewhat ‘true’ at the same time: a particle can be in multiple positions at once, or a bit could be 0 and 1 simultaneously. When we say ‘at the same time’ we mean that, to predict an IntroduCtIon to the Quantum World 17 any cause and effect, we need to keep track of all these possibilities. To illustrate a superposition, I sometimes picture a quantum particle splitting into many opaque copies of itself, spread out over space, where the degree of transparency determines how likely the particle is to be found there: the darker it is, the more presence it has at that location. To throw in some more examples of superpositions: an electron can move at a velocity of 10m/s and 100m/s at the same time (which obviously also leads to a superposition in its location). More relevant for us: a computer memory might store the numbers 5 and 11 ‘simultaneously’ or even 46 different Microsoft Excel spreadsheets ‘at once’. An important building block to make this all work is the qubit, which is any kind of hardware that can store bit values 0 and 1, and any possible superposition of these two. If we have a bunch of qubits together, we’ll call it a quantum memory. Let us illustrate the weirdness of superpositions with an example where the 46 spreadsheets each take 1 megabit (Mb) to store. A regular, classical hard drive would allocate the first Mb to a first spreadsheet, then another Mb to store the second, and so forth. In total, it would use 46 Mb. The quantum memory has an additional option to store the spreadsheets in superposition: using the qubit-equivalent of just 1Mb (one million qubits) it would encode all the data in just that limited amount of memory. Whereas 1 Mb of classical memory can fit just one spreadsheet, a quantum memory of 1 Mb can represent several of them, all thanks to the unique properties of quantum physics. However, as we’ll see later, there is a catch to storing all that data so compactly. How can you possibly describe a world where particles and computer memories are in superposition? For now, let’s focus on an isolated particle. We specify its state using a lengthy list, where for each possible position, we store a number called the amplitude, which is related to how likely the particle is to be found at that location. In other words, the state describes precisely to what extent a particle is at position x = 0 , to what extent at position x = 1 , and so forth, for every possible location that the particle can be at. And indeed, this list could be infinitely long! Luckily, when dealing with computers, we work with simpler objects. A quantum bit 18 IntroduCtIon to Quantum ComputIng for BusIness needs just two amplitudes, which denote the extent to which the bit is ‘0’ or ‘1’, respectively. The amplitudes used to describe quantum states feel somewhat analogous to probabilities, which can similarly tell us the likelihood that, for example, a particle can be found at a particular location. However, there is a fundamental difference. Probabilities in the classical world help us deal with information we don’t have: surely, the particle is already at some location, but perhaps we just don’t know which location yet. Quantum mechanics is different. Even if we know every tiny detail about the location of a particle, we still need to describe it as a superposition. Fundamentally, the location is not determined yet. Hence, there is literally no better way to describe the particle than by tracking this convoluted superposition. Amplitudes are also more finicky to deal with than probabilities because these numbers can become negative (and for math experts, they can even be complex numbers). The second weird phenomenon is how quantum measurements work. Why do we never observe an electron at two places at the same time? Why do I never find a car both moving and standing still? In quantum mechanics, as soon as we measure the location of a particle, it instantly jumps to a single location at random – making its location fully determined. Similarly, when we measure a qubit, it jumps to either ‘0’ or ‘1’. When we measure the data in a quantum memory, we may find any one of the 46 spreadsheets that were stored. A measurement essentially changes a system into a normal, classical state. The effect of a measurement is intrinsically random (and hence, our world is not deterministic!). But this doesn’t imply that we cannot understand quantum mechanics. We can calculate the probabilities of measurement outcomes with incredible precision as long as we know the state before the measurement. It is important to note that we cannot learn anything about the world without measuring – it is our only way to obtain data about physical objects. Any observation, even a slight peek at our system, is a measurement in quantum mechanics. Additionally, measurements are destructive in the sense that they change the state of the world. We fundamentally cannot ‘look’ at a particle without disturbing it. In fact, measurements delete all the rich data encoded in a superposition! If a particle was initially at position x = 0 ,  x = 3 and x = 10, all simultaneously, then upon measurement, it jumps to one of these three options. To give you a bit of jargon, we call this instantaneous change a ‘collapse.’ From that moment, it is 100% at a fixed location: if, at first, we measure the particle to be at x = 3 , then any an IntroduCtIon to the Quantum World 19 subsequent measurement will give the same result, until some other force moves it again. In the context of a quantum computation, this means that we should carefully choose when we perform any measurements – we cannot just peek at the data at any moment we like, or we risk disturbing a superposition. This also means that a single piece of quantum memory cannot store an immense number of spreadsheets at the same time – at least, you wouldn’t be able to retrieve each of them. To store 15 Mb worth of classical data, we need 15 Mb worth of qubits. Hence, quantum computers are not particularly useful for storing classical data. The fact that a measurement changes the state of the world poses a serious problem for the engineers who are building quantum computers. No matter what material we construct our qubits from, they will surely interact with other nearby particles, and some of these interactions could act like destructive measurements. We call this effect decoherence, and, as we will see later, this forms one of the core challenges to large-scale quantum computation. At this point, quantum data doesn’t seem particularly useful. Why would we want to deal with superpositions if they lead to all this uncertainty? The important advantage stems from the way in which a quantum computer can process quantum data. Using quantum mechanics, a device can manipulate data in ways that a classical computer could never do. That leads us to the third unique phenomenon. A quantum computer can manipulate the data it stores using so-called quantum gates, or simply ‘gates’ for short. These are rapid bursts of some physical forces that change the state of one or more qubits. They can turn a classical-looking state into a quantum superposition or vice versa. They can act like logical operations, like the AND and OR gates that are used in classical electronics, but also like new quantum logic that has no classical counterpart. From a functional perspective, a quantum gate takes one or more qubits as input, changes their internal state, and then outputs the same number of qubits (with their altered states). In other words, the number of physical objects remains unchanged, but the overall state changes. As an example, you may think of our prototypical magnet that was initially pointing ‘up’, but a quantum gate might flip this to ‘down’. There are many such gates possible, each having a different effect on their input. We like to give them names in capital letters, such as X, Z, H, and CX. Importantly, a quantum gate is deterministic, meaning that its input-output behaviour is always the same, as opposed to the quantum measurements we saw earlier. 20 IntroduCtIon to Quantum ComputIng for BusIness The canonical way to describe a quantum computer program is by defining a sequence of quantum gates, where for each gate, we also indicate what qubits are supposed to be the gate’s input. At the end of the computation, we measure all qubits. An example of such a program, using the standard Quantum Assembly (QASM) language, is given below. Together, these steps can be graphically displayed in a quantum circuit, as shown here on the right. Quantum circuits represent each qubit with a horizontal line and indicate time flowing from left to right. Whenever a box with a letter is displayed over a qubit line, then the corresponding gate should be applied. This isn’t unlike the way we read sheet music! You may notice that sometimes, two or more gates can be performed in parallel as long as they act on different qubits. When we run a circuit on an actual quantum computer, the final measurements lead to probabilistic outcomes. We get to see a bunch of ones and zeroes: one classical bit for each qubit. If the circuit is a good quantum algorithm, then, with high probability, these classical bits will tell us the answer we are looking for. But even then, we might need to redo the computation a few times and take (for example) the most common result as our final answer. If you are completely confused at this point, you are not alone. The whole business of quantum superposition and quantum operations is incredibly complex and is not something you could possibly master after reading a few pages. Scientists who have studied the subject for many years are still an IntroduCtIon to the Quantum World 21 frequently baffled by deceptive paradoxes and counter-intuitive phenomena. On the other hand, we hope that the functionality of quantum circuits makes some sense: we define a list of instructions and feed them into a machine that can execute them. We don’t have to know precisely what’s going on under the hood! There is one remaining quantum phenomenon to cover – one that comes with a mysterious flair surrounding it. We’re talking about quantum entanglement, which we’ll describe using the following example. Imagine that we have two qubits, which we can transport independently from each other without disturbing the data they store. Together, the qubits can represent the states 00, 01, 10, or 11, or any superposition of these. According to quantum mechanics, we can create a very specific state where the pair of qubits is simultaneously 00 and 11. Now, imagine that computer scientist Alice grabs one of the qubits, takes it on her rocket ship, and flies it all the way to the dwarf planet Pluto. The other qubit remains on Earth in the hands of physicist Bob. Upon arriving on Pluto, Alice measures her qubit and finds outcome ‘1’. A deep question is: what do we now know about Bob’s qubit? Since the only possible measurement outcomes were 00 and 11, the other qubit can only be measured as ‘1’ from now onwards. It essentially collapses to be 100% in the state ‘1’. But how could the Earth-based qubit possibly know that a measurement occurred on Pluto? What mechanism made it collapse? According to Einstein’s theory of relativity, information cannot travel faster than the speed of light, which translates into a few hours between Earth and Pluto. Nevertheless, measuring the qubits intwo faraway locations will always give a consistent result, even when the two qubits are measured at exactly the same time. This paradox reveals, once again, how confusing quantum mechanics can be. However, the story above is perfectly consistent with both quantum mechanics and the theory of relativity. The core principle is that no information can be sent faster than light between Alice and Bob. For example, can you see why Bob has no way of detecting when Alice performs her measurement just by looking at his entangled qubit? In the most common interpretation of quantum mechanics, the Earth qubit does indeed change its state instantaneously when Alice measures her qubit, although there is no way to exploit this effect for fast messaging. More generally, entanglement is the phenomenon where two or more faraway qubits can have correlated measurement outcomes that are classically impossible. There is a fascinating further discussion about the philosophy behind entanglement, but we’ll leave that to other sources. What matters 22 IntroduCtIon to Quantum ComputIng for BusIness to us is that entanglement leads to new functionalities that we can exploit. We will discover what these are in the chapter on quantum networks. So, there you have it: four surprising phenomena you may hear frequently in quantum technology conversations. To summarise: – Superposition: the phenomenon where a qubit is both 0 and 1 at the same time. – Quantum measurement: measuring a quantum memory destroys super - position. The result we obtain is probabilistic. – Quantum gates: deterministic changes to the state of qubits, which generalise classical logic gates like OR, AND, NOT. A list of several quantum gates (together with the qubits they act on) forms a quantum circuit. – Entanglement: qubits separated over a long distance can still share unique properties. 1.3 What does a quantum computer look like? Most large-scale computing today happens in data centres, where we don’t care much about the specifics of the devices that do our calculations. We also expect that future quantum computers will mostly be tucked away in the ‘cloud’, making their appearance and inner workings largely irrelevant to most users. However, for this optional chapter, we can take the opportunity to view what today’s cutting-edge hardware looks like. There are many different ways to build a quantum computer, each based on distinct physical systems and principles. Here, we describe the example of so-called superconducting qubits, a relatively mature platform used by companies like IBM, Google, and Rigetti and several academic institutes. Research institute QuTech in Delft, the Netherlands, was kind enough to provide photos that allow us to look inside their labs. We will see that only a tiny part of the computer is actually ‘quantum’, whereas most of the machine consists of classical machinery that’s required to keep the computer working. The real quantum magic happens on a chip, not unlike the computer chips used in your laptop or phone. The qubits are formed by tiny electronic circuits where the flow of electrical current is restricted to just one out of two states: the ‘bit’ states 0 and 1. Since this is a quantum system, the current can also be in a superposition – picture all the electrons in the wire participating both in flow ‘0’ and flow ‘1’ simultaneously! This only works when the chip is cooled down to unimaginably low temperatures, down to around 10 millikelvin – a hundredth of a degree above absolute zero. At these temperatures, the electronic circuits become superconducting, such an IntroduCtIon to the Quantum World 23 that an initial current can flow indefinitely. This is important because any damping of the current would cause unwanted disturbance to the qubit state. The temperature constraint is why the quantum chip is placed in a massive dilution refrigerator, a cylinder of about half a metre in diameter and over a metre tall, which specialises in keeping the quantum chip cool. In the future, larger quantum computers may need even bigger fridges or combine several of these close together. Deeper parts of the fridge have increasingly low temperatures, allowing us to cool in stages. An example could be to cool a first environment to 35 Kelvin (-283 °Celsius or -396.7 °Fahrenheit), followed by subsequent stages to ~3K, 900mK, 100mK, until the final stage of ~10mK is reached. Engineers typically suspend the fridge on the ceiling so that the higher temperatures are on top, and the ultracold quantum chip is placed at the very bottom. The internals are shaped accordingly: several layers of gold disks are hung below one another, one disk for each temperature zone. A large number of wires run between the disks, transporting signals between the ceiling and the lowermost areas. The whole structure forms the iconic metal chandelier that you often see in images, although it would all be covered by a boring metal case when the fridge is in operation. To make the qubits do something useful, like executing a quantum gate or performing a measurement, we need to send signals into the chip. Just like with classical computers, a ‘signal’ is a voltage difference between a quantum chip. photo credits: marc Blommaert for Qutech. 30 IntroduCtIon to Quantum ComputIng for BusIness meaning that numbers up to 18,446,744,073,709,551,615 can be processed. Each of these elementary steps can be something like addition, multiplication, a comparison, etc., and we have powerful tools to weave these basic operations together to form efficient software. Now, quantum computers are supposed to be even faster, right? Well, it’s not hard to find support for that claim: news headers by techradar2 and Iflscience3. You may be disappointed to hear that, as of 2024, quantum computers cannot even add or multiply numbers of more than 3 or 4 bits. And even if they could, their rate of operation would by no means reach several GHz, but more likely several MHz (a few million operations per second) at best. In other words, they’re more than a thousand timesslower.To make things worse, the information in quantum computers is extremely fragile and needs to be constantly checked and corrected using so-callederror correction.This is a form of overhead that could make quantum computers another several orders of magnitude slower. Even in the far future, when quantum computers are more mature and more reliable, we still expect them to be much slower than the classical chips at that time. How does this rhyme with the news about ever-faster quantum computers? And why are we still interested in these slow machines? As we claimed before, we hope to do certain computations in afundamentally different way. Let’s look at a beautiful analogy that Andy Matuschak and Michael Nielsen bring up in their online course Quantum Country4. the BaCkground: Why are We so enthusIastIC aBout Quantum teChnology? 