Quantum simulation of spin systems on quantum computers
Abstract
Quantum simulation represents a formidable challenge for classical computers due to the intricate behavior of quantum systems. Digital quantum computers aim for precise approximations of a wide range of quantum systems. Within the realm of quantum simulation, the study of spin systems plays a pivotal role, providing insights into complex properties challenging to model through classical means. This work focuses on investigating chiral spin systems through the development of a dedicated quantum circuit for chirality measurement. In this Masters Dissertation, we present an overview of the current state-of-the-art in quantum simulation of chiral spin systems and introduce our approach to addressing this challenge. Scalar spin chirality, a three-body physical observable, holds a critical position both in classical magnetism, where it characterizes non-coplanar spin textures, and in quantum magnetism, serving as an order parameter for chiral spin liquids. In the context of quantum information, scalar spin chirality serves as a witness to genuine tripartite entanglement. In this study, we delve into various methodologies to tackle the problem at hand, subjecting them to comparison. The objective is to identify the most suitable approach that demands fewer quantum resources. Our best proposed method introduces an indirect measurement scheme based on the Hadamard test, designed to estimate the scalar spin chirality for general quantum states. We apply this innovative approach to measure chirality in two specific types of quantum states: the generic one-magnon states of a ferromagnet and the ground state of a model characterized by competing symmetric and antisymmetric exchange interactions. Our research findings highlight the practicality of achieving a single-shot determination of scalar chirality for chirality eigenstates, leveraging the power of quantum phase estimation with a single auxiliary qutrit. This novel methodology extends beyond providing a solution to the chirality measurement problem; it also unifies the theory of chirality in both classical and quantum magnetism. The implications of our work offer valuable insights and pave the way for future quantum research endeavors in this domain.
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University of Minho School of Engineering Irving Leander Reascos Valencia Quantum Simulation of spin systems on Quantum Computers october 2023
University of Minho School of Engineering Irving Leander Reascos Valencia Quantum Simulation of spin systems on Quantum Computers Masters Dissertation Master’s in Engineering Physics Information Physics Dissertation supervised by Ph.D. Nuno Miguel Machado Reis Peres Ph.D. Joaquín Fernández-Rossier october 2023
Copyright and Terms of Use for Third Party Work This dissertation reports on academic work that can be used by third parties as long as the internationally accepted standards and good practices are respected concerning copyright and related rights. This work can thereafter be used under the terms established in the license below. Readers needing authorization conditions not provided for in the indicated licensing should contact the author through the RepositóriUM of the University of Minho. License granted to users of this work: CC BY https://creativecommons.org/licenses/by/4.0/ i
Acknowledgements I would like to express my heartfelt gratitude to several individuals and institutions who have been instrumental in my journey throughout my Master’s program. Your support, guidance, and encouragement have made a profound impact on my academic and personal development. I sincerely appreciate my supervisors: Joaquín Fernández-Rossier, Ernesto Galvão, Bruno Murta, and Nuno Peres, who have been invaluable throughout my journey. Your invaluable support, insights, and guidance have been essential. I’m deeply grateful for your mentorship, igniting my passion for my Master’s subjects. Your wisdom will greatly impact my academic future. To my family, I am eternally grateful for your unwavering support and trust. Irving, my father, and Silvia, my mother, allowed me to pursue my dreams by coming to Portugal. Even when you had to return to Ecuador, you believed in me and gave me the chance to stay in Portugal, continue my studies, and grow as an individual. My brothers, Andrés and Alejandro, who shared this journey with me, your presence has been invaluable. Our shared moments, whether playing basketball, exploring Minecraft, or simply spending time together, have been treasured. Lastly, to my beloved girlfriend, Brisell, or ”Bris” as I affectionately call you, you’ve been my rock and unwavering support, helping me overcome the toughest days. Your presence in my life has been a source of strength and inspiration, and the love and happiness we’ve shared have been the driving force behind my accomplishments. I’d also like to acknowledge my friends, who have been a source of motivation and camaraderie throughout my academic journey. To Barros, Gabriel, Magda, Rodrigo, Sara, and Tiago, the TEKKENLAB group, I extend my heartfelt thanks for the memorable moments we’ve shared. I’m gratefu to the International Iberian Nanotechnology Laboratory (INL) for the research opportunity during my Master’s program and to the Calouste Gulbenkian Foundation for the New Talents grant, which connected me with a remarkable scholarly community. Your collective support has made this journey enriching and fulfilling experience, and I am truly grateful for each and every one of you. ii
Statement of Integrity I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. University of Minho, Braga, october 2023 Irving Leander Reascos Valencia iii Assinado por: Irving Leander Reascos Valencia Num. de Identificação: 33217377 Data: 2023.10.31 14:47:35+00'00'
Abstract Quantum simulation represents a formidable challenge for classical computers due to the intricate behavior of quantum systems. Digital quantum computers aim for precise approximations of a wide range of quantum systems. Within the realm of quantum simulation, the study of spin systems plays a pivotal role, providing insights into complex properties challenging to model through classical means. This work focuses on investigating chiral spin systems through the development of a dedicated quantum circuit for chirality measurement. In this Masters Dissertation, we present an overview of the current state-of-the-art in quantum simulation of chiral spin systems and introduce our approach to addressing this challenge. Scalar spin chirality, a three-body physical observable, holds a critical position both in classical magnetism, where it characterizes non-coplanar spin textures, and in quantum magnetism, serving as an order parameter for chiral spin liquids. In the context of quantum information, scalar spin chirality serves as a witness to genuine tripartite entanglement. In this study, we delve into various methodologies to tackle the problem at hand, subjecting them to comparison. The objective is to identify the most suitable approach that demands fewer quantum resources. Our best proposed method introduces an indirect measurement scheme based on the Hadamard test, designed to estimate the scalar spin chirality for general quantum states. We apply this innovative approach to measure chirality in two specific types of quantum states: the generic one-magnon states of a ferromagnet and the ground state of a model characterized by competing symmetric and antisymmetric exchange interactions. Our research findings highlight the practicality of achieving a single-shot determination of scalar chirality for chirality eigenstates, leveraging the power of quantum phase estimation with a single auxiliary qutrit. This novel methodology extends beyond providing a solution to the chirality measurement problem; it also unifies the theory of chirality in both classical and quantum magnetism. The implications of our work offer valuable insights and pave the way for future quantum research endeavors in this domain. Keywords Quantum Simulation, Magnetism, Quantum Entanglement, Magnetic Order iv
Resumo A simulação quântica desafia os computadores clássicos devido ao comportamento complexo dos sistemas quânticos. Os computadores quânticos digitais buscam aproximar uma ampla variedade de sistemas quânticos. Na simulação quântica, a análise de sistemas de spin é essencial para compreender propriedades complexas difíceis de modelar com abordagens clássicas. Este estudo concentra-se em investigar sistemas de spin com quiralidade por meio do desenvolvimento de um circuito quântico dedicado para medição. Nesta Dissertação de Mestrado, apresentamos um panorama atual da simulação quântica de sistemas de spin com quiralidade e introduzimos nossa abordagem para abordar esse desafio. O operador de quiralidade, um observável físico de três corpos, desempenha um papel crítico no magnetismo clássico, caracterizando texturas de spin não coplanares, e no magnetismo quântico, atuando como um parâmetro de ordem para líquidos de spin com quiralidade. No contexto da informação quântica, a quiralidade serve como testemunha de entrelaçamento tripartite genuíno. Neste estudo, exploramos várias metodologias para abordar o problema em questão, comparando-as com o objetivo de identificar a mais eficiente em termos de recursos quânticos. O nosso melhor método proposto é um esquema de medição indireta baseado no teste de Hadamard, criado para estimar a quiralidade do spin em estados quânticos gerais. Aplicamos esta abordagem para medir a quiralidade em dois tipos específicos de estados quânticos: os estados genéricos de um magnon de um ferromagneto e o estado fundamental de um modelo com interações de troca simétricas e antissimétricas em competição. As descobertas de nossa pesquisa destacam a praticidade de alcançar uma determinação única da quiralidade do spin para estados com quiralidade, aproveitando o poder da estimativa de fase quântica com um único qutrit auxiliar. Essa metodologia inovadora vai além de fornecer uma solução para o problema de medição de quiralidade; ela também unifica a teoria da quiralidade tanto no magnetismo clássico quanto no magnetismo quântico. As implicações de nosso trabalho oferecem conhecimentos valiosos e abrem caminho para futuras esforços de pesquisa quântica nesse domínio. Palavras-chave Simulação Quântica, Magnetismo, Entrelaçamento Quântico, Ordem Magnética. v
Contents 1 Introduction 1 1.1 Motivation ...................................... 2 1.2 Minimal Background ................................. 5 1.2.1 Quantum Simulation ............................. 5 1.2.2 Classical Spin States ............................. 6 1.2.3 Quantum treatment ............................. 8 1.2.4 Scalar Spin Chirality ............................. 10 1.3 Objectives ...................................... 11 1.4 Outline of the Document ............................... 12 2 Theoretical Background and Methodology 14 2.1 Quantum Simulation ................................. 14 2.1.1 Analog and Digital Quantum Simulators .................... 14 2.1.2 Designing a Quantum Simulation ....................... 16 2.2 Digital Quantum Computers .............................. 18 2.3 Quantum Circuits and Techniques ........................... 21 2.3.1 Lie-Trotter-Suzuki Formula .......................... 21 2.3.2 Linear Combination of Unitaries (LCU) .................... 22 2.3.3 Quantum Fourier Transform (QFT) ...................... 24 2.3.4 The Hadamard Test ............................. 25 2.3.5 Quantum Phase Estimation (QPE) ...................... 27 2.3.6 Efficient Implementation of |W⟩States .................... 29 2.4 Spin Systems ..................................... 31 2.4.1 Spin ..................................... 31 2.4.2 Spin Operators and Their Eigenstates ..................... 34 vi