31 Imagine that you’d like to travel from Morocco to Spain, which are separated by a small piece of sea called the Strait of Gibraltar. If your technology does not allow you to cross the sea, then you’d need to take a large detour, all the way through North Africa, past the Arabian Peninsula, and through Europe, before you can reach your destination. This represents the steps taken by a classical computer. In the same analogy, a quantum computer grants you the ability to traverse both land and sea (much like a hovercraft) so that you can take a much more direct route. The beauty of quantum computation is that we have a fundamentally different way to travel (do computations), which can sometimes bring us to our destination using a shorter route (doing fewer computational steps). Even with a much slower vehicle (computer), one may arrive at the destination sooner. In fact, the quantum advantage often grows as problems become larger and more complicated. The analogy also shows that quantum computers do not always have an advantage: you would not want to travel from Amsterdam to Berlin by hovercraft. Unfortunately, in many cases, we don’t yet know what the fastest means of transportation is. It is still an active area of research to completely map out the landscape over which quantum and classical computers can travel and to determine which problems allow a speedup, and which don’t. For this reason, we don’t expect that classical computers will be replaced any time soon. Instead, classical and quantum processors will live side by side, and programmers will pick whichever tool is better suited to solve a certain problem. The situation could be similar to how we use graphical processing units (GPUs) today, which offer tremendous the BaCkground: Why are We so enthusIastIC aBout Quantum teChnology? 33 speedups for the training of artificial intelligence models but are not made to replace regular classical processors (CPUs). Perhaps we should even give quantum computers a similar abbreviation, like ‘QPU’ for Quantum Processing Unit. In the analogy with the Strait of Gibraltar, the precise route that you travel denotes the chosenalgorithm.In the field of computer science, an algorithm is astep-by-step list of instructionsthat describes how a computational problem should be solved. The‘steps’here should be sufficiently simple so that it is completely unambiguous how to do them. They could be operations such as adding, multiplying, or comparing two numbers. Needless to say, the fewer steps the algorithm requires, the better. By exploiting quantum mechanics, a quantum computer introduces new basic steps that are impossible to perform on a classical computer. For example, the previous chapter introduced quantum logic gates that generalise operations like AND and OR. Using these building blocks, we can formulate quantum algorithms that take much fewer steps than the best classical algorithm ever could! In the end, the time needed to solve a problem can be very roughly calculated as: “Time to solve a problem” = “time per step” × “number of steps required” The ‘time per step’ is a property of the hardware that you use. Clearly, a faster CPU will lead to faster solutions. The ‘number of steps required’ is dictated by the algorithm. The latter is precisely how quantum computers can offer spectacular speedups. As long as the improvement in the ‘number of steps required’ compensates for the disadvantage in ‘time per step’, a quantum computer can help us solve problems in less time! A recurring theme in this book is the search for industrially relevant quantum algorithms. This turns out to be more challenging than it seems at first sight. Quantum algorithms are built on deep and complex mathematics, rely on counter-intuitive quantum phenomena, and require inventive new methods to tackle a problem. Simple tweaks to existing classical algorithms are rarely sufficient. In fact, for most problems, no quantum speedups have been identified at all, despite the best attempts by scientists worldwide. We might go as far as to say that, even if we had a large-scale quantum computer today, its value would be limited. For this reason, the ongoing development of novel algorithms is exceedingly important. 34 IntroduCtIon to Quantum ComputIng for BusIness 2.4 From algorithm to software In the end, simply finding a good algorithm is not enough: it has to be turned into software, a piece of language that explicitly tells a computer how to execute the step-by-step instructions. The difference between ‘algorithms’ and ‘software’ is subtle. An algorithm is a purely mathematical description that describes precisely how numbers should be manipulated. It could tell which two numbers must be multiplied, what function must be evaluated, or how an image must be transformed. However, different computers can use different types of processors and memory, and an algorithm does not describe how these operations are done on a specific computer. This is where software comes into play. It describes precisely what hardware operation must be called, where each number is stored in memory, and how an image is represented in binary. As an analogy, you may think of the algorithm as a recipe to bake the perfect chocolate cookie. The algorithm should unambiguously describe what should happen to the ingredients: in what order they should be mixed, how long they should be heated at what temperature, etc. However, to build a factory that produces these cookies, you need to be even more specific: Where is the sugar stored? Out of what pipe does the dough flow? How are cookies laid next to each other in the oven? Fundamentally, core scientific breakthroughs come from finding new algorithms. Once a new algorithm is found, it can be re-used many different times on any capable machine (assuming a good software developer will turn it into appropriate code!). In this book, we care less about quantum software and more about quantum algorithms. Firstly, the algorithms tell us precisely the functionality that quantum computers can offer. Moreover, we don’t yet know how a mature quantum computer will be programmed or how quantum hardware and software will change in the following years. On the other hand, once a new algorithm is found, it can be cherished forever. Now that we have come to appreciate algorithms, it is natural to ask which quantum algorithms we know of. What problems do quantum computers solve well? And how do these algorithms compare to their classical equivalents? This will be the topic of the next chapter. the BaCkground: Why are We so enthusIastIC aBout Quantum teChnology? 35 2.5 Further reading The Map of Quantum Computing (YouTube)– a 30-minute overview video by domain of science that forms a great supplement to this book. Chris ferrie’s book What You Shouldn’t Know About Quantum Computers debunks several myths about quantum computers, presented in an accessible way. are you looking for a much more extensive and technical source that covers pretty much everything there is to know about quantum computers? french consultant olivier ezratty has written a 1500+ page book, Understanding Quantum Technologies. 2.6 Notes 1. See e.g. https://www.marketsandmarkets.com/Market-Reports/Quantum-HighPerformance-Computing-Market-631.html and https://www.mordorintelligence.com/ industry-reports/cloud-high-performance-computing-hpc-market. 2. Wyciślik-Wilson, S.E. (2019) ‘Google creates quantum chip millions of times faster than the fastest supercomputer’, TechRadar. https://www.techradar.com/news/googlecreates-quantum-chip-millions-of-times-faster-than-the-fastest-supercomputer. 3. Dunhill, J. (2021) ‘Chinese Scientists Create Quantum Processor 60,000 Times Faster Than Current Supercomputers’, IFLScience. https://www.iflscience.com/chinesescientists-create-quantum-processor-60000-times-faster-than-current-supercomputers-61475. 4. Matuschak, A. and Nielsen, M. (2019) ‘Quantum Country’. https://quantum.country. 3 The applications: What problems will we solve with quantum computers? At a glance the most important application areas are: 1. the simulation of material properties and chemical processes; 2. cracking cryptography; 3. using quantum networks to distribute cryptographic keys; and 4. solving large-scale optimisation and aI problems. getting utility out of a quantum computer is not straightforward. It requires an algorithm that beats all other known methods (even those that run on very fast classical computers), and it must tackle a problem with real-world relevance. especially in optimisation and aI, we have not found a convincing ‘killer application’ yet. In the previous chapter, we saw that quantum algorithms can solve certain problems in fewer steps, allowing a large-scale quantum computer to com - plete specific tasks much faster than any classical computer could. However, the precise speedup depends strongly on the task at hand. Therefore, the most important question in this field is: for which problems do quantum computers offer a meaningful advantage? The Quantum Algorithm Zoo 1 lists pretty much all known quantum algorithms. It has become an impressive list that cites over 400 papers. Unfortunately, upon closer inspection, it’s hard to extract precisely the useful business applications, for a few reasons. Some algorithms solve highly artificial problems for which no real business use cases are known. Others may make unrealistic assumptions or may only offer a speedup when dealing with an outrageously large amount of data (that we never encounter in the real world). Nevertheless, scrolling through it is definitely recommended. For this book, we take a different approach. We focus specifically on algorithms with plausible business applications. To assess their advantage, we split our main question into two parts: – What applications offer a quantum speedup? – How large is this speedup in practice? 38 IntroduCtIon to Quantum ComputIng for BusIness 3.1 What applications offer a quantum speedup? We foresee four major families of use cases where quantum computing can make a real impact on society. We briefly discuss each of them here. For more details, we dedicate a more in-depth chapter to each application family in Part2. 1. Simulation of other quantum systems: Molecules, materials, and chemical processes Most materials can be accurately simulated on classical computers. However, in some specific situations, the locations of atoms and electrons become notoriously hard to describe, sometimes requiring quantum mechanics to make useful predictions. Such problems are the prototypical examples of where a quantum computer can offer a great advantage. Realistic applications could be in designing new chemical processes (leading to cheaper and more energy-efficient factories), estimating the effects of new medicine, or working towards materials with desirable properties (like superconductors or semiconductors). Of course, scientists will also be excited to simulate the physics that occur in exotic circumstances, like at the Large Hadron Collider or in black holes. Simulation is, however, not a silver bullet, and quantum computers will not be spitting out recipes for new pharmaceuticals by themselves. Breakthroughs in chemistry and material science will still require a mix of theory, lab testing, computation, and, most of all, the hard work of smart scientists and engineers. From this perspective, quantum computers have the potential to become a valued new tool for R&D departments. 2. Cracking a certain type of cryptography The security of today’s internet communication relies heavily on a cryptographic protocol invented by Rivest, Shamir, and Adleman (RSA) in the late 70s. The protocol helps distribute secret encryption keys (so that nobody else can read messages in transit) and guarantees the origin of files and webpages (so that you know that the latest Windows update actually came from Microsoft, and not from some evil cybercriminal). RSA works thanks to an ingenious mathematical trick: honest users can set up their encryption using relatively few computational steps, whereas ‘spying’ on others would require one to solve an extremely hard problem. For the RSA cryptosystem, that problem isprime factorisation,where the goal is to the applICatIons: What proBlems WIll We solve WIth Quantum Computers? 39 decompose a very large number (for illustration purposes, let’s think of 15) into its prime factors (here: 3 and 5). As far as we know, for sufficiently large numbers, this task takes such an incredibly long time that nobody would ever succeed in breaking a relevant code – at least on a classical computer. This all changed in 1994 when computer scientist Peter Shor discovered that quantum computers happen to be quite good at factoring. The quantum algorithm by Shor can crack RSA (and also its cousin calledelliptic curve cryptography, abbreviated to ECC) in a relatively efficient way using a quantum computer. To be more concrete, according toa recentpaper,2 a plausible quantum computer could factor the required 2048-bit number in roughly eight hours (and using approximately twenty million imperfect qubits). Note that future breakthroughs may further reduce the stated time and qubit requirements. Fortunately, not all cryptography is broken as easily by a quantum computer. RSA and ECC fall into the category ofpublic key cryptography,which delivers a certain range of functionalities. A different class of protocols issymmetric key cryptography,which is reasonably safe against quantum computers but doesn’t provide the same rich functionality aspublic keycrypto. The most sensible approach is replacing RSA and ECC with so-calledpost-quantum cryptography(PQC): public key cryptosystems resilient to attackers with a large-scale quantum computer. Interestingly, PQC doesnotrequire honest users (that’s you) to have a quantum computer: it will work perfectly fine on today’s PCs, laptops, and servers. At the time of writing, a complex migration lies ahead of pretty much every large organisation in the world, which comes in addition to many existing cybersecurity threats. The foundations have been laid: thanks to the American National Institute of Standards and Technology (NIST), cryptographers from around the globe came together to select the best quantum-safe alternatives, culminating in the publication of the first standards in August2024. These are the new algorithms that the vast majority of users will adopt. Unfortunately, many governments and enterprises run a great amount of legacy software that is hard to update, making this a complex IT migration that could easily take 5–15 years, depending on the organisation. There’s a serious threat that quantum computers will be able to run Shor’s algorithm within such a timeframe, so organisations are encouraged to start migrating as early as possible. A new type of cryptography comes with its own additional risks: the new standards have not yet been tested as thoroughly as the nearly fifty-year-old RSA algorithm. Ideally, new implementations will behybrid, meaning that 46 IntroduCtIon to Quantum ComputIng for BusIness Here is a rough overview of quantum speedups as we understand them today, categorised by their type of asymptotic speedup: Cracking RSA / ECC (Shor’s algorithm) Some chemistry and material science Brute-force search (Grover’s algorithm) Differenal equaons, Lasso, … NP-complete problems Sorng Loading a large amount of data from a drive Exponenal Polynomial No speedup Heurisc (unknown) Annealing (opmizaon) Variaonal Quantum Circuits Some chemistry and material science Binary opmizaon Neural Networks Support Vector Machines – The ‘exponential’ box is the most interesting one, featuring applications where quantum computers seem to have a groundbreaking benefit over classical computers. It containsShor’s algorithmfor factoring, explaining the towering advantage that quantum computers have in codebreaking. We also believe it contains some applications inchemistry and material science, especially those relating to dynamics (studying how molecules and materials change over time). – The’polynomial’box is still interesting, but its applicability is unclear. Recall that a quantum computer would need much more timeper step– and, moreover, it will have considerable overhead due toerror correction. Does a polynomial reduction in the number of steps overcome this slowness? According to arecent paper,10 small polynomial speedups (as achieved byGrover’s algorithm) will not cut it, at least not in the foreseeable future. – For some computations, a quantum computer offersno speedup.Examples include sorting a list or loading large amounts of data. If this were the complete story, then most people would agree that quantum computing is a bit disappointing. It would be a niche product for hackers and a tiny community of physicists and chemists who study quantum mechanics itself. – Fortunately, there is yet another category: many of the most exciting claims come from theheuristicalgorithms. This term is used when an algorithm might give a suboptimal solution (which could still be useful) or when we cannot rigorously quantify the runtime. Such algorithms are common on classical computers: neural networks fall in this category, and these caused a significant revolution in AI. Unfortunately, it is unclear what the impact of currently known heuristic quantum algorithms will be. the applICatIons: What proBlems WIll We solve WIth Quantum Computers? 47 In summary, the potential for economic value varies greatly across quantum algorithms. The case of factoring has a clear and convincing speedup, but is only useful for codebreaking (where we hope that impact is limited thanks to the adoption of quantum-safe cryptography). In contrast, machine learning and optimisation do tackle a broad palette of relevant problems, but the speed advantage of a quantum computer remains uncertain in this field. The applications of chemistry and material science fall somewhere in the middle, with some relevant areas ofapplicability and concrete indications of a practical speed advantage. 3.3 Where is the killer application? Is there hope that we’ll find new quantum algorithms with a large commercial or societal value? For a quantum algorithm to be truly impactful, we require two properties: 1. [Useful] The algorithm solves a problem with real-world significance (for example, because organisations can work more efficiently or because it helps answer a scientific question). 2. [Better/faster] Using this particular algorithm is the most sensible* choice from a technical perspective,** even when compared to all other possible methods. Throughout this book, we will use the term quantum utility when both properties are convincingly satisfied. The precise definition can be a bit finicky, so before we start searching for utility, we need to get some technical details out of the way. * What is ‘sensible’ (2) depends strongly on the context of the real-world problem (1). In most cases, we care about how fast a problem is solved, but one should also take into account the total cost of developing the software, the cost of leasing the hardware, the energy consumption, the probability of errors, and so forth. For example, a high-frequency trader might be happy with a 2% faster algorithm even if the costs are sky-high and there’s a decent chance of failure, whereas a hospital could dismiss a 200x faster quantum approach if the costs don’t outweigh the benefits. Indeed, what is ‘sensible’ is highly subjective. In practice, we can relax this requirement somewhat and focus primarily on speed, which is a sufficiently complex figure of merit on its own. Ideally, the quantum algorithm should enjoy anexponentialspeedup or at least a large polynomial speedup. 