26 Figure 26a illustrates the schematic representation of the unitary vector n characterized by the polar angle φand azimuthal angle θ. Additionally, Figure 26b depicts a schematic representation of a general coherent state |n⟩in the Bloch sphere, demonstrating the relationship between the state and the classical vector n................ 71 27 Estimation of scalar spin chirality ˆχin spin-wave states defined on an N-site ring. The maximal value of ˆχacross all trios of spins-1 2and all Nspin-wave states is shown in black (Equation 5.10). The same values were estimated using the Hadamard test in Fig. 25 with 10,000 samples for each N, shown in blue. The horizontal dashed line marks the threshold for genuine tripartite entanglement Tsomokos et al. [2008]. While the scalar spin chirality surpasses this threshold only for N= 3, the concurrence fill Xie and Eberly [2021], a measure of tripartite entanglement, confirms its existence for all system sizes in the plot. The black and red solid lines emphasize the decay of ˆχand the concurrence fill with the system size N......................... 75 28 The figures display the simulation results obtained using QuSpin for a ring with N= 10 particles, J= 1, and D=−Jtan 2π N. In these simulations, the operator Siwas measured for each particle i∈[1, N]in two systems: one without a local magnetic field (Bx= 0, 28a) and the other with a local magnetic field (Bx=−1, 28b) to achieve a broken-symmetry ground state. The dashed lines in the figures represent the analytical solutions obtained through classical analysis. ..................... 78 29 The figures display the simulation results obtained using QuSpin for a ring with N= 10 particles, J= 1, and D=−Jtan 2π N. In these simulations, the correlator Sα 1Sα i, α ∈ {x, y, z}was measured for each particle i∈[2, N]in two systems: one without a local magnetic field (Bx= 0, 29a) and the other with a local magnetic field (Bx=−1, 29b) to achieve a broken-symmetry ground state. The dashed lines in the figures represent the analytical solutions obtained through classical analysis. In subfigure 29a, the dashed lines represent the analytical classical solutions for ⟨Sα 1Sα i⟩, where α∈ {x, y}................................... 79 xiii
30 Simulation results for the chirality measurement of the Hamiltonian in Equation 5.11 for Bx= 0 (30a) and Bx=−0.1(30b) with varying B∈[0,1]. The Quantum Ground State (QGS) was obtained through exact diagonalization using QuSpin. The Classical Ground State (CGS) was obtained by minimizing the classical energy E(θ, q)(Equation 5.23). In the case where Bx= 0, the ground state is a superposition of an infinite set of spin spirals, resulting in a classical chirality of zero. However, in the case with Bx= −0.1, the system leads to a broken-symmetry ground state that allows the existence of classical chirality. Black curve: Exact quantum chirality ⟨ˆχ149⟩. Blue dots: Estimated quantum chirality using the cycle test circuit in Qiskit. Green: Classical chirality χcla 149 computed from the box product of classical spin vectors si= (⟨Sx i⟩,⟨Sy i⟩,⟨Sz i⟩)over the QGS (see Equation 5.24). Red: Classical chirality χcla 149 calculated using Equation 5.22 with values of θand qmin from the CGS. ..................... 81 31 Classical spin vectors for three different values of B. The left figure corresponds to B= 0, and its spin configuration is a coplanar spin spiral with zero chirality. In the middle figure, for B= 0.17, the maximum quantum chirality is observed, and the nonzero classical chirality is apparent due to the non-coplanar spin configuration. The right figure, for B= 1, shows zero chirality, with the spins aligned with the external magnetic field. The chirality values were presented in Figure 30. ................. 82 xiv
List of Tables 1 Common Quantum Gates and Their Representations. There are specific cases for the phase gate P(ϕ), namely: Pπ 2corresponds to the Sgate, and a phase of ϕ=π 4 gives the Tgate. ................................... 20 2 Single-Qubit Rotation Gates .............................. 21 3 The table summarizes the results obtained by simulating the circuit shown in Figure 22 that apply the LCU method to estimate ⟨ˆχ2⟩for each chirality eigenstate. The error is given by ε=8 √Nshoots = 0.08, where Nshoots = 10 000................ 55 4 The table summarizes the results obtained by simulating the circuit shown in Figure 23 that uses the Hadamard Test and the LCU method to estimate ⟨ˆχ⟩for each chirality eigenstate. The error is given by ε=4 √Nshoots = 0.04, where Nshoots = 10 000.. . . 57 5 Estimated eigenvalues χ=λusing QPE. The estimation error is limited by the precision determined by the number of qubits. With n= 2 qubits, the binary fraction error is bounded by δmax =1 2n+1 = 0.125. Consequently, the chirality estimation error is ϵ= 3δmax = 0.375. The measured value of φ 22, providing the best approximation of the binary fraction representing ϕ=−λ 3, was obtained by selecting the result with the highest probability from simulations with different input states (see Figure 24). ..... 63 6 Estimated eigenvalues χ=λusing QPE with n= 5 qubits. The estimation error is limited by the precision determined by the number of qubits. With n= 5 qubits in the ancillary register, the maximum error in the estimated value of ϕis δmax =1 2n+1 ≈0.02. Consequently, the error in the chirality estimation is ϵ= 3δmax ≈0.05. The table summarizes the results obtained for the estimated eigenvalues using the enhanced QPE setup. ........................................ 63 xv
7 The table shows the results of the noiseless simulation of the cycle test (see Figure 25), for the chirality eigenstates as input. The estimation was obtained with Nshots = 10,000 trials. The error bounds for the chirality and chirality squared are ϵχ=2 √3Nshots ≈0.01 and ϵχ2=2 3 1 √Nshots ≈0.007, respectively. ...................... 66 8 The table presented below provides an overview of the resources needed for performing the chirality estimation. It compares the number of ancillary qubits, circuit depth, and the total number of gates. Specifically, the count of CNOT gates is explicitly displayed. . 67 9 The table below compares the number of ancillary qubits and the number of shots Nshots required to obtain the chirality estimation with a bounded error of ε= 0.01. For the QPE method, the number of shots is associated with a probability of obtaining the correct result in a single shot, and this probability is explicitly stated in the table. ........ 69 xvi
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Chapter 1 Introduction Some real-world phenomena are difficult to study through observation or experimentation due to their rarity or the necessity of complex and sophisticated equipment. Therefore, a common approach is to study them through simulation, using more accessible resources. These simulations replicate the system’s behavior, making its analysis possible. For instance, it is possible to simulate the launch of a spacecraft by modeling all the variables, such as gravitational forces, the propulsion system, atmospheric conditions, and so on, in order to predict and find the optimal launch trajectory to complete the mission. In this example, the necessity of simulation is clear as it provides a way to have confidence in the success and safety of the mission without spending resources or risking lives. Similar to the previous example, simulating other aspects of physics helps us better understand our universe. Simulation can be seen as a tool to find answers to unresolved questions. Therefore, simulating quantum mechanics is very important due to its applications in fields such as materials science, highenergy physics, and quantum many-body problems Daley et al. [2022]. This dissertation focuses on simulating quantum systems, which are systems where quantum phenomena cannot be neglected. Quantum systems have the capability to exhibit a unique state called superposition, wherein they combine the properties of multiple classical states simultaneously Daley et al. [2022]. This leads to the exponential growth of the Hilbert space, which is the space where the system is described. For example, if we consider a system composed of nquantum particles with two allowed states, the dimension of the Hilbert space for this system is 2n. To simulate this system on a classical computer, which is based on transistors and stores classical information using bits, 2ncomplex numbers would be needed to describe a complete superposition of the system. For a sufficient number of particles, a classical computer with the necessary resources does not exist. This limit is known as the classical limit and is reached for n= 50 particles, where 250 ≈1015 complex numbers would need to be stored. Additionally, for n= 300, the number of complex numbers needed to be stored exceeds the number of atoms in the visible universe Lloyd [1996], Tacchino et al. [2020], Daley et al. [2022]. 1
As it is possible to see, simulating quantum systems is challenging, even impossible, for classical computers. The late Physics Nobel Prize laureate Richard Feynman was aware of this problem and, in a lecture in 1982, conjectured the possibility of having a suitable class of quantum machines that could mimic another quantum system. Thus, Feynman introduced the concept of a programmable quantum system, also known as a quantum computer. This approach allows us to overcome the problem of the enormous size of the Hilbert space, because the quantum computer is governed by the same rules of quantum mechanics as the desired system to simulate Feynman [1982]. Recalling the previous example of a system of ntwo-state particles, if the quantum computer is made up of particles with two allowed quantum states, called qubits for short, then the number of resources needed scales linearly with the scale of the system, just needing nqubits. The importance of studying techniques for simulating quantum systems using the capabilities of quantum computers is evident, and this is the research field of the present work. Hence, the purpose of this introductory chapter is to provide a detailed explanation of the motivation behind the research goal and the significance of this master’s thesis. Additionally, the final section outlines the structure of this document to guide readers and provide a comprehensive understanding of the content covered. 1.1 Motivation In quantum mechanics, each physical system is governed by an operator known as the Hamiltonian, represented as H. The Hamiltonian encompasses the system’s total energy, which includes its interactions, and it establishes the dynamical rules that the system must adhere to. Consequently, when considering a specific state of the system denoted by ψ(t), which corresponds to a particular configuration of its constituent elements, the evolution of this state is determined by the system’s Hamiltonian. The connection between these quantities is described by the Schrödinger Equation (SE) presented below Sakurai and Commins [1995], Cottam and Haghshenasfard [2020]. Hψ(t) = iℏ∂ ∂tψ(t)(1.1) In this context, ℏrepresents the Planck constant. The Schrödinger Equation (SE) is a differential equation discovered by Erwin Schrödinger, and it governs the dynamics of the wave function ψ(t). However, in many practical scenarios, this equation can be transformed into a time-independent form where the operator Hcan be represented as a matrix, and accordingly, the wave function ψ(t)can be expressed as a complex vector ψ. Consequently, in the time-independent Schrödinger Equation, the equation trans2