48 IntroduCtIon to Quantum ComputIng for BusIness ** We explicitly look for technical perspectives. Otherwise, one might also say that using a quantum algorithm is commercially the best option because it creates good PR or because it keeps the workforce excited. Then, perhaps, the first utility has already been reached! However, this is not the computational revolution that we’re looking for, so we explicitly exclude such non-technical reasons in property (2). Similarly, we don’t want to worry too much about legal issues (‘it doesn’t comply with regulations’) because it feels somewhat artificial to dismiss a quantum algorithm for such reasons. Supremacy, advantage, utility Around 2019 and 2020, the terms quantum supremacy and quantum advantage were popularly used when quantum computers did, for the first time, beat the best supercomputers in terms of speed (property 2).11, 12 This involved an algorithm that was cherry-picked to perform well on a relatively small and noisy quantum computer whilst being as challenging as possible for a conventional supercomputer. Quantum advantage was mostly a man-on-the-moon-type scientific achievement, showcasing the rapid progress in hardware engineering and silencing the sceptics who still thought quantum computing wouldn’t work. There was no attempt to have any practical value (1). As a natural next step, the race is on to be the first to run something useful whilst leaving classical supercomputers in the dust. This led IBM to coin the term quantum utility, 13 which we adapted above. In the following years, we can expect the leading hardware and software manufacturers to maximise the amount of ‘utility’ that they could possibly squeeze out of medium-sized quantum computers, whilst competitors will use their best classical simulations to dispute these claims. The first battles have already been fought: in June2023, IBM claimed to simulate certain material science models better than classically possible,14 quickly followed by two scientific responses that showed how easy it was to simulate the same experiment on a laptop.1516 It seems to us that such healthy competition is good for the field overall. It should lead to increasingly convincing and rigorous quantum utility, from which the end-users will eventually profit! In parallel, there is a rapidly expanding number of press releases by startups and enterprises that claim to create business value by solving industrial problems on today’s hardware, often without sharing many details. These approaches typically start with a relevant problem in mind and hence score well on usefulness (1). However, it is questionable whether the applICatIons: What proBlems WIll We solve WIth Quantum Computers? 49 quantum algorithms were indeed the best option (2), and most reports we’ve seen hardly bother to show any argumentation in this direction. Such claims should only be taken seriously if a rigorous benchmark against state-of-the-art classical techniques is included. Do known algorithms provide utility? With the quantum utility criteria in mind, we can revisit the algorithms that were discussed before. (1) useful (2) Better than classical optimisation: rigorous but slow algorithms ✓? optimisation: fast algorithms in search of a use cases ? ✓ optimisation: heuristic algorithms ✓? simulation of molecules and materials ✓? Breaking rsa ✓ ✓ Several ‘rigorous but slow’ algorithms, most notably Grover’s algorithm, have an extensive range of industrial applicability. However, it seems that, in practice, other (classical) approaches solve such problems faster. The quadratic speedup will be insufficient in the near term, and it’s unclear if it will be in the future. Then, we have several exponential speedups, like the algorithm for topological data analysis, for which no practical uses have been found (despite many scientific and industrial efforts). Most optimistic outlooks focus on heuristic algorithms, for which the speed advantage will become clear with maturing hardware. Nevertheless, we judge that no optimisation algorithms can tick both boxes for quantum utility yet. Even for simulation of molecules and materials, it is not straightfoward to pinpoint precisely where we can find utility. Classical computers are already incredibly fast, and excellent classical algorithmic techniques have been developed. Scientist Garnet Chan even givestalks that are suggestively titled ‘Is There Evidence of Exponential Quantum Advantage in Quantum Chemistry?’.17The case for quantum simulation is subtle, and we elaborate on this matter in the chapter on applications in chemistry and material science. To the best of our knowledge, codebreaking (Shor’s algorithm) is the only impactful algorithm that has little competition from classical methods. Hopefully, most critical cryptography will be updated well before a quantum computer arrives, making large-scale deployment of Shor’s algorithm 50 IntroduCtIon to Quantum ComputIng for BusIness relatively uninteresting. Either way, the application of codebreaking is not quite the positive innovation that quantum enthusiasts are looking for. Could the nature of quantum mechanics be such that exponential speedups are only found in codebreaking, chemistry, and a bunch of highly artificial toy problems, but nowhere else in the broad spectrum of practically relevant challenges? Most people would argue that such a scenario is unlikely. There are still high hopes that either some of the caveats with existing algorithms will be addressed or that new breakthrough algorithms will be discovered. How optimistic you are about quantum computing should depend on (at least) the following questions: – How impactful will heuristic and to-be-discovered algorithms be compared to classical algorithms? In other words, what is the algorithmic potential of quantum computing? – How will quantum hardware develop relative to classical hardware? Ultimately, the commercial success of quantum computers depends strongly on these questions. If we allow ourselves to do some more hypothetical dreaming, we imagine that the following future scenarios could be possible, on a spectrum of optimism versus pessimism: Starting on the pessimistic side, if one believes that optimisation algorithms turn out to be lacklustre, then quantum computing might remain a niche for academics. However, depending on the utility of more widely applicable algorithms, one might predict that quantum computers will be installed in special-purpose computing facilities or, even more optimistically, that they the applICatIons: What proBlems WIll We solve WIth Quantum Computers? 51 become increasingly common additions to data centres (much like GPUs today). Where would you place yourself in this figure? 3.4 Further reading ‘The Potential Impact of Quantum Computers on Society’18 (ronald de Wolf, 2017) is an accessible overview of known algorithms, together with an assessment of how we can ensure a mostly positive net effect on society. ‘Quantum Algorithms: An Overview’19 (ashley montanaro, 2016) is a more technical overview paper that describes a selection of impactful algorithms in greater detail. professor scott aaronson warns us to ‘Read The Fine Print’ of optimisation algorithms.[appeared inNature physics, with paywall] professor sanker das sarma warns of hype within the field of quantum optimisation and machine learning. (technical) a quantitative analysis of grover’s runtime compared to today’s supercomputers. (scientific paper) amazon researchers lay out a comprehensive list of end-to-end complexities of nearly every known quantum algorithm. 52 IntroduCtIon to Quantum ComputIng for BusIness 3.5 Notes 1. Jordan, S. (2024) Quantum Algorithm Zoo. https://quantumalgorithmzoo.org. 2. Gidney, C. and M. Ekerå (2021) ‘How to Factor 2048 Bit RSA Integers in 8 Hours Using 20 Million Noisy Qubits’, Quantum, 5, p.433. https://doi.org/10.22331/q-2021-04-15-433. 3. Harrow, Aram W, Avinatan Hassidim, and Seth Lloyd (2008). ‘Quantum Algorithm for Linear Systems of Equations’. Physical Review Letters, 103 (15) 150502. https://doi. org/10.1103/PhysRevLett.103.150502 4. Aaronson, S. (2015). Read the fine print. Nature Physics, 11(4), 291–293. https://doi. org/10.1038/nphys3272. Open access: https://www.scottaaronson.com/papers/qml.pdf. 5. Liu, Y., Arunachalam, S., & Temme, K. (2021). A rigorous and robust quantum speedup in supervised machine learning. Nature Physics, 17(9), 1013–1017. https://doi. org/10.1038/s41567-021-01287-z 6. Qiskit. ‘How Ewin Tang’s Dequantized Algorithms Are Helping Quantum Algorithm Researchers’. Qiskit (blog), 15March2023. https://medium.com/qiskit/how-ewintangs-dequantized-algorithms-are-helping-quantum-algorithm-researchers3821d3e29c65. 7. With the symbol ~ we mean ‘roughly proportional to’. It allows us to write down an approximation of a function, making them easier to read, throwing away some details are not important here. 8. You may find even sources stating that Shor’s algorithm takes a time proportional to n2log(n). Such scaling is theoretically possible but relies onasymptotic optimisationsthat are unlikely to be used in practice. 9. Technically, the best algorithms for factoring, like the general number field sieve, have a scaling behaviour that lies between polynomial and exponential. Hence, the speedup of Shor’s algorithm is technically a bit less than ‘exponential’ – a more correct term would be ‘superpolynomial’. Still, this book (and many other sources) continue to use the term ‘exponential speedup’ to emphasise the enormous scaling advantage over polynomial speedups. 10. Babbush, R. et al. (2021) ‘Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage’, PRX Quantum, 2(1), p.010103. https://doi.org/10.1103/PRXQuantum.2.010103. 11. Zhong, H.-S. et al. (2020) ‘Quantum computational advantage using photons’, Science, 370(6523), pp.1460–1463. https://doi.org/10.1126/science.abe8770. 12. Arute, F. et al. (2019) ‘Quantum supremacy using a programmable superconducting processor’, Nature, 574(7779), pp.505–510. https://doi.org/10.1038/s41586-019-1666-5. 13. Technically, IBM has a subtly different interpretation. In a blog post (see https:// www.ibm.com/quantum/blog/what-is-quantum-utlity), they define ‘utility’ as: ‘Quantum computation that provides reliable, accurate solutions to problems that are beyond the reach of brute force classical computing methods, and which are otherwise only accessible to classical approximation methods’. In other words: a quantum computer doesn’t have to outperform any classical algorithm, it merely has to compete with the silly approach of brute-force search – which is almost never the best algorithm in practise. This definition seems heavily focused on claiming utility as soon as possible. Nevertheless, if we look at the big picture, we seem to have a similar notion of ‘advantage for end-users’ in mind, so I’m happy to adopt the term ‘utility’ anyway. 14. Kim, Y. et al. (2023) ‘Evidence for the utility of quantum computing before fault tolerance’, Nature, 618(7965), pp.500–505. https://doi.org/10.1038/s41586-023-06096-3. 15. Begušić, T. and Chan, G.K.-L. (2023) ‘Fast classical simulation of evidence for the utility of quantum computing before fault tolerance’. arXiv. https://doi.org/10.48550/ arXiv.2306.16372. the applICatIons: What proBlems WIll We solve WIth Quantum Computers? 53 16. Tindall, J. et al. (2024) ‘Efficient Tensor Network Simulation of IBM’s Eagle Kicked Ising Experiment’, PRX Quantum, 5(1), p.010308. https://doi.org/10.1103/PRXQuantum.5.010308. 17. Chan, G. (2022) ‘Is There Evidence of Exponential Quantum Advantage in Quantum Chemistry?’ Berkeley Quantum Colloquium, 12April. https://www.youtube.com/ watch?v=DZPH7ENcRLU. 18. De Wolf, R. (2017) ‘The Potential Impact of Quantum Computers on Society’, Ethics and Information Technology, 19(4), pp.271–276. https://doi.org/10.1007/s10676-0179439-z (open access: https://arxiv.org/abs/1712.05380). 19. Montanaro, A. (2016) ‘Quantum Algorithms: An Overview’, npj Quantum Information, 2(1), pp.1–8. https://doi.org/10.1038/npjqi.2015.23. 4 Timelines: When can we expect a useful quantum computer? At a glance the earliest commercial quantum applications will need several million qubits, according to the most rigorous studies. assuming an exponential growth similar to moore’s law, we predict that the first applications could be within reach around 2035–2040. The billion-dollar question in our field is: When will quantum computers outperform conventional computers on relevant problems? In the previous chapter, we defined the requirements more precisely and coined the term ‘utility’ for such an achievement. Unfortunately, nobody can confidently answer this question today, and past predictions often proved inaccurate. Moreover, a relevant quantum computer won’t just appear from one day to the next: there’s a continuous evolution where these devices will become increasingly capable. In this chapter, we will show how we can make a rough prediction about future timelines and discuss what will happen on the path towards large-scale quantum computation. Note as an important disclaimer, this chapter is highly subjective. It’s not hard to arrive at different conclusions simply by choosing other sources and making different assumptions. We did our utmost best to rely on the most up-to-date information, combining the views of the most widely accepted papers, and making assumptions that align with the view of most experts to present a balanced perspective. 4.1 What parameters are relevant? Compared to currently available technology, we’d require a fundamental improvement to these specifications: – Number of qubits 62 IntroduCtIon to Quantum ComputIng for BusIness such tiny machines would bring an exponential advantage over enormous supercomputers. Now that the field is coming of age, many are becoming more careful. To illustrate, when looking back at a 2021 report, consultancy firm BCG chivalrously admits:8 Our assumptions for near-term value creation in the NISQ era, however, have proved optimistic and must be revised. The most serious recent claim about NISQ utility comes from the IBM team in a paper titled ‘Evidence for the utility of quantum computing before fault-tolerance,’9 in which a quantum simulation of a specific physical system was performed using 127 noisy qubits. However, their arguments were quickly refuted by further studies that simulated IBM’s impressive quantum experiment on a conventional laptop.10 Maryland-based professor Sankar Das Darma expresses the view of many academics in his opinion article ‘Quantum computing has a hype problem’. 11 He stresses that ‘the commercialisation potential [of NISQ] is far from clear’, pointing out that claims of speedups in finance, machine learning and drug discovery have so far come with highly unsatisfying evidence. That certainly doesn’t mean that NISQ utility is ruled out. Most experts seem to keep an eye on the developments of NISQ applications but will agree that, as yet, no utility for NISQ machines has been found. To illustrate, an overview article about pharmaceutical applications 12 has a careful but suggestive message: Most NISQ algorithms […] rely heavily on classical optimisation heuristics, and the actual run time is difficult to estimate. Furthermore, recent results suggest that in NISQ approaches, the number of measurements required to achieve a given error scales exponentially with the depth of the circuit. For these reasons, here we focus our discussion exclusively on fault-tolerant quantum computers. Similarly, a recent overview13 of quantum chemistry seems to remain agnostic with regard to NISQ advantage while pointing out that fault-tolerance has a higher chance of succeeding: [I]t is difficult to predict when or if algorithms on near-term noisy intermediate-scale quantum devices will outperform classical computers for useful tasks. But it is likely that, at some point, the achievement of large-scale quantum error correction will enable the deployment of a host of so-called error-corrected quantum algorithms. tImelInes: When Can We expeCt a useful Quantum Computer? 63 In this book, we choose to follow the view of most scientists and stick to the well-understood use cases for early fault-tolerant quantum computers that we discussed previously. Nobody can rule out new breakthroughs that allow NISQ utility, but it seems unwise to count on these. A potential scientific leap could completely stir up our fragile prediction – but so would unexpected backlashes in hardware development or even unforeseen funding stops. 4.3 How long until we have million-qubit machines? Now that we’ve set our target to roughly a million qubits, we’d like to estimate when such hardware will be available. We highlight the following sources: 1. Road maps and claims of hardware manufacturers; 2. Surveys to experts; 3. Extrapolation of Moore’s law. What do manufacturers say? Below, we see the qubit numbers that several manufacturers have already realised (solid disks) and what they will produce in the future according to their public road maps (opaque plusses). Note that the vertical axis is logarithmic, displaying a broad range from around 10 to 10,000 qubits. A lower number of qubits by no means indicates that these computers are worse. In fact, the machines with the lower numbers of qubits on this graph have an important edge in other parameters, such as gate accuracies and qubit connectivity. Besides their road maps, companies sometimes make more daring claims in media interviews or at presentations at large events. Based on the application targets above, it should come as no surprise that manufacturers aim for around a million qubits as a ‘moonshot’ accomplishment. Back in 2020, IBM claimed that it would reach the 1 million qubit target by 2030.14 Around the same time, journalists interpreted Google’s pronouncements as meaning that it would do this even faster (around 202915). The start-up PsiQuantum, which made waves thanks to record-high investments of over a billion dollars for their photonic quantum chips, went as far as claiming that it would have a million qubits by 2025.16, 17 It seems that these claims were too ambitious. In 2024, with only a year to go and no publicly presented product progression, PsiQuantum shifted its 1 million qubit road map to 2027.18 IBM took an even more conservative 64 IntroduCtIon to Quantum ComputIng for BusIness step, and it’s now claiming that it will have just 100,000 qubits in 203319 (although this machine should meet the error correction capabilities that we assumed in the previous sections). Although this delay sounds disappointing, hardware manufacturers are still making impressive progress, not least because the number of available qubits grows faster than one would predict according to Moore’s law for classical chips! Trapped-ion machines tend to have fewer qubits but higher gate accuracies. Perhaps this is why IonQdisplays its road map in a different format: they aim to achieve 1024so-called algorithmic qubitsby 2028. 20 This means that IonQ will haveat leastthis number of qubits, but it also guarantees sufficient gate accuracy to run reasonably long circuits. It’s unclear whether error correction will be used for this. Competitor Quantinuum recently announced a more concrete road map,21 predicting around 100 logical qubits in 2027. These should bring the effective gate errors down by roughly a factor of 10. Looking ahead to 2029, Quantinuum projects thousands of physical qubits that form hundreds of logical qubits. This might not be enough to run the algorithms discussed earlier, but it’s not too far off either. the largest number of qubits demonstrated by a selection of hardware manufacturers, shown for different years. opaque plusses indicate manufacturers’ road maps. data taken from publicly available sources up until august2024. tImelInes: When Can We expeCt a useful Quantum Computer? 65 What does Moore’s law say? One could assume that quantum computers will ‘grow’ at a similar rate as classical computers. Moore’s law states that the number of transistors in a dense integrated circuit grows exponentially: the number doubles roughly every two years. This has been a surprisingly accurate predictor for the development of classical IT. If we apply Moore’s law to quantum, then boosting qubit numbers from around a thousand to one million would take around twenty years – predicting that the one million qubit mark won’t be passed until 2044. Clearly, most hardware manufacturers are more optimistic. If we assume the number of qubits doubles each year, then one would predict that one million qubits will be available in ten years. While doubling a quantum computer’s size each year is already a daunting challenge, companies like IBM, Pasqal, and QuEra set the bar even higher for themselves, hoping to double every 7–9months. What do experts say? The Global Risk Institute conducts annualsurveys asking experts to state thelikelihoodthat quantum computers will pose a significant threat to public key cryptography 5 years from now. Similarly, respondents also estimate the likeliness 10, 15, 20, and 30 years away. 66 IntroduCtIon to Quantum ComputIng for BusIness This essentially boils down to the question: when will a quantum computer run Shor’s algorithm to crack RSA-2048?We previously saw that around 20 million qubits would be needed for this (although experts may take into account that this number can still be lowered). We consider this an important source because many important authorities in the field (like professors and corporate leaders) take part in this study. The results from December2023, 22 gathered from 37 participants, are displayed below. results of the december2023 expert survey by global risk Institute. figure credits: m. mosca, mpiani, www.globalriskinstitute.org. How to read this graph? let’s look at the column labelled ‘5 years’. a total of 24 correspondents indicate that there is less than 1% probability that quantum computers pose a security threat in the next five years. a single person is quite pessimistic and assigns a >70% chance that this will happen. on average, experts say that there’s a fairly small likelihood that quantum computers will pose a threat to cryptography in the next five years. further to the right, the ratios shift. looking at 20 years from now, the majority of experts believe that quantum computers pose a serious threat, with over half of them assigning a likelihood of 70% or more. tImelInes: When Can We expeCt a useful Quantum Computer? 67 It appears that the majority of experts believe that the tipping point is between 10–20 years from now. Somewhere between 15 and 20 years away, there’s a point where the median participant assigned roughly 50% chance to see a quantum computer capable of breaking cryptographic codes.However, we should take into account a significant uncertainty: even experts make wildly varying estimates, so there’s no obvious conclusion from this data. These experts are almost certainly aware of hardware manufacturer’s road maps, as we shall see below. 4.4 Putting it all together The graph on the next page sums up our earlier findings. Assuming that qubit numbers will grow exponentially (and that all other parameters will keep up accordingly), we can consider several scenarios. A pessimistic scenario would be that the number of qubits ‘merely’ follows the classical version of Moore’s law, and qubit numbers double only once every two years (dotted line). Then, we would have to wait until well past 2040 to reach 100,000 qubits. An even worse scenario would be if we cannot achieve exponential growth, which would stretch the timelines even further. An extremely optimistic outlook would follow the blue dashed line (which extrapolates the progress by IBM, doubling their qubits every ~9months). If one also believes in practical applications with much less than a million qubits, then these could be available by 2030. An intermediate perspective is to assume that the number of qubits doubles annually. Interestingly, this seems to approximately align with IBM’s latest claims and the typical expert opinion. Depending on the application, it would mean that quantum chemistry simulation and codebreaking can be within reach between ~2033 and 2040. To conclude, our estimates strongly depend on the assumptions that you’re willing to accept (who would’ve thought!). Do you believe that improving algorithms and error correction techniques will allow for applications with much less than a million qubits? How quickly do you believe that the hardware will improve? If you were to force me to make a prediction, I’d say the first applications will arise around 2035, with the understanding that there’s a considerable margin for error. As a final remark, a full utility-scale quantum computer requires much more than just some number of qubits. To reach the first useful applications, we likely require simultaneous progress in algorithmics, software, gate accuracies, error correction techniques, fridges, lasers, and many other 68 IntroduCtIon to Quantum ComputIng for BusIness tImelInes: When Can We expeCt a useful Quantum Computer? 69 important subfields of quantum computing. Hopefully, all these disciplines will find the required breakthroughs that will sustain the exponential growth of quantum computing hardware. 4.5 Further reading scientist samuel Jaques (Waterloo) makes insightful graphs that combine the number of qubits and the error rates, and puts them in the perspective of applications requirements. 4.6 Notes 1. Technically, quantum gates are continuous operations, so numbers like fidelity are defined slightly differently. Still, the picture of discrete bit flips is not too far off and will lead to the same conclusions, so we prefer this more accessible explanation. 2. Gidney, C. and Ekerå, M. (2021) ‘How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits’, Quantum, 5, p.433. https://doi.org/10.22331/q-2021-04-15-433. 3. Lee, J. et al. (2021) ‘Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction’, PRX Quantum, 2(3), p.030305. https://doi. org/10.1103/PRXQuantum.2.030305. 4. Goings, J.J. et al. (2022) ‘Reliably assessing the electronic structure of cytochrome P450 on today’s classical computers and tomorrow’s quantum computers’, Proceedings of the National Academy of Sciences, 119(38), p. e2203533119. https://doi. org/10.1073/pnas.2203533119. 5. Beverland, M.E. et al. (2022) ‘Assessing Requirements to Scale to Practical Quantum Advantage’. arXiv. https://doi.org/10.48550/arXiv.2211.07629. 6. See https://www.youtube.com/watch?v=-UrdExQW0cs&t=1024s, starting at 17:04. 7. McKinsey Digital (2024) ‘Quantum Technology Monitor’. https://www.mckinsey. com/capabilities/mckinsey-digital/our-insights/steady-progress-in-approaching-thequantum-advantage. 8. Bobier, J.-F. et al. (2024) The Long-Term Forecast for Quantum Computing Still Looks Bright, BCG Global. https://www.bcg.com/publications/2024/long-term-forecast-forquantum-computing-still-looks-bright. 9. Kim, Y. et al. (2023) ‘Evidence for the utility of quantum computing before fault tolerance’, Nature, 618(7965), pp.500–505. https://doi.org/10.1038/s41586-023-06096-3. 10. Begušić, T. and Chan, G.K.-L. (2023) ‘Fast classical simulation of evidence for the utility of quantum computing before fault tolerance’. arXiv. https://doi.org/10.48550/ arXiv.2306.16372. 11. Das Sarma, S. (2022) ‘Quantum computing has a hype problem’. https://www.technologyreview.com/2022/03/28/1048355/quantum-computing-has-a-hype-problem/. 70 IntroduCtIon to Quantum ComputIng for BusIness 12. Santagati, R. et al. (2024) ‘Drug design on quantum computers’, Nature Physics, 20(4), pp.549–557. https://doi.org/10.1038/s41567-024-02411-5. 13. Cao, Y. et al. (2019) ‘Quantum Chemistry in the Age of Quantum Computing’, Chemical Reviews, 119(19), pp.10856–10915. https://doi.org/10.1021/acs.chemrev.8b00803. 14. Hackett, R. (2020) IBM plans a huge leap in superfast quantum computing by 2023, Fortune. https://fortune.com/2020/09/15/ibm-quantum-computer-1-million-qubitsby-2030/. 15. Finke, D. (2020) ‘Google Goal: Build an Error Corrected Computer with 1 Million Physical Qubits by the End of the Decade’, Quantum Computing Report, 5September. https://quantumcomputingreport.com/google-goal-error-corrected-computer-with1-million-physical-qubits-by-the-end-of-the-decade/. 16. Wang, B. (2020) ‘PsiQuantum Targets Million Silicon Photonic Qubits by 2025’, 23April. https://www.nextbigfuture.com/2020/04/psiquantum-targets-million-silicon-photonic-qubits-by-2025.html. 17. What will million-qubit computers look like in a few years? (2022) ICV TAnK-icv. https:// www.icvtank.com/newsinfo/629365.html. 18. Finke, D. (2024) ‘PsiQuantum Receives $940 Million AUD ($620M USD) to Install a 1 Million Qubit Machine in Australia by 2027’, Quantum Computing Report, 30April. https://quantumcomputingreport.com/psiquantum-receives-940-million-aud-620musd-to-install-a-1-million-qubit-machine-in-australia-by-2027/. 19. Baker, B. (2023) IBM Details Road to 100,000 Qubits by 2033, IoT World Today. https:// www.iotworldtoday.com/industry/ibm-details-road-to-100-000-qubits-by-2033. 20. Chapman, P. (2020) ‘Scaling IonQ’s Quantum Computers: The Roadmap’, IonQ, 9December. https://ionq.com/posts/december-09-2020-scaling-quantum-computerroadmap. 21. Quantinuum accelerates the path to Universal Fully Fault-Tolerant Quantum Computing (2024) Quantinuum. https://www.quantinuum.com/blog/quantinuum-accelerates-the-path-to-universal-fault-tolerant-quantum-computing-supports-microsofts-ai-and-quantum-powered-compute-platform-and-the-path-to-aquantum-supercomputer. 22. Mosca, M. and Piani, M. (2023) Quantum Threat Timeline Report 2023. https://globalriskinstitute.org/publication/2023-quantum-threat-timeline-report/. 5 Four myths about quantum computing This chapter relies on a bit of quantum physics jargon. See the chapter ‘An introduction to the quantum world’ for a quick introduction. 5.1 Myth 1: Quantum computers find all solutions at once This myth is likely the most technical, and builds on a misinterpretation of the concept of superposition. A single qubit can be in two states at the same time (0 and 1), two qubits can represent four states (00, 01, 10, 11), and three qubits are potentially in eight unique configurations simultaneously. As we increase the number of qubits, this number of coexisting states scales exponentially! This means that a mere 1000 qubits can effectively ‘store’ 2 1000 unique values, all at the same time. That’s an incomprehensibly large number, much more than there are atoms in the visible universe. Even the fastest computers in the world couldn’t loop through all these states in a lifetime. Each of these states can be interpreted like a file on a computer, be it an Excel spreadsheet, a web page, a CAD drawing, or whatever kind of data we choose to work with. A smart computer scientist can also devise a way to make 1000 bits represent ‘solutions’ to a problem. For example, imagine that we want to find an optimal aeroplane wing that generates incredible lift while requiring as few materials as possible. Using quantum superposition, we might represent 2 1000 such wings simultaneously. We picked the example of aeroplane wings because simulating their aerodynamic properties requires a pretty hefty computation. Let’s assume that we have written such a computer program that accurately simulates any wing. Let’s call that program f . It will output 1 if the wing works well (according to whatever metric), and 0 otherwise. Surely, the program takes a very large number of computation steps, which we’ll call T. The program will need some input, denoted by x , which is a 1000-bit description of all the relevant properties of a hypothetical aeroplane wing. In other words, the computer program computes f ( x ) = 1 if x is a fantastic wing, and f ( x ) = 0 if it’s rubbish. Now, a quantum computer should be able to execute any classical function, right? We should be able to run f on a quantum computer, but now we have the unique feature that the 1000-qubit input can represent a humongous number of potential aeroplane wings at the same time. By doing a mere T computational steps, we can check the properties of 2 1000 solutions! Part2 More about the applications 6 Applications in chemistry and material science Perhaps the most credible application of quantum computers is to study quantum physics itself. This deepens our understanding of microscopic systems like molecules, atoms, or even sub-atomic particles, ultimately leading to the discovery of new drugs, materials, and chemical production methods. At first sight, there seems to be a significant advantage compared to conventional computers, which struggle to store the complex quantum state of systems with many particles. As far back as 1981, physicist Richard Feynman ended a conference talk with a famous quote, hinting at the opportunities of quantum computing:1 I’m not happy with all the analyses that go with just the classical theory, because nature isn’t classical, dammit, and if you want to make a simula - tion of nature, you’d better make it quantum mechanical. Since then, scientists have become increasingly adept at accurately controlling quantum systems. Today, universities boast a wide spectrum of analogue quantum experiments that help us understand nature under exotic circumstances. We’re now lining up our tools to take these simulations to the next level: studying nature with digital quantum machines. In this chapter, we will assess how quantum computers can impact the fields of chemistry and material science. That makes this chapter more technical, and we’ll assume some (very) basic background in chemistry and physics. We discuss the most relevant algorithms, evaluate claims about quantum computing’s benefits in the fight against climate change, and analyse why the nitrogenase enzyme receives such widespread attention. 6.1 What problems in chemistry and material science will we solve? The computational problems that chemists care about typically come in two flavours: static and dynamic problems. The most studied problem is the static variant, where the goal is to find the arrangement(s) of particles with the lowest possible energy. We call such an arrangement the ground state. These states are relevant because we usually find systems in (or close to) their 82 IntroduCtIon to Quantum ComputIng for BusIness lowest energy states in nature. In the context of molecules, the atomic nuclei are relatively heavy, while the lightweight electrons move much faster and are more prone to be entangled or in a quantum superposition. Therefore, chemists tend to make approximations that allow them to focus primarily on the positions and spins of the electrons: the electronic structure problem. The other main problem is about dynamics: given some initial configuration of particles, how do they reconfigure themselves after a certain amount of time? This is often referred to as a system’s (time) evolution. Both problems are informally referred to as quantum simulation. We often receive the question of why it’s so hard to simulate quantum mechanics on a classical computer. Intuitively, this hardness arises when we deal with many particles that exhibit large amounts of superposition and entanglement, such that the location of one particle is heavily dependent on the (undecided) position of many other particles. We call such states strongly correlated. Classical computers struggle because they need to keep track of all the possible locations that particle A can be, but also all the locations of particle B, and the same for particle C, etc. As the number of particles grows, the number of possible configurations of these particles increases exponentially. This means that the number of relevant amplitudes (see the chapter on quantum physics) that a classical computer needs to process grows very quickly. Even with a mere one hundred particles, brute-force simulation is far beyond the capabilities of the world’s best supercomputers. It is a common misconception that quantum computers straightforwardly offer an exponential advantage compared to classical computers for all chemistry problems. An influential recent paper reports2: [W]e conclude that evidence for such an exponential advantage across chemical space has yet to be found. While quantum computers may still prove useful for ground-state quantum chemistry through polynomial speedups, it may be prudent to assume exponential speedups are not generically available for this problem. Note that this comment is specifically about finding ground states, which, arguably, remains the most relevant problem in chemistry. There is still ample evidence that quantum computers offer an exponential speedup for time evolutions. There is more bad news for quantum computers. Over the years, computational chemists have found brilliant approximations, hacks, and optimisations to work around the classical computer’s bottlenecks, raising a high bar before a quantum computer can meaningfully compete. For nearly applICatIons In ChemIstry and materIal sCIenCe 83 every problem in chemistry, there appears to be a clever trick to solve it somewhat efficiently on a classical machine. For a killer application, we likely need to search in a fairly specific niche, right at the sweet spot where classical methods struggle while a quantum computer excels. It is not entirely clear how large this niche is, and it is an active research area to identify more systems where classical methods fall short. One promising area involves multi-metal systems, where multiple metal ions are close together. Such systems are present in biologically relevant enzymes such as P450 and FeMoco.3 Another is in heterogeneous catalysis, where the catalyst and reagents/products are in a different phase of matter.4 The first practical users of quantum simulation algorithms will most likely be scientists who study the fundamentals of quantum systems. Physicists are already employing devices that are similar to early quantum computers to mimic certain classes of materials. We wouldn’t call these devices computers yet, but rather analogue simulators. One of the first actual applications of a fully digital quantum computer could be to analyse theoretical models of quantum materials, such as the famous Hubbard model.5 The first error-corrected quantum computers will hopefully find