forms into an eigenvalue problem, where the eigenvalue represents the energy Eof the system Sakurai and Commins [1995], Cottam and Haghshenasfard [2020]. Hψ =Eψ (1.2) Therefore, solving this problem involves two main steps. The first step is constructing the Hamiltonian and its matrix representations, which can be done straightforwardly using classical resources. Solving the eigenvalue problem in quantum systems is notably challenging, mainly due to the exponential growth of their Hilbert space with the number of particles (degrees of freedom) involved Feynman [1982], Daley et al. [2022]. This requires substantial classical resources, which increase exponentially, as reflected in the size of the Hamiltonian matrix. This challenge is particularly significant for large-scale systems. As mentioned earlier, a potential solution is to employ quantum resources, which scale linearly with the system’s size. This approach allows us to study larger systems beyond the classical limit by simulating another quantum system Feynman [1982], Lloyd [1996], Daley et al. [2022]. The process of using quantum resources to simulate desired quantum systems is referred to as quantum simulation Lloyd [1996], Tacchino et al. [2020], Daley et al. [2022]. The primary approach to tackle this challenge involves leveraging Quantum Computers (QCs). The standard paradigm for QCs is the digital quantum circuit model, where the basic unit of representation mimics the classical unit of information known as a bit. In analogy, the minimal unit of quantum information is called a qubit Nielsen and Chuang [2002]. Just as classical digital circuits are implemented by utilizing a minimal set of gates to perform required operations, quantum digital circuits have their own set of quantum gates. These gates, similar to their classical counterparts, are employed in a circuit model to construct quantum algorithms Nielsen and Chuang [2002]. In digital QCs, there exists a group of gates known as universal set of quantum gates, which allows the implementation (or approximation with arbitrary accuracy) of any unitary operation Nielsen and Chuang [2002], Childs [2017]. This set typically consists of oneand two-qubit operations. Thus, the process of simulating a quantum system using this type of architecture relies on implementing quantum circuits that approximate the desired unitary operations. While it may seem intuitive to grasp the concept of simulating a physical configuration using another one, the actual implementation is not straightforward. Therefore, it is necessary to develop methods that enable the imitation of the target system and obtain relevant information from it. However, there is an additional constraint associated with the physical implementation of quantum simulators. These devices are governed by the rules of quantum mechanics and leverage its advantages, but they are also 3
subject to certain challenges. The primary issue is the ”coherence” time, which represents the duration during which the computation remains accurate Nielsen and Chuang [2002], Preskill [2018]. Beyond this time, noise predominantly affects the information due to interactions between the device and its environment, as well as within the device itself. John Preskill introduced the concept of Noisy Intermediate Scale Quantum (NISQ) technology to describe the currently available devices Preskill [2018]. Consequently, when designing a quantum simulation, it is crucial to consider the method’s performance in terms of utilizing the fewest possible resources, such as the number of quantum subsystems and the runtime. Many intriguing problems nowadays lie in the study of quantum systems due to their vast range of applications. Consequently, it is imperative to find ways to investigate these physical configurations. However, the range of quantum systems is incredibly broad. Therefore, this thesis will specifically focus on one type of quantum system: spin systems. In simpler terms, a spin system refers to a collection of quantum systems that possess an intrinsic property known as spin, which will be explained in more detail later. For instance, an electron is a particle with charge and a spin of 1/2 Sakurai and Commins [1995], Parkinson and Farnell [2010]. There is a class of materials referred to as insulators, which, as the name suggests, do not allow the flow of electric charge within the material. However, within the category of insulating materials, there is another class known as magnetic insulators. While these materials are electrically insulating, meaning the charge is constrained, they still exhibit a degree of freedom associated with spin Wu and Hoffmann [2013]. The behavior of these materials is described in terms of the dynamics of the spin. This gives rise to spin models, which provide a framework for understanding their properties and interactions. Examples of spin models include: the Ising model, the Heisenberg model or the XY model Parkinson and Farnell [2010], Pires [2021]. These models capture various aspects of spin interactions and have been widely studied in the field of condensed matter physics. The significance of studying this type of quantum entities arises from their diverse range of applications, particularly in the field of quantum magnetism, which aims to comprehend their magnetic properties Parkinson and Farnell [2010]. Therefore, the primary motivation behind this dissertation work is to make significant advancements in our understanding of utilizing the capabilities of Quantum Computers (QCs) for performing quantum simulations of relevant cases involving magnetic insulators. 4