their place in industrial R&D settings. One of the first application areas could be to better understand theaforementioned multi-metal systems, which are relevant in thecalculations of ligand binding affinities in drugs and in understanding the mechanism behind the biological production of ammonia. We address the latter example at the end of this chapter. Another exciting area could be to explore the mechanism behind Type-II superconductivity and to search for materials that become superconducting at even higher temperatures.6 It is hard to say what the impact of quantum computers will be beyond such niche areas, as this will depend strongly on the usefulness of small polynomial speedups and unpredictable breakthroughs in quantum algorithms. We see a broad palette of other impactful applications that have been proposed, such as photocatalytic reactions (for example, efficiently splitting water to produce hydrogen fuel), 7 carbon capture mechanisms, 8 the study of efficient solar cells, 9 and the development of higher-capacity batteries.10 6.2 Algorithms for quantum chemistry We describe three of the most important quantum simulation algorithms. The first is the Trotter-Suzuki method, sometimes called ‘Trotterisation’, 84 IntroduCtIon to Quantum ComputIng for BusIness which simulates time evolution. In this case, we assume that some correct initial state of the world is encoded in the qubits of some quantum computer. The Trotter-Suzuki method is guaranteed to return a good approximation of the state at a later time, again encoded in the qubit registers. The second algorithm is quantum phase estimation (QPE), which reports the energy of a certain quantum state and can be used to produce a system’s ground state. As a subroutine, it requires some time evolution method, like Trotter-Suzuki. Unfortunately, QPE can only provide information about a certain state if it receives an input that is already a reasonable approximation to this state. Especially in the context of describing low-energy configurations, this shifts the problem to producing good candidate ground states. The most popular algorithm for creating states with certain properties (like very low energies) is the variational quantum eigensolver (VQE). This is an example of a variational quantum circuit: a series of gates that can be gradually changed until the output matches certain requirements. Just like other variational approaches, it is a heuristic algorithm, lacking rigorous guarantees that it will produce the desired output in a reasonable time. However, it is a popular method today thanks to its ease of use and the ability to work with small, noisy computers. Creating a good approximation to a ground state is, in general, NP-hard. This means that it is extremely unlikely that a rigorous algorithm exists that can find the ground state of any quantum system. On the other hand, there is good hope that more heuristic methods (just like VQE) will be found that work well on certain subsets of systems. In fact, such heuristic methods already form the workhorse of classical computational chemistry, with tools such as Density functional theory (DFT), Configuration Interaction (CI) and Quantum Monte Carlo (QMC). These work for small systems but are often too slow to study large systems such as proteins or drugs. 11 A workaround is to apply these methods to just a small part of the target system, employing faster but less accurate methods to oversee the larger whole. An example of a basic workflow to find a ground state on a quantum computer could be as follows. The first step is to train a VQE to output states with low energy. 12 These might not be the exact ground states, but they will hopefully be similar (in jargon, they have a large overlap with the ground state). As a second step, we append a QPE circuit, which will not only report the energy of the VQE states, but also has a fair probability of changing these states into perfect ground states (in jargon: it projects onto the ground state). Running the VQE + QPE combination a few times will almost certainly give the lowest energy states, assuming the VQE produces proper approximations of it. applICatIons In ChemIstry and materIal sCIenCe 85 Further reading on simulation algorithms Various more technical and sophisticated methods exist, for which we refer to other more technical sources. These require expert knowledge of quantum chemistry. ‘Introduction to Quantum Algorithms for Physics and Chemistry’ (2012),13 a pedagogical book chapter. ‘Quantum Algorithms for Quantum Chemistry and Quantum Materials Science’ (2020),14 a scientific overview article. 6.3 A hype around quantum computing for climate change Some businesses make spectacular claims about how quantum computing could be a cornerstone in solving climate change, thanks to the boost to R&D on batteries, carbon capture, and more efficient chemical factories. However, rarely do we see any evidence – most seem to assume that quantum computers simply spit out blueprints for revolutionary sustainable technologies. McKinsey takes the biscuit with their report titled ‘Quantum computing just might save the planet’.15 The article rightfully selects some of the most impactful technologies to reduce CO2 emissions, like electrification of transport, improved solar panels, and even vaccines that reduce methane emissions by cattle (indeed, due to cow farts). The article concludes that the selected innovations could reduce global warming from 1.7–1.8 °C by 2050 down to just 1.5 °C. It is a mystery to us why they throw in quantum computing because there is no mention whatsoever about why specifically quantum algorithms would be the key enabling factor. This exemplifies what we see more frequently in popular articles: quantum computers are depicted simply as insanely fast computers that will magically solve the barriers to other new technologies on our wishlist. What are the true prospects for quantum computing in the context of climate change? Sceptics may point out that technological innovations alone will not be sufficient to avert a climate disaster – we will remain agnostic 86 IntroduCtIon to Quantum ComputIng for BusIness in this debate. A much more concrete issue is the mismatch in timelines. Climate experts agree that, to limit global warming to no more than 1.5° C, we need to act relatively soon. Imperial College London concludes on their website,16 referencing the 2014 IPCC report: Limiting warming to 1.5°C will only be possible if global emissions peak within the next few years, and then start to decline rapidly, halving by 2030. Our chapter on timelines shows that it is exceedingly unlikely that significant quantum utility is possible anywhere before the 2030s. Additionally, it will take several years before a computational discovery is sufficiently mature for large-scale deployment. For this reason, we don’t see quantum computers as a good investment against climate change, but rather as a long-term development that can help us tackle other problems that humanity will face in the future. Do we really have no concrete applications in climate science? Well, we do have some concrete leads. In the search for a killer application in chemistry, perhaps the most-studied topic is the enzyme Nitrogenase. Its active site is precisely a multi-metal system that classical methods struggle with, and as we’ll soon see, it appears in reputable plans for decarbonisation. To understand the relevance of this molecule, we need to dive into the world of food production. 6.4 A case study of a potential killer application: FeMoco Today’s agriculture relies heavily on the use of artificial fertilisers. Without large-scale use of supplementary nutrients, we would not be able to sustain intensive farming practices and feeding our world’s huge population would be problematic. In fact, abouthalf of the nitrogen atomsin our body have previously passed a fertiliser factory! Unfortunately, the production of fertiliser involves enormous energy consumption and carbon emissions. The main culprit is the ingredient ammonia (NH3), of which we use as much as230 Mton per year. Although our air consists mainly of molecular nitrogen (N2), plants cannot directly absorb this. Instead, they rely on bacteria (or, in the case of artificial fertiliser, humans) to perform so-called nitrogen fixation, breaking the strong triple bond of molecular nitrogen and converting this into ammonia. Microorganisms can convert this into further nitrogen-containing compounds that the root system can absorb. applICatIons In ChemIstry and materIal sCIenCe 87 Pretty much all of the world’s ammonia production facilities follow the so-called Haber-Bosch process, where hydrogen gas (H2) and nitrogen gas (N2) react together to form ammonia. This method has the benefit that it can be implemented in large, high-yield production lines but comes with the disadvantage of its staggering energy consumption. The inefficiency stems from two essential steps: first, producing sufficiently pure hydrogen and nitrogen gasses, and later, separating the H2 and N2 molecules into individual atoms. Breaking N2 is especially challenging due to its strong triple bond. As an effect, factories operate at extreme conditions, with high temperatures (~400 degrees Celsius) and high pressure (over 200 atmospheres), driven mainly by natural gas. As much as1.8% of the world’s CO2 emissionis caused by factories performing such reactions, consuming around 3–5% of the world’s natural gas production! Can’t this be done more efficiently? We strongly suspect so. Certain bacteria are also capable of making ammonia, but in a seemingly more efficient way, without high temperatures or high pressure. It would be extremely valuable to copy this trick. To imitate bacteria, we need to better understand a particular substance, the FeMo cofactor (in short: FeMoco), which acts as a catalytic active site during ammonia production. A perfect simulation of FeMoco is not possible on classical computers, as the structure of roughly 120 strongly reacting electrons rapidly becomes intractable. In 2016,researchers from ETH Zurich the chemical structure of the femo cofactor of the nitrogenase enzyme. figure credits: smokefoot for www. wikimedia.org. 94 IntroduCtIon to Quantum ComputIng for BusIness In asymmetric cryptography, more often calledpublic key cryptography (PKC), each participant has two keys: apublic keyand aprivate key. Thepublic keycan be shared with anyone, while theprivate keymust be kept secret. That’s why we use the suggestive colours green (save to share) and red (keep private!). If Alice wants to send an encrypted message to Bob, she usesBob’spublic keyto encrypt the message. The message can only be decrypted using Bob’s private key, ensuring that only Bob can read the message. The setting with two keys offers more functionality. For example, using public key cryptography, Alice could securely send a secret key to Bob that they can subsequently use for symmetric cryptography, which is faster in practice. When public key cryptography is built for this purpose, we call it a key encapsulation mechanism (KEM). Furthermore, the protocol works in ‘reverse’. Alice can use herprivate keyto encrypt a message, which then anyone in the world (including Bob) can decrypt using the correspondingpublic key. Bob should then be confident that Alice is the only person who could have encrypted this message. Indeed, something encrypted with theprivate keycanonly be decrypted with thepublic key, and vice versa. The encrypted message is much like a signature that only Alice can produce. This forms the basis of digital signatures and certificates. the ImpaCt on CyBerseCurIty 95 You can see public key cryptography in action whenever you visit a web page. Your browser (like Chrome or Firefox) will display that the connection is secure, which means that it verified that the digital signature is valid, amongst other things. This guarantees authenticity (the page came from a registered server) and integrity (the site arrived unchanged). It should come somewhat as a surprise that public key cryptography is even possible at all! It’s a small miracle that encryption and decryption with two totally different keys can be made to work, thanks to some powerful mathematics. However, it turns out that the delicate relationship between the two keys is also a weak spot… How good are quantum computers at cracking cryptography? Symmetric-key cryptographyis quite safe against quantum hackers. The biggest problems are brute-force attacks, where an attacker effectively tries every possible secret key. Using a key size of 128 bits, the total number of possible keys is 2128– that’s an incomprehensibly large number, much more than the number of atoms in a human body. We know that Grover’s algorithm speeds up brute-force search by reducing the number of attempts from2128 to its square root, which is264. This is something that cryptographers are not happy about, but considering the slowness and extra overhead that comes with quantum computers, this doesn’t seem to be a problem in the foreseeable future. Still, to be on the safe side, it is recommended to double key lengths, hence, to use the same algorithm with 256-bit keys. Changing this in existing IT infrastructure is relatively straightforward, although one 96 IntroduCtIon to Quantum ComputIng for BusIness shouldn’t underestimate the time and costs for such changes within large organisations. The situation is entirely different withpublic key cryptography.The most-used algorithms today,RSAandECC, can be straightforwardly broken by a large quantum computer. We discussed the details of Shor’s algorithmearlier and saw that around 20 million qubits and eight hours are needed to retrieve a secret RSA key. Fortunately, there exist PKC systems that are believed to be safe against quantum computers, and an obvious way forward is to start using these. We call such systemspost-quantum cryptography, and despite the confusing name, they’re built to work on conventional computers. We discuss the rabbit hole of migrating to new cryptographyin a different chapter. Unfortunately, even today’s communication could be at risk due to a practice calledharvest now, decrypt later.Encrypted messages that are sent over a network can be intercepted and stored for many years, until a quantum computer can efficiently decrypt the messages. Even though we use public key encryption mainly to establish temporary keys for symmetric cryptography, a smart attacker could still retrace all the intermediate steps and retroactively spy on our communication. It is unclear at what scale storage of sufficiently detailed internet data is genuinely happening, but it seems plausible that security agencies of larger nations are already doing this. The following table summarises how our cryptosystems are threatened: Symmetric Public-key Quantum networks today (aes, … ) today (rsa, eCC) pQC Qkd Safe against classical computers ✔ ✔ ✔ ✔ Safe against quantum computers ✔* *with double key lengths Unsafe ✔ ✔ Why don’t we switch to symmetric cryptography? Public key cryptography solves a very fundamental problem: how can Alice and Bob agree on a secret key before they have a means of encryption in the first place? They cannot just send a new key over the internet without any form of encryption, because anyone would be able to read this. This is the fundamental problem ofkey distribution. Let us look at the functionality offered by the two types of cryptography: the ImpaCt on CyBerseCurIty 97 Symmetric Public-key Quantum key distribution Confidentiality (privacy) only with pre-shared keys ✔ ✗ Authentication / Integrity only with pre-shared keys ✔ ✗ Establishing secret keys ✗ ✔ ✔* *only when another mechanism takes care of authentication. If only we could somehow give Alice and Bob pre-shared keys in a secure way, we would resolve most of these problems. Without public key cryptography, there are other options: – Trusted courier. Alice and Bob could meet every other week to exchange USB drives with secret codes. – Trusted third party. Alice and Bob could both trust a large ‘key server’. If both share a secret key with the key server, they can securely ask the server to generate a new secret key that they can use together. – Quantum key distribution. We discuss this solution further below. Unfortunately, trusted couriers or trusted third parties are rarely an attractive alternative to public key cryptography, especially when scaling up to networks with thousands or millions of connected users. Couriers are simply too slow for today’s standards, and single trusted parties would pose a particularly interesting target for attackers. 7.3 What solutions exist? There is a clear need for post-quantum cryptography to replace commonly used cryptosystems like RSA and ECC. Fortunately, back in 2016, the American National Institute of Standards and Technology (NIST) started a competition to select a new cryptosystem, which should balance safety and practical usability (for example, it should not be too slow or memory-inefficient). They invited experts from around the globe to propose cryptographic algorithms, which peers assessed. Four rounds and several broken algorithms later, NIST selected a first set of winners that are suitable for large-scale use. As of August2024, the first three PQC algorithms are now official NIST standards. 98 IntroduCtIon to Quantum ComputIng for BusIness Even though this effort was coordinated by an American institute, the process was backed and carried out by cryptographers from around the world. A broad majority of cybersecurity experts have confidence in NIST’s competition and recommend the final standards. National security organisations from other countries like BSI (Germany) and ANSSI (France) may prefer different algorithms but have also explicitly stated that this does not mean that they consider NIST’s standards unsafe. The results of the competition are as follows. Firstly, NIST selected one Key Encapsulation Mechanism that can be used to establish secret keys over an unencrypted connection – remember the problem of communicating with a web shop that you had never encountered before. Functionality NIST Name Problem family Documentation Original name key encapsulation mechanism ml-kem module-lattice based fIps 203 Crystals-kyber Secondly, NIST selected three different Digital Signature Algorithms. These are used for authentication and integrity – remember how we don’t want our messages to be altered in transit or how we want to prevent malware injected in software updates. Functionality NIST Name Algorithm family Documentation Original name digital signatures algorithm ml-dsa module-lattice based fIps 204 Crystalsdilithium digital signatures algorithm slh-dsa stateless hash-Based fIps 205 sphInCs+ digital signatures algorithm fn-dsa fast-fourier transform over ntru-lattice based fIps 206 falCon You might wonder why three algorithms were selected. Unfortunately, all three standards come with downsides, for example, because the keys can take up more memory or because the performance (time to sign or verify) is worse. The real-world impact will differ per use case. ML-DSA is the main cryptosystem recommended for general use, whereas SLH-DSA and FN-DSA may be beneficial in specific circumstances. the ImpaCt on CyBerseCurIty 99 Are the new standards considered safe? The short answer is yes: the new PQC standards are considered ready for use, and choosing algorithms such as ML-KEM or ML-DSA is widely regarded as a sound decision. There may be exceptions in specific high-security scenarios, but if you are operating in such a context, you are likely already aware of these nuances. However, there seems to be some uncertainty within the cryptographic community regarding whether the new PQC standards will be as reliable as our trusted RSA or ECC. The new standards have not yet stood the test of time, and it is possible that unexpected weaknesses – whether minor implementation flaws or fundamental vulnerabilities – may still be present. To illustrate, a PQC method called SIKE1 was in the race to become a new NIST standard and made it all the way to the fourth round until it was proven unsafe. To mitigate any unexpected vulnerabilities in the new standards, most authorities recommend a hybrid implementation that combines the strengths of both conventional and post-quantum PKC. Moreover, organisations are generally advised to invest in cryptographic agility, a broad term used to describe the ability to easily update cybersecurity defences. The above may sound somewhat negative, but we don’t expect the slightly lower trust to stand in the way of adoption. Cryptographic algorithms themselves are rarely the weakest point, so it seems wise to focus on other potential vulnerabilities instead. What about Quantum Key Distribution (QKD)? Quantum key distribution is also presented as a solution for key exchange, making it a potential alternative to RSA, ECC and ML-KEM. Still, many security authoritieswarnagainstadoptingQKD today. Although the idea is promising, today’s hardware is still immature. Moreover, QKD doesn’t provide any functionality for digital signatures, thus we will need the migration to PQC anyway. It is somewhat of a pity that QKD is not so mature yet, because it would be a viable weapon against Harvest Now, Decrypt Later. Nevertheless, since a quantum threat could be here as soon as the early 2030s, experts warn that companies and governments should fix their PQC first. At a later stage, QKD can be considered as an add-on for further security. What about Quantum Random Number Generators (QRNG)? Good random number generators are exceptionally important in cryptogra - phy, and QRNGs could provide a good alternative to thehardware random number generatorsthat are widely used today. 100 IntroduCtIon to Quantum ComputIng for BusIness However, all they do is generate random numbers – that doesn’t make any protocol in itself quantum-safe. As a general warning:products with ‘quantum’ in the name do not automatically protect against Shor’s algorithm! 7.4 Conclusion Cryptography is strongly intertwined with quantum computing through Grover’s algorithm, Shor’s algorithm, and Quantum Key Distribution. Security experts recommend that there is an obvious way forward: – Replace current public key cryptography with new, quantum-safe protocols (PQC); – Double key lengths in symmetric cryptography. Especially the first bullet is a major challenge. There are many legacy systems on the internet that can not be updated so easily. Billions of devices are all interconnected, so updating one device may cause incompatibilities somewhere else. Moreover, PQC protocols will likely require more CPU power, memory, and bandwidth than today’s trusted methods. Companies may need to update the core code of hundreds or even thousands of applications. Lastly, the new protocols haven’t been tested as extensively as our conventional methods, so it is not unlikely that new security issues will be found. Before they are even built, quantum computers are already causing headaches to cryptographers and cybersecurity managers. 7.5 Further reading Cloudflare’s resource page ‘The State of the Post-Quantum Internet‘ explains many aspects of the migration to post-quantum cryptography. the nsa publishes recommendations on which cryptographic algorithms should be used and sketches a concrete timeline about when governmental security systems should be updated. the ImpaCt on CyBerseCurIty 101 The PQC Migration Handbook is a free guide for corporate managers on how to tackle the upcoming cryptography migration, written by dutch research organisations tno, CWI, and the secret service aIvd. In the context of harvest now, decrypt later, the urgency to migrate depends on how long your data should remain confidential, according to mosca’s theorem. 7.6 Note 1. Goodin, Dan. ‘Post-Quantum Encryption Contender Is Taken out by Single-Core PC and 1 Hour’. Ars Technica, 2August2022. https://arstechnica.com/informationtechnology/2022/08/sike-once-a-post-quantum-encryption-contender-is-koed-innist-smackdown/. 8 Applications of quantum networks If we’re building computers that deal with qubits, superposition, and entanglement, wouldn’t these computers also need some way to send qubits to each other? This is the dream of the quantum internet: a network parallel to our well-known classical internet that allows the transmission of qubits. There is a bit of a paradox here. On the one hand, a full-blown quantum internet that stretches across the globe is very, very far away – it will require quantum repeaters to bridge longer distances, purification mechanisms to repair imperfections, and many more technologies that we’re only just figuring out. On the other hand, it is often said that quantum networks have a higher Technology Readiness Level than computing. That sounds like a contradiction, right? The main explanation is that there are some applications for small-scale ‘imperfect’ quantum networks, particularly in the context of cryptography. In a sense, quantum networking applications have always been ahead of quantum computing. Already in 1984, long before quantum computers were seriously considered, quantum pioneers Charles Bennett and Gilles Brassard discovered a method to securely negotiate a secret key (think of a password) between two distant parties based on sending individual photons. Their result is now famously known as theBB’84 protocol. Similarly, the commercialisation of network technologies has long been ahead of computing. Early quantum startups like MagiQ Technologies and ID Quantique were founded around the start of this century, and their first commercial networking products were brought to the market in 2003 and 2004. This technology, where a quantum network is used to generate a secret key at two endpoints, is called Quantum Key Distribution (QKD) – an application that we will address in much more detail below. 8.1 The promises of the quantum internet There is a long list of arguments why we should be excited about the quantum internet. Here are some of the applications that we hear most frequently: – Clustering quantum computers: By connecting multiple smaller computers, one might build a much larger computer with more combined memory, allowing it to tackle more complex problems. – Securing classical communication.The main contender here is Quantum Key Distribution (QKD), sometimes dubbed the ‘unhackable’ network.This 110 IntroduCtIon to Quantum ComputIng for BusIness optImIsatIon and aI: What are CompanIes doIng today? 111 Big O notation (see the Box ‘What does asymptotic runtime mean?’ in the chapter on applications), making it straightforward to recognise and compare the efficiency of algorithms. From the perspective of asymptotic scaling, a broad spectrum of quantum algorithms existsthat could speed up optimisation tasks. Scientifically, it is downright fascinating that these algorithms can provide such advantages, using the laws of exotic physics to save trillions of computational steps. However, this book is about quantum computing for business, so while we appreciate the marvels of nature, at the end of the day, we want to know what the most practical way to solve our problems is. No matter what abstract mathematics says, all we care about is the actual wall clock time for our specific niche of problems. At this point, the competition from classical computers becomes fierce. Today’s processors from companies like AMD or Nvidia are so incomprehensibly fast that a quantum algorithm must be quite special before it can overcome the relative slowness of a quantum computer. Moreover, quantum computers will have a fair amount of overhead from error correction that conventional computers don’t have to worry about. If we’re looking at wall clock time, the race between quantum and classical is much tighter! Even when we compare classical algorithms, asymptotic complexity isn’t always the best indicator. For example, the Coppersmith-Winograd algorithm can multiply huge matrices relatively efficiently – asymptotically, it’s much faster than the naïve brute-force methods used today. Large matrices are abundant in computationally hungry fields like engineering and AI, so one might expect Coppersmith-Winograd to be widely adopted. Nevertheless, it appears that hardly any professional software implementations actually use this algorithm, nor any of its relatives.1 It turns out to be difficult to work withand enabling its speedup requires even larger matrices than we handle today. Asymptotic complexity is a useful tool, but no silver bullet. Moreover, the theory of asymptotic complexity is unsuitable when comparing heuristic algorithms. For example, a class of problems that we call ‘NP-complete’ is hard to solve in theory, while we have software tools like Gurobi and CPLEX that solve such problems quite well on a daily basis. The only truly fair comparison is benchmarking. It involves standardised tests to indicate the performance of an algorithm or a machine. The tests could be as simple as a set of reference problems that should be solved as quickly as possible. For example, supercomputers are commonly compared through the LINPACK benchmark, whereas algorithms for the Traveling Salesman Problem can be tested in TSPlib. The field of AI has been playing this game for a long time, focusing on fuzzy problems like producing natural 112 IntroduCtIon to Quantum ComputIng for BusIness English texts or recognising what’s on an image – stuff that’s hard to formally define in mathematics. For example, neural network architectures for image recognition cannot be taken seriously until they have been tested on standardised datasets like MNIST and ImageNet. To assess the advantage of quantum computers, we’ll need to compare them to classical machines in similar benchmarks. Unfortunately, today’s hardware is far from adequate, and, so far, the best comparisons are based on resource estimates and heuristic arguments. Today, it seems nearly impossible to prove the utility of a quantum optimisation algorithm. Nevertheless, it is not hard to find articles that boldly claim a businessready speedup with just a few thousand noisy qubits, and we strongly recommend being sceptical about such sources. There are many ways in which such results can be misleading. For example, many articles merely report that a quantum computer can solve a problembut fail to quantify how fast or accurate it is in comparison to the best-known classical method. These articles can still have very suggestive titles that make one believe that a quantum computer is faster. Sometimes, researchers compare their quantum algorithm only to ‘weak’ contenders, like classical brute force search or a simplified algorithm that’s rarely used in practice. Such situations are likely to occur when analysing some obscure datasetor solving a problem that nobody has seriously looked at before. Occasionally, a quantum algorithm is benchmarked against a classical machine learning model trained by the same researchers. Optimising AI methods is finicky, and such reports make us sceptical aboutwhether the classical method was treated just as carefully as the quantum approach. All of these examples indicate the importance of testing quantum algorithms against well-studied classical approaches. This all sounds quite negative, but we still see it as a positive development when companies perform early explorations of quantum algorithms, often testing accessible algorithms like variational circuits on sector-specific toy problems. Quantum computing can be incredibly complex, and it will take time to gain experience, train a qualified workforce, and tackle all the barriers that stand in the way of taking a quantum algorithm to production. It would be best for the field if everyone is honest when the outcome of a proof-of-concept is primarily a set of learned lessons, without inflating the result as a revolutionary speedup. To conclude, quantum algorithms will need to prove their worth in standardised benchmarks, similar to how leading AI methods are assessed today. While we are waiting for the hardware to mature, the most relevant information comes from rigorous resource estimates. One should be careful with claims purely based on an algorithm’s performance on relatively small-scale problems. optImIsatIon and aI: What are CompanIes doIng today? 113 Further reading the scientific paper ‘Better than classical? the subtle art of benchmarking quantum machine learning models’ performs a systematic test on several quantum machine learning models. olivier ezratty proposes a framework to assess quantum computer case studies. metriq is a platform that collects several early quantum benchmarks. (technical) the Quantum economic development Consortium (Qed-C) proposes benchmarks based on several optimisation tasks. microsoft azure features a resource estimator that helps gauge the number of qubits and theamount of time needed to run certain quantum algorithms. 9.2 Where should we look for a new killer application? Well, we simply don’t know! However, some useful technical hints may be: – We’d most likely require anexponential, a large polynomial,or someheuristicspeedup. This is much more likely achieved on problems where we don’t already know very efficient classical algorithms. – When reading data is a limiting factor (for example, in big data applications), quantum computers appear to be relatively slow. Getting the data into a quantum computer seems to take at least as long as processing the data on a much cheaper supercomputer. This holds, for example, when searching through a large database, but also for data-intensive simulations like weather forecasting. 114 IntroduCtIon to Quantum ComputIng for BusIness – Similarly, if the desired output is a large amount of data (such as a very large list or table), then a quantum computer is likely not efficient. Most quantum algorithms look at a global property of a function or dataset that can be encoded in a very small output (like DeutschJozsa or Shor’s algorithm when interpreted as finding the period of a function). – Some people would say that if quantum computers are not ‘faster’, perhaps they might solve a problem ‘more accurately’ (for example, they might produce a more reliable forecast). However, when we look at speedups, then accuracy is already taken into account: we compare the number of needed to achieve a given accuracy. – Classical computers are already incredibly fast, and the bottleneck for many real-world computational problems is not in a computer’s clock speed. If an application does require a supercomputer today, then it’s unlikely that anyone will invest in a quantum computer soon. 9.3 Examples of results in different sectors To gain further understanding of the commercial applications of quantum computers, we reach a point where we can no longer provide any generic wisdom. The best way to understand this field is by looking at various examples. In this section, we present three industries that are commonly mentioned in the context of quantum applications: pharmaceuticals, finance, and energy. For each of these, we briefly highlight typical use cases and discuss one or two technical reports. The reports are picked for no particular reasonexcept that they should provide a decent amount of technical information – much more than a typical press release or blog post would. Moreover, these reports cover a broad spectrum of results, tackling different problems, featuring different types of companies, and taking different perspectives on the degree of utility that quantum computers would offer. We limit ourselves to use cases in optimisation and AI, because quantum simulation and cybersecurity are already covered in more depth in different chapters. Note the application areas and use cases highlighted here are speculative: there is no hard guarantee that quantum computers will offer significant advantages optImIsatIon and aI: What are CompanIes doIng today? 115 for these applications. We selected the examples below because they have notable potential, meaning that further investigation is justified (and will likely happen in the following years). moreover, this section is meant to give examples, and it’s far from exhaustive. Pharmaceutical industry & health The pharmaceutical sector seems willing to make long-term investments, mainly because IP and patents can be very profitable. Indeed, the larger corporations file some 50–100 ‘quantum’ patents each year. 2 Part of the enthusiasm is justified because computational chemistry R&D is part of their core business. The broader health industry, including parties like hospitals and manufacturers of medical equipment, may have less focus on quantum simulations but are still frequently mentioned. Some of the most studied themes include: – Computer-aided drug discovery, where a (quantum) computer simulates how a proposed drug reacts with compounds in the human body. In particular, quantum-mechanical interactions may be relevant when estimating the binding strength between a drug and biological compounds; – Optimising strategies for drug synthesis; – Simulation of the molecular spectra expected in NMR or spectroscopy experiments. Even though the chemical nature of drug design lends itself well to exponential speedups, some restraint is warranted. The most important quantum speedups are expected for strongly correlated systemsthat exhibit large amounts of superposition and entanglement. A recent overview article states the following about drug design:3 [Classical methods] offer good-enough accuracy for most systems. This is because most oral drugs are small closed-shell organic molecules (they need to pass through the gut wall to be absorbed) which generally lack strong correlation. This leads them to conclude: [I]f the advantage of quantum computers is limited to strongly correlated systems, they might have limited practical significance in drug design. 