1.2 Minimal Background Since most of the necessary physics background for this work has been covered within the dissertation, this section aims to provide a concise introduction to some important concepts required to comprehend the objectives of this thesis. The purpose is to offer a brief overview and ensure that readers have the necessary foundation to grasp the goals and significance of the research presented in this thesis. 1.2.1 Quantum Simulation The first concept to introduce is the general structure of a quantum simulation. As mentioned previously, a quantum simulation aims to replicate a desired natural quantum system, referred to as the target system, in another system known as the simulator (see Figure 1)Feynman [1982], Daley et al. [2022]. This process can be divided into three stages: preparation, evolution, and post-processing Daley et al. [2022]. ... ... (a) Target System |0⟩ |Ψ(0)⟩UH(t) |0⟩ |0⟩. . . |0⟩ Preparation Evolution |Ψ(t)⟩Post-processing (b) Quantum Simulator Figure 1: Schematic representation of the General Structure of a Quantum Simulation. Figure 1a depicts the target system, which represents the natural system under analysis. Figure 1b illustrates the mapping of the target system into the quantum simulator, along with the corresponding states of the simulation. In the preparation stage, the simulator is initialized to match the initial state |ψ(0)⟩of the target system. This involves mapping the initial state of the target system onto the resources of the simulator |Ψ(0)⟩. Next, in the evolution stage, the quantum state encoded in the simulator undergoes time evolution. 5
[2021]. An objective of this work is to explore the potential of utilizing the cycle test method to estimate the chirality of a given system. In order to achieve the research objectives, several training objectives needed to be fulfilled. Firstly, it was crucial to develop a comprehensive understanding of the basic theory of quantum magnetism, including the description of spin operators and various spin models. This involved an in-depth study of relevant literature to become familiar with the theoretical frameworks and the underlying methods in quantum mechanics. Key topics included the description, construction, diagonalization, and classical simulation of quantum models expressed in terms of spin operators. By acquiring proficiency in these fundamental concepts, a solid foundation was established for investigating the physics underlying scalar spin chirality. Furthermore, expertise in quantum circuit design and quantum simulation was indispensable for exploring and refining the intended measurement strategies. This entailed acquiring knowledge of various quantum gates, quantum algorithms, and tools for quantum simulation. As it will be necessary to choose the best approach to estimate the scalar spin chirality a comparative analysis will be conducted to assess the advantages and disadvantages of different approaches in terms of required quantum resources, with the aim of identifying the most suitable method for NISQ (Noisy Intermediate-Scale Quantum) computers. Once the optimal algorithm is determined, its effectiveness will be validated by applying it to calculate the chirality of diverse spin systems. By undertaking this research, the aim is to contribute to the advancement of efficient techniques for measuring the chirality of spin systems, thereby overcoming the existing limitations of current quantum methodologies. 1.4 Outline of the Document This dissertation comprises seven chapters. The first chapter, which serves as the introduction to this work, has already been presented. Chapter 2delves deeper into the thesis’s background and methodology. Chapters 3through 5present the results and the process of acquiring them. The final chapter, chapter 6, provides a summary of the work, draws conclusions, and outlines future research possibilities. In the following text, each chapter will be briefly explained. Chapter 2provides the foundational theoretical background for this dissertation, introducing key concepts such as quantum simulation, digital quantum computation, quantum circuit components, quantum algorithms, and quantum mechanics in the context of quantum magnetism. This chapter also presents the state-of-the-art in the field, serving as the foundation for our research. 12
The methodology section 2.7 outlines the research methods used in this dissertation. It introduces Python as the primary programming language and discusses its importance in scientific computing. The section focuses on two main methodologies: analytical calculations and computer-assisted simulations. Analytical calculations provide theoretical insights into proposed quantum circuits, while simulations are vital for empirical testing. The Chirality Eigenstate Preparation Chapter 3focuses on the initial phase of quantum simulation, where we prepare the initial state, specifically the eigenstates of the chirality operator, and discuss their implementation in quantum computers. Then, Chapter 4builds on the quantum algorithms introduced in the theoretical background (Chapter 2). We apply quantum circuits to propose and test various indirect methods for measuring chirality within a given state, with a focus on finding the most effective approach in the era of Noisy Intermediate-Scale Quantum (NISQ) computing. Chapter 5utilizes the most promising NISQ method developed in the previous chapter to indirectly measure the chirality of various systems. This includes classical magnetic states represented by separable coherent spin states, spin-wave models, and spin spirals. Finally, Chapter 6summarizes the conclusions drawn from our research and outlines potential directions for future work and research in the field. 13