116 IntroduCtIon to Quantum ComputIng for BusIness Nevertheless, there are still plentiful computational challenges that classical computers haven’t solved, both in the areas of quantum simulation and optimisation. Whether quantum computers will address just a small niche of strongly correlated systems or prove to have broader applicability is still an open question. Example results Exploring the Advantages of Quantum Generative Adversarial Networks in Generative Chemistry The paper is based on Generative Adversarial Networks (GAN), where two neural networks are trained simultaneously. One network is a ‘discriminator’, which has to detect whether a structure (graph) of a molecule derives either from a fixed dataset or whether it is created by the other network, the ‘generator’. By training both networks in parallel, they become increasingly adept at their task, such that eventually, the generator mimics natural molecule structures very accurately. The paper constructs the GANs partially from variational quantum circuits (VQC) and sees improvements in some benchmarks. Note that this has only been tested for relatively small molecules. My subjective view is that this looks like an overall interesting approach. The abstract does get us sceptical due to a claim that the authors ‘demonstrate the quantum advantage of a VQC in the discriminator of GAN’ because the VQC performs certain tasks better than a classical neural network while using fewer internal parameters. A comparison to just one selfwritten classical contender is never fair. Moreover, a quantum model with fewer parameters can still take more time and resources to train or optimise. Press release: https://zapata.ai/news/zapata-foxconn-insilico-medicineuniversity-toronto-quantum-generative-ai-for-drug-discovery/. Paper reference: Kao, Po-Yu, Ya-Chu Yang, Wei-Yin Chiang, Jen-Yueh Hsiao, Yudong Cao, Alex Aliper, Feng Ren, et al. ‘Exploring the Advantages of Quantum Generative Adversarial Networks in Generative Chemistry’. Journal of Chemical Information and Modeling 63, no.11 (12June2023): 3307–3318. https://doi.org/10.1021/acs.jcim.3c00562. Organisations involved: Insilico Medicine, Foxconn, Zapata. optImIsatIon and aI: What are CompanIes doIng today? 117 Hybrid Quantum Image Classification and Federated Learning for Hepatic Steatosis Diagnosis In this work, the authors train a neural network to assess photos of livers with the aim of diagnosing non-alcoholic fatty liver disease (NAFLD). They compare a standard (classical) convolutional neural network with a ‘hybrid’ model that contains a variational quantum layer. The paper claims that the quantum version is more accurate by 1.8percentage points. My personal evaluation would be quite positive if it weren’t for an important detail that the quantum layer uses just five qubits. It seems unlikely that such an architecture would outperform classical methods in a fair comparison, especially because simulating five qubits is trivial for a classical computer. A possible explanation is that the classical network wasn’t properly optimised (and the paper doesn’t share the necessary details to check this). This hypothesis seems supported by one of the paper’s own plots, where the classical model’s accuracies drop when it gains access to more training data. This shows why it’s important to compare algorithms on well-studied benchmarks. Press release: https://www.einpresswire.com/article/735111499/quantumalgorithm-outperforms-current-method-of-identifying-healthy-livers-fortransplant. Paper reference: Lusnig, Luca, Asel Sagingalieva, Mikhail Surmach, Tatjana Protasevich, Ovidiu Michiu, Joseph McLoughlin, Christopher Mansell, et al. ‘Hybrid Quantum Image Classification and Federated Learning for Hepatic Steatosis Diagnosis’. Diagnostics 14, no.5 (6March2024): 558. https://doi. org/10.3390/diagnostics14050558. Organisations involved: Terra Quantum, University of Trieste See also: (scientific overview article) ‘drug design on quantum computers’, https://www.nature.com/articles/ s41567-024-02411-5 (open access: https://arxiv.org/ abs/2301.04114). (scientific overview article) ‘Quantum Computing for molecular Biology’, https://doi.org/10.1002/cbic.202300120. 118 IntroduCtIon to Quantum ComputIng for BusIness Finance There is an extensive body of literature on applications in the financial services sector. Our intuition tells us that this is mainly thanks to two top-down reasons: small algorithmic improvements can quickly lead to large monetary gains, and institutions like banks have relatively long investment horizons, making them more willing to invest in technologies that could be several years away. Unfortunately, at this point, there is little evidence for rigorous exponential speedups in this sector, so the focus is primarily on polynomial and heuristic improvements. Some of the most commonly studied themes include: – Optimising investment portfolios (for high profit and low risk); – Analysing risk and studying future market scenarios; – Estimating the price of complex assets, such as options; – Fraud detection. Example results Quantum Deep Hedging A hedge is an investment chosen specifically to offset the potential for loss in other investments. For example, a bank with many assets in a volatile market might also invest in a sector that typically moves in theopposite direction. The problem can be cast in a conventional reinforcement learning framework, where a computer program makes virtual investment decisions and receives rewards depending on its performance, allowing it to learn better strategies. Deep hedging is an existing classical method to train a good software agent using deep (multi-layer) neural networks. This paper investigates the potential of quantum computers in this area. Amongst other things, the authors replace certain network layers with quantum variants. Compared to the classical approach, they achieve comparable scores while using fewer trainable parameters. They also produce qualitatively different investment strategies, hence offering something unique compared to the conventional approach. The new methods are tested on Quantinuum’s H1–1 and H1–2 trapped ion computers using up to 16 qubits. Our subjective interpretation is that this is an interesting and sound paper that focuses on rigorous analysis rather than extravagant claims. As a downside, we are not aware of any standardised benchmark in this field, nor is there evidence that the quantum approach could lead to faster computation times (as the reduction in parameters suggests). optImIsatIon and aI: What are CompanIes doIng today? 119 Press release: https://www.jpmorgan.com/technology/news/jpmorganchase -qcware-evolve-hedging-for-a-quantum-future. Paper reference: Cherrat, El Amine, Snehal Raj, Iordanis Kerenidis, Abhishek Shekhar, Ben Wood, Jon Dee, Shouvanik Chakrabarti et al. ‘Quantum Deep Hedging’. Quantum 7 (29November2023): 1191. https://doi. org/10.22331/q-2023-11-29-1191. organisations involved: Jpmorgan Chase, QCWare, université de paris Quantum portfolio optimisation by Citi Innovation Labs and Classiq The portfolio optimisation problem is as follows. You receive a list of possible stocks you may invest in and a probabilistic outlook of their expected gains and volatility (i.e. the riskiness of the stock). The gains can be correlated. Given that you’re allowed only to take a certain amount of risk, what would be the optimal set of stocks to invest in? In this work, the authors optimise assets using the Quantum Approximate Optimisation Algorithm, an example of a variational quantum circuit. There are no methodological innovations, but the authors do a good job of combining existing building blocks into a full end-to-end implementation: the algorithm is written in a high-level software package (by Classiq), using real-world data (by Yahoo finance) in a standard Python data processing pipeline (using Pandas), and running the resulting quantum program through the cloud (through AWS, albeit on a classical simulation in this case). There is no comparison with any classical methods. In our subjective interpretation, this is more a marketing outing (showcasing the technical wit of the parties involved) than a newsworthy result. Nonetheless, several news outlets picked this up, most likely thanks to the large companies involved. Press release: https://www.classiq.io/insights/citi-and-classiq-advancequantum -solutions -for-portfolio-optimization-using-amazon-braket. Blog reference: ‘Citi and Classiq Advance Quantum Solutions for Portfolio Optimization Using Amazon Braket | AWS Quantum Technologies Blog’, 7February2024. https://aws.amazon.com/blogs/quantum-computing/ citi-and-classiq-advance-quantum-solutions-for-portfolio-optimization/. 10 Quantum hardware Conventional computer hardware is extremely reliable. Professional servers are supposed to run non-stop for years without any hardware failures. If you take a new product out of a box, you can be reasonably sure that it will work precisely as advertised – and if does not, it should be straightforward to replace. Moreover, classical IT is extremely well-standardised. No matter what supplier you buy a computer from, you can be reasonably sure you can run your favourite applications on them. Thanks to such high reliability and clear compatibility, it is rather easy to compare different machines, for example, by looking at speed (e.g. floating-point operations per second, FLOPS) and memory size. We will see that this is radically different for quantum computers. Devices make mistakes, have limited functionalities, and memory is scarce compared to classical computing standards. Several manufacturers focus on niche applications, making trade-offs in certain features to enhance performance in others. In this chapter, we take a high-level perspective at quantum computing hardware. We address the two most important aspects: – What functionality does a device have? – What type of qubits are used? 10.1 Different functionalities The figure below shows three different functionalities that quantum computers can have (top, red), along with some examples of products on the market (yellow), built from different building blocks. This list is by no means complete! It should, at best, give an indication of the current state of the art. Let us start by taking a closer look at the functionalities. Our biggest dream is to have a‘universal quantum computer’. The word ‘universal’ indicates that it can execute any quantum algorithm (or, technically, it can approximate any algorithm’s output to arbitrary precision). For comparison, your laptop, phone, and even a modern coffee machine are universal classical computers, making them capable of running any classical application you can think of: spreadsheets, 3D games, data encryption, and so on. Similarly, a proper universal quantum computer is suitable for any quantum application, regardless of whether it is already known today or invented in the future. 128 IntroduCtIon to Quantum ComputIng for BusIness The definition of ‘universal’ is blind to some details, such as memory limitations (it assumes you will never run out of RAM), and omits tedious details about software compatibility (a PlayStation game won’t run on an Xbox). In our high-level overview, such details are unimportant: the main point is that there also exist devices that cannotrun just any algorithm. Does a universal computer need to be ‘gate-based’? no, there are various computational models that are universal. there are different ways to make a ‘universal quantum computer’. the most popular way is to use agate-basedapproach, where elementary operations (‘gates’) change the data stored one or two qubits at a time. this perspective is most intuitive for those used to conventional logical circuits (with and, or and not gates), and most quantum algorithms are presented in this language. other alternatives includeadiabaticcomputation andmeasurement-basedcomputation, which can theoretically run any algorithm written for a gate-basedcomputer without issues and vice versa. Currently, gate-based computers are by far the most widespread and appear to be the most popular approach in the race towards a million-qubit quantum computer: nearly all large tech companies rely on this architecture. there is one important exception. some photonics startups are working towards measurement-based computing, as this overcomes the challenges in performing ‘entangling’ quantum gates with photons. In the following, we will focus mostly on gate-based computers. Quantum hardWare 129 No matter what architecture or qubit type you pick, today’s technology will only allow you to run relatively short computations. This is due to the inherent imperfections in qubit construction and control methods. The imperfections cause errors to accumulate, so after some number of steps, the result is almost surely corrupted and unusable. For longer computations, fixing errors on the fly is essential, using so-callederror correction. At the time of writing, we live in the so-called NISQ era, withNoisy Intermediate-Scale Quantum devices. Many are theoretically fully universal, except that they are limited both in the number of qubits and, most of all, in the number of steps they can execute. Companies like IBM, IonQ, Quantinuum, and Pasqal all have NISQ computers available to test over the cloud. A universal computer is a jack-of-all-trades, but it excels at nothing. Engineers can makespecial-purpose devices that improve in certain areas (like the number of qubits or clock speed) by omitting certain functionalities. Aquantum simulator specialises in mimicking the behaviour of a particular class of materials or molecules. The precise capabilities can be described in the mathematical language of a ‘Hamiltonian’ that specifies which materials qualify. For example, Harvard-spinoff QuEra offers a quantum simulator over the cloud that mimics a quantum Ising model. 1 Today’s simulators (like QuEra’s) are fairly similar to a universal NISQ computer, missing only a few essential ingredients, and similarly having restrictions due to noise. Although they look similar, they are not designed to run conventional (gate-based) algorithms. The jargon around simulators can be a bit confusing. Firstly, the term ‘quantum simulation’ is also used when a classical computer tries to calculate the output of a quantum algorithm. To differentiate, some prefer the term ‘emulation’ for such classical approaches.Secondly, we often hear a distinction between ‘analogue’ and ‘digital’ simulation. Ironically, both approaches tend to discretise information over discrete qubits (which we call digital). In practice, the terms are rather used to distinguish between continuous and discrete time steps. An analogue simulation would use longer, continuous operations on the qubits, whereas a digital simulation uses quantum gates that act in short, discrete bursts on the qubits. Another special-purpose device is thequantum annealer,popularised mainly by the Canadian scale-up D-Wave. These special-purpose devices can solve a specific class of optimisation problems that goes by the name ofQUBO: quadratic unconstrained binary optimisation. There is a welldeveloped theory of mapping various industrial problems into the QUBO Quantum hardWare 131 formalism, making annealers fairly versatile machines. However, quantum annealers will never be able to take advantage of the various other quantum algorithms out there: even with enough qubits, we won’t see them cracking codes using Shor’s algorithm. Further reading d-Wave’s introduction to its quantum annealing platform scale-up pasqal reports on a material science simulation with 196 qubits. In another article, they explain why an ‘analogue’ quantum simulation has its advantages. Quera makes a 256 qubit simulatoravailable over the Cloud. 10.2 Different building blocks Another important question concerns the materials used to create qubits. Scientists have cooked up several competing approaches, such as superconducting materials, photons, individual atoms, or ions, each with their own strengths and weaknesses. When comparing different qubits, we use the terminology of qubit implementation, the qubit type, or (what we prefer) qubitplatform. The conventional computer electronics industry has settled on a single choice of material and manufacturing process: essentially, all computer chips are made using lithography on silicon wafers. On the contrary, there is an ongoing race between wildly different qubit platforms, and it is still unclear which will eventually be the winner — or whether we will converge to a single winner at all. There is fascinating physics behind the different hardware types, but we won’t delve into that in this non-technical book (would you care otherwise what material your classical CPU is made of?). However, as soon as you want 132 IntroduCtIon to Quantum ComputIng for BusIness to test a prototype quantum program on real-world NISQ hardware, you probably want to learn more details. Interested readers are invited to take a look at the references below. It is interesting to note that all these different functionalities (universal computers, annealers, and simulators) can, in principle, be built using any type of qubit. Returning to the figure at the top, you can see that specific qubit platforms have been used for multiple purposes, and it’s likely that the empty fields will also be populated in the future. 10.3 Further reading different types of qubits explained by sifted.eu different types of qubits at IQC Waterloo different types of qubits on Wikipedia a mooC about different hardware types by tu delft 10.4 Note 1. Gemelke, N. and Lukin, A. (2022) Hamiltonian simulation on QuEra’s 256-qubit Aquila machine, QuEra. https://www.quera.com/events/hamiltonian-simulation-on-queras256-qubit-aquila-machine. 11 Error correction At a glance to run long computations, we need to dramatically reduce the likelihood of error in each computational step – not just a little bit, but by a factor of millions. error correction is the most effective method to achieve extremely low error probabilities. It combines a small number of ‘physical’ qubits (think of several hundred) into a single ‘logical’ qubit that suppresses errorsexponentially. logical qubits are still not perfect: the ‘number of steps’ that they can survive is an important specification that determines whether they can a particular application. It’s 2024 and we’re seeing a major shift in the road maps of quantum computer manufacturers. Several companies no longer put their bare qubits in the spotlight, but instead focus on logical qubits. Error correction seems to be an essential component of large-scale quantum computing, adding yet another facet in which these devices differ from their classical counterparts. Although this is a relatively advanced topic, we find it so important that it deserves a dedicated chapter in this book. As with many aspects of quantum computing, error correction can be rather confusing. A statement (that is incorrect!), which we often hear is: Logical qubits (or error-corrected qubits) are resilient to errors that occur during a computation. Once we have logical qubits, we can increase the length of our computations indefinitely. What’s the problem here? Well, not every logical qubit is created equally. In the near future, we expect to see logical qubits that are perhaps 2x more accurate than today’s bare hardware qubits, and later 10x, and in the future perhaps 1000x. Error correction is a trick toreducethe probability of errors, but it will not eliminate errors completely. In the following decade, we expect gradual improvements, hopefully down to error rates of 10-10and below. 