Chapter 2 Theoretical Background and Methodology In the introduction of this dissertation, key concepts were introduced to provide an initial understanding of the thesis and its significance. This chapter serves to further elucidate the theoretical foundation upon which the outcomes of this project are built. This chapter will delve into several crucial topics, including quantum simulation utilizing quantum computers as simulators, the current state-of-the-art of quantum algorithms, Hamiltonians for spin systems, the concept of scalar spin chirality, and the forefront of quantum simulation concerning these specific types of systems. 2.1 Quantum Simulation As discussed in the previous chapter, simulating quantum systems is a challenging task for classical computers due to the rapid increase in the Hilbert space as the system size grows. For classical computers, this task becomes practically impossible when using greedy algorithms to simulate such systems, like attempting to simulate all interactions and store the complete information of the wave function. Even with more complex classical algorithms, for sufficiently large systems, the required amount of resources becomes unfeasible Feynman [1982], Lloyd [1996], Daley et al. [2022]. Consequently, a solution was proposed by Richard Feynman in the early 1980s. This solution involves using one quantum system (referred to as a simulator ) to simulate another target quantum system Feynman [1982]. 2.1.1 Analog and Digital Quantum Simulators In quantum simulation, there are two main classes based on the type of simulator used: analog quantum simulators (AQS) and digital quantum simulators (DQS). Both approaches share the fundamental idea of emulating a quantum system of interest using another quantum system. To differentiate between the two, consider a target quantum system governed by the Hamiltonian H, which encodes its dynamics and interactions. The target system evolves from an initial configuration |ψ(0)⟩to a final state |ψ(t)⟩over a 14
time tunder the time evolution operator UH(t)Daley et al. [2022]. The key distinction between the two approaches lies in their methods. An analog quantum simulator aims to replicate the dynamics and interactions of the target system. In this case, the simulator’s Hamiltonian Happroximates the target Hamiltonian H(H≈ H). Conversely, a digital quantum simulator seeks to approximate the system’s dynamics. Here, the unitary time evolution of the simulator approximates that of the target system (UH(t)≈UH(t))Daley et al. [2022]. Both approaches have their own set of advantages and disadvantages. Analog Quantum Simulators (AQS) offer benefits for specific many-body problems, leveraging particular types of quantum platforms that yield good outcomes when used to emulate the target system. For example, trapped ions are well-suited for studying interacting spin systems Monroe et al. [2021], superconducting circuits excel at investigating various interactions like strong photon-photon interactions Wilkinson and Hartmann [2020], and cold atoms are employed to simulate condensed matter systems Henriet et al. [2020]. Flannigan et al. [2022]’s analysis indicates that the first practical instance of quantum advantage in simulation can be achieved through special-purpose analog devices enabling direct model implementation. However, an inherent drawback of AQS is its limited generality. This limitation stems from the fact that AQS are confined to replicating only the types of interactions that the simulator is capable of emulating Daley et al. [2022]. As a result, AQS may not provide a comprehensive solution to a wide range of problems. Therefore, the DQS aims to be a general-purpose quantum simulator by approximating unitary operations, rather than replicating specific interactions. This is accomplished through the use of digital quantum computers (DQC) which employ a universal set of quantum operations or gates Tacchino et al. [2020], Daley et al. [2022]. This gate set allows for the approximate representation of any unitary operation, enabling DQS to simulate a wide range of quantum systems. However, a significant challenge arises with this approach. To solve any quantum system accurately, a fault-tolerant DQC is required. This implies that the quantum computer can perform computations even in the presence of errors Daley et al. [2022]. Unfortunately, such fault-tolerant DQCs are not currently available. As Preskill [2018] has noted, our current state of quantum computing is characterized by Noisy Intermediate Scale Quantum (NISQ) devices. These devices have limitations in terms of coherent computation time and susceptibility to environmental noise, which restricts the scope of applications that can be effectively handled. However, it is important to highlight that even within the NISQ regime, there exists a variety of applications where these devices can provide valuable outcomes across different domains. For instance, recent work by Kim et al. [2023] demonstrated the measurement of precise expectation values for circuit volumes exceeding the capabilities of brute-force classical computation. This showcases the practical utility 15
of quantum computing even in the absence of fault-tolerant capabilities. These achievements are made possible through the application of various error mitigation techniques that are particularly relevant when using real quantum devices. Although this work does not delve into the specifics of these error mitigation algorithms, for further details, please refer to the original paper by Kim et al. [2023]. This work will specifically concentrate on the Digital Quantum Simulation (DQS) case, and the upcoming sections will introduce quantum circuits falling within the realm of Digital Quantum Computing (DQC). 2.1.2 Designing a Quantum Simulation The process of simulating a quantum system using a quantum simulator can be broken down into three primary stages: preparation, evolution, and post-processing. These stages are fundamental to quantum simulation, regardless of the specific type of quantum simulator employed. The subsequent discussion will delve into each of these stages in greater detail. Preparation In the first stage of quantum simulation, qubits are prepared to be in the desired initial state, often corresponding to the system’s ground state. This involves defining the basis states of the target system and determining how this information will be encoded within the quantum computer. For instance, representing a single spin-1 2particle in a qubit is straightforward due to its inherent qubit nature. Additionally, for encoding fermions 1, the Fock space representation is commonly employed. In this representation, each qubit signifies an occupation