134 IntroduCtIon to Quantum ComputIng for BusIness 11.1 What is error correction? In quantum error correction, we combine some number (think of hundreds or thousands) of‘physical’hardware qubits into a virtual‘logical’qubit. The logical qubits are the information carriers used in an algorithm or application. Error correction methods can detect whenever tiny errors occur in the logical qubit, which can then be ‘repaired’ with straightforward operations. Under the assumption that the probability of hardware errors is sufficiently low (below a certain error threshold), the overall accuracy improves exponentially as we employ more physical qubits to make a logical qubit. Hence, we obtain a very favourable trade-off between the number of usable qubits and the accuracy of the qubits. Doesn’t measuring a quantum state destroy the information in the qubits? Indeed, if we naively measure all the physical qubits, we destroy potentially valuable information encoded in the qubits. However, quantum error correction uses an ingenious way to measure only whether or not an error occurred. It learns nothing about the actual information content of the qubit. It turns out that this way, the data stored in the logical qubit is not affected. Why are errors so much of a problem?How do errors screw up our computations? In short, even tiny errors are a problem because we want to perform an astonishing number of quantum operations successively — think of billions or trillions of them. Let’s make this more concrete. A computer program is essentially a sequence of‘steps’, each of which a computer knows how to perform. We say that a program or algorithm has awidth,which is the number of qubits it requires. It also has adepth,which is the number of consecutive steps that need to be performed. You may interpret one step in early hardware as a single quantum gate (although, in practice, gates may be performed in parallel, making theimpact of errors slightly more complicated). error CorreCtIon 135 Width (number of bits) Depth (number of steps) Set a = 450 Compute b = a * 2 Compute c = a * b Compute d = c + a … … The concept of ‘width’ is pretty straightforward: if the computer doesn’t have enough memory, it cannot run the program. Dealing with ‘depth’ is harder. To run a program of 10 9 steps, we need to limit errors to roughly the inverse, say, a probability of 10-9per step. If the error is larger, it becomes extremely unlikely that the quantum computer will produce the correct outcome. These are not hard numbers: a computer with 10-10error would be a significant improvement (resulting in much fewer mistakes), and a computer with 10 -8 error might be pushed to also find the correct answer after many tries. However, as the imbalance between depth and error grows, the probability of finding a correct outcome is reducedexponentially. We illustrate this in more detail in the box below. To illustrate, why do we need such small error rates? let’s look at a simple model of a computer, which is not unlike what happens inside a quantum computer or a modern (classical) Cpu. as above, the computer is supposed to work through a list of instructions. We can consider various specifications of a computer: – the available memory, measured in bits (or perhaps megabytes or gigabytes, if you like). error CorreCtIon 143 LDPC codes are now rapidly gaining attention. They build on a large body of classical knowledge and could have (theoretically) more favourable scaling properties over the surface code. Which code will eventually become the standard (if any) is still completely open. What are the main challenges? Firstly, we would need justslightlymore accurate hardware. We mentioned a certain accuracythresholdearlier: state-of-the-art hardware seems to be close to this threshold but not comfortably over it. Secondly, error correction also requires significant classical computing power, which needs to solve a fairly complex ‘decoding’ problem within extremely small time bounds (within just a few clock cycles of a modern CPU). Classical decoding needs to become more mature, both at the hardware and the software level. It is likely that purpose-built hardware will need to be developed, which for some platforms might be placed inside a cryogenic environment (placing stringent bounds on heat dissipation). Theoretical breakthroughs can still reduce the requirements of classical processing. Lastly, it turns out that ‘mid-circuit measurements’ are technically challenging. Without intermediate measurements, one might retroactively detect errors, but one cannot repair them. We should also warn that manyrelated terms exist, such as ‘error mitigation’ and ‘error suppression’. They might be useful for incremental fidelity improvements, but they don’t bring an exponential increase in depth like proper error correction does. 11.4 Conclusion The bottom line is that one shouldn’t naively take ‘logical qubits’ as perfect building blocks that will run indefinitely. A logical qubit is no guarantee that a computer has any capabilities; it merely indicates that some kind of error correction is applied (and it doesn’t say anything about how well the correction works). A much more interesting metric is the probability of error in a single step (in jargon: the fidelity of an operation), which gives a reasonable indication of the number of steps that a device can handle! 144 IntroduCtIon to Quantum ComputIng for BusIness 11.5 Further reading ‘The Quantum Threat Timeline Report’asked several experts what they find the most likely approach to fault-tolerance (section 4.5). British startup riverlane builds a hardware chip that decodes which error occurred on logical qubits. they provide an accessible press release and a more technical scientific article. Craig gidney (google) has amore technical blog poston why adding physical qubits will remain relevant in the following decades. (technical) somescientificwork speaks of ‘early fault-tolerant’ quantum computing, such as: ‘early fault-tolerant Quantum Computing’, discussing how we can squeeze as much as possible out of limited devices. ‘Assessing the Benefits and Risks of Quantum Computers’ takes a similar width x depth approach as we do here, but uses it to assess what applications will be within reach first. 12 What steps should your organisation take? In the previous chapters, we discussed theuse cases, thethreats,and thetimelines of quantum technologies. We will now look at the strategic perspective of a typical non-quantum enterprise. We will assume a typical large-scale organisation that does not sell IT products per se, but relies heavily on computing infrastructure to optimise its operations, supervise processes, communicate with suppliers and clients, and potentially invest in computer-aided R&D. While these organisations may be excited about the potential of quantum computing, they may also feel vulnerable – whether due to competitors advancing ahead or due to hackers attacking legacy cryptography. We outline the typical process an organization undertakes in three steps. The first steps, like growing expertise, finding adequate staff, and doing first proof-of-concept studies, will be largely sector-independent. Further steps can become more organisation-specific, and we will highlight several tools for tailored assessment Cryptography Quantum applications 1. No-regret moves Appoint a working group Assess the urgency of PQC Read up and learn Create awareness 2a. Preparation steps Find impactful use-cases Sketch a road map 3a. Implementation 2b. Preparation steps Create an inventory Form a migration plan 3b. Implementation Migrate to post-quantum cryptography 12.1 Common first steps Step 1: Start with no-regret moves Most companies start with early steps aimed at better understanding the situation. These can be done with very little financial risk. 146 IntroduCtIon to Quantum ComputIng for BusIness Some must-do actions: – Appoint a quantum lead or a quantum working group tasked with following the developments. – Read up and learn. If you’ve come this far in this book, you’re already doing a fantastic job.We have a separate chapter on further learning resources. – Create internal awareness. Many employees will enjoy inspirational talks, tours or demonstrations that academics or quantum manufacturers can provide. Optionally: – Put quantum on the agenda with senior management. – Involve collaborators, suppliers and vendors, and make your interest in quantum known. It is to your benefit if suppliers are well-prepared. – Participate in a workshop, hackathon, or similar event. In terms of more concrete follow-up actions, it makes sense to split your quantum journey into two different categories: a. Preparing forquantum applications,where the goal is to leverage quantum technologies to gain some competitive advantage (for example, by strengthening your R&D, further optimising your logistics, improving a product, etc). b. Migrating toquantum-safe cryptography, where the goal is to keep your IT secure against attackers with a quantum computer. These endeavours serve very different purposes and are likely spearheaded by different departments. Hence, it seems logical to break these down into separate projects. We discuss further steps in both directions separately. 12.2 Prepare to use quantum applications Step 2a: Explore use cases At this stage, most organisations will want to make low-regret moves that get them prepared to leverage quantum technologies fairly soon after practical utility becomes available. Some of the bottlenecks could be the lack of in-house knowledge, a limited available workforce, or a long timeline to integrate quantum applications in production environments. Must do: – Identify the most impactful use cases in your sector. – Sketch a road map for the coming years. What steps should your organIsatIon take? 147 Optionally: – Start concrete proof-of-concept projects. Right now, these are unlikely to offer practical utility and will likely tackle just a toy problem. However, these help build experience in setting up quantum projects and can uncover ‘unknown unknowns’. For staff with a strong physics or mathematics background, it is relatively accessible (and fun!) to get acquainted with quantum programming packages andimplement a first test algorithm. – Find strategic partners. Organisations can save costs by collaborating on early, pre-competitive exploration. – Create PR! We notice that many companies are actively promoting their early results on quantum applications, even if these do not offer significant advantages yet. – Hire staff with a strong background in quantum technologies who understand the market, have the right skills to lead proof-of-concept studies, and can offer advice for strategic decisions. Step 3a: Implementing actual applications, whenever ready From here onwards, it gets increasingly difficult to give concrete advice, as priorities may depend on your business and on the way the field of quantum computing will progress. Several sources will simply tell you do ‘develop a long-term strategy’ or similar. Others highlight the need to ‘remain agile’ to quickly adapt to this rapidly evolving field. For inspiration or a dot on the horizon, you may think towards a competence centre for quantum computing, similar to how many companies have special departments for data science and/or AI. A concrete task could be to elaborate on the list of impactful use cases from the previous step, benchmarking the performance of various quantum and classical software tools. Another task could be to professionalise an earlier proof-of-concept project, bringing it closer to implementation in a production environment. Identifying fruitful use cases From a top-down perspective, it is a good exercise to identify your current needs in high-performance computing.What do you currently spend your computing budget on? Are there any areas where new tools in computation or modelling could provide serious business value (for example, by being faster, tackling bigger problems, or delivering higher accuracy)? Which quantities would you ideally have calculated but are beyond the reach of current computers?This results in a longlist of use cases where new computational tools are worth further investigation. The next step would 148 IntroduCtIon to Quantum ComputIng for BusIness be to research to what extent a quantum computer (or whichever other new computational tool) offers any advantage. We recommend this top-down approach because it can lead to conclusions sooner, especially because it avoids studying use cases that are not worth your time (for example, because additional computational power provides little value). It is also possible to take a bottom-up approach. Looking at the available quantum algorithms, which would speed up processes in your existing IT? Would any of them provide value for your business? This more technical perspective requires some in-depth quantum expertise but can definitely be worth the effort, especially if you have people with the right skills available. the Quantum application lab is a collaboration between various dutch research organisations. they invite end-users to explore the benefits of quantum computers in projects that last anywhere between three and twelve months, ranging between a first exploration of use cases to advanced development of quantum prototype software. several example projects can be found on their website: www.quantumapplicationlab.com. Further reading scientists propose a framework to discover which real-world problems are potentially accelerated by quantum computers. Consultant olivier ezratty proposes a framework to assess the maturity of quantum computing case studies. (youtube) a recording of Quantum.amsterdam’s online seminar ‘What do companies get out of quantum projects today?’ What steps should your organIsatIon take? 149 What does an R&D collaboration with academia look like? several end-users have started collaborations with universities to better understand the use cases of quantum computing. this is often a win-win situation, as companies can learn from renowned experts at relatively low costs, whereas academics benefit from additional funding and showcasing that their research has practical interests. moreover, many countries provide subsidies for so-called ‘public-private partnerships’. Below, we sketch a personal experience with the process of starting such a partnership. you will most likely be dealing with a university’s tech transfer office (tto), which specialises in making in-house knowledge available externally. as a first step, it is important to agree on the scope of the project: what are the research questions, what are the expected outcomes, how long will the project run, and so forth. Ideally, this would be a discussion between an expert from your organisation and a university’s (assistant) professor. the professor will most likely take a supervising role, as the actual work is often executed by a junior researcher employed as a phd candidate or a postdoctoral (pd) researcher. phd programmes take relatively long, 3–5 years depending on your locale, and it may take some time before the first results come in. postdoc projects often take 1–3 years and can lead to results sooner, but as of 2024, it can be much harder to hire a postdoc with the right competencies. When the topic and duration of the project are clear, it is important to discuss details around intellectual property (Ip), often done by legal experts. for universities, it is important that researchers can keep building upon the project’s Ip in an academic setting. moreover, they will demand that the results can be published in scientific journals. at the same time, a paying company will want sufficient options to patent new discoveries and will require exclusive use of the Ip within their sector. these demands do not necessarily conflict with each other, and in principle, it should be possible to find an arrangement that satisfies both parties. a straightforward way to ensure that the company learns from the academic developments is by organising meetings or workshops throughout the collaboration project, in which the ongoing r&d is discussed with company staff. the occasional dialogue with company staff is arguably more important than a shiny final report or paper, which risks disappearing in someone’s drawer. 12.3 Migrating to post-quantum cryptography This section relies on technical knowledge from the previous chapter on cybersecurity. What steps should your organIsatIon take? 151 Step 2b: Prepare your migration Cryptography is a completely different beast, with a more concrete goal, and more urgent timelines for most organisations. Contrary to the applications in the previous section, the cryptography migration is not optional. Fortunately, most organisations face the same problem, and there is ample research on effective steps. The core challenge is to upgrade all existing public key cryptography to Post-Quantum Cryptography (PQC) in the next decade, which could be spread over hundreds or thousands of different applications. Many businesses, especially those dealing with critical infrastructure, may additionally deal with regulators who may or may not have guidelines ready. Moreover, IT transitions can be incredibly slow – it is not uncommon to see plans that cover five or even ten years.1 Authorities seem to agree that the following initial steps should be taken urgently by all large organisations. – Create awareness: make sure that the quantum threat is well-understood in your security departments and among IT managers and product owners throughout the organisation. – Create an inventory of cryptographic assets used within the organisation. This should include both software and hardware and should clearly specify the used algorithms, whether developed in-house or purchased from a vendor. Some parties refer to a ‘cryptographic bill of materials’ (CBOM). – Determine the risk and urgency of PQC migration. Most organisations already perform regular risk assessments of their IT infrastructure. Additionally, organisations should assess whether they classify as an urgent adopter of PQC (see below). – Create a migration plan. This is a more complex step, which should at least prioritise which assets must be migrated first and indicate whether the migration of all urgent systems can be realistically achieved in time, before the arrival of cryptographically relevant quantum computers. For more details, we recommend following thePQC Migration Handbook, a free guide written by the Dutch secret service AIVD and research organisations CWI and TNO.Security authorities in other countries have made similar guidance available.