basis state, either |1⟩(occupied) or |0⟩(unoccupied) Claudino [2022]. Subsequently, the quantum algorithm for preparation establishes the initial computational configuration. Several algorithms are available for preparing the initial state of a quantum system for simulation. One such method is the Quantum Phase Estimation (QPE) algorithm, which estimates the eigenvalues of a unitary operator Uby iteratively applying it Kitaev [1995]. This technique will be further elucidated in the upcoming section 2.3 on quantum algorithms. Additionally, QPE can be leveraged to prepare a desired eigenstate of the same unitary operator U, as measurement of the eigenvalue causes the input state to collapse into the associated eigenstate Murta et al. [2023]. Another example is the Variational Quantum Eigensolver (VQE), which employs variational techniques to approximate the ground state of a given Hamiltonian by minimizing the measured energy Peruzzo 1Fermions adhere to Fermi-Dirac statistics, which dictate that a collection of Nidentical fermions is entirely antisymmetrical when any pair is interchanged. Moreover, fermions possess half-integer spin and adhere to the Pauli exclusion principle, much like electrons Sakurai and Commins [1995]. 16
et al. [2014], McArdle et al. [2019]. A different approach is outlined in Carbone et al. [2022], where a recursive method for constructing total spin eigenfunctions through variational means is demonstrated. Furthermore, Murta and Fernández-Rossier [2021] introduces algorithms for encoding the Gutzwiller wave function, while Murta et al. [2023] suggests techniques for encoding Valence-Bond solid states on NISQ devices. Ultimately, the choice of a specific circuit may hinge upon the particular problem under consideration and the desired outcomes. Evolution While certain quantum simulation algorithms might focus exclusively on the initial state preparation of a system, others extend to encompass the state’s evolution based on a provided Hamiltonian. The manner in which the model is represented on a quantum computer, along with the corresponding quantum operations, depends on the chosen basis state representation. For instance, consider a scenario as depicted in Anselme Martin et al. [2022], where the Hamiltonian is described using fermionic operators. In such cases, it might be necessary to convert it into Pauli operators through techniques like the Jordan-Wigner or Bravyi-Kitaev transformations Bauer et al. [2020]. Various alternative approaches are also available for achieving this objective. The target Hamiltonian Hfor quantum simulation is often represented as a sum of local interactions H=Pjhj, and the system’s unitary evolution is achieved through exponentiating this operator, UH(t) = e−iHt. When the local interactions commute, the overall time evolution operator can be readily decomposed into a sequence of local time evolution operators, UH(t) = Qje−ihjt. However, in many scenarios, these local interactions do not commute, which requires the use of techniques like Trotterization to approximate the total evolution operator Claudino [2022] or employing the Linear Combination of Unitaries (LCU) method Childs and Wiebe [2012]. These methods will be elaborated upon in more detail in Section 2.3. Once this unitary operation has been transformed into permissible gate operations, a quantum computer can employ it to simulate the evolution of the system. Post-processing The final stage of a quantum simulation involves extracting information about the system’s ultimate state, such as expectation values for specific observables Childs [2017]. It’s important to emphasize that in a quantum computer, only the qubits are subjected to measurement, and this process merely discloses the computational basis 2state to which the qubit has collapsed Nielsen and Chuang [2002]. Consequently, 2The eigenstates of the Z(Pauli Z) operator, represented by |0⟩and |1⟩, are commonly referred to as the computational basis states. 17
it might be necessary to repeat the simulation numerous times and analyze the resulting data statistically. For measuring expectation values, we apply operators and process the acquired data. A common technique for obtaining operator expectation values is Hamiltonian averaging. This method entails repeating measurements while altering the basis depending on the specific Pauli operator being measured for each qubit McClean et al. [2014]. Additionally, tomography techniques can be employed to gain deeper insights into the simulated system, employing a similar approach Cheng and Lou [2020]. The mentioned techniques correspond to direct measurement approaches, and an exhaustive characterization of the output state is typically justified when multiple expectation values need to be computed. However, there are alternative methods that involve indirect measurements, which may be more suitable in certain cases than the commonly used direct measurement techniques. The main advantage of these indirect measurement techniques is that they only measure the specific information of interest, making them more compatible with NISQ devices. A notable example of such an indirect measurement procedure is the Hadamard test Cleve et al. [1998], which employs a single ancillary qubit. Quantum phase estimation (QPE) extends this concept to multiple ancillary qubits Kitaev [1995], Nielsen and Chuang [2002]. 2.2 Digital Quantum Computers As previously mentioned, digital quantum computers (DQC) aim to be general-purpose quantum computers. They achieve this by employing a collection of quantum operations that form a universal set, regardless of the particular quantum hardware in use. This universal set of gates allows for the simulation of a a wide range of quantum systems Daley et al. [2022]. The universal set of gates is defined to approximate any unitary operator Uwith an arbitrary error εusing a sequence of ngates Uifrom the universal set Nielsen and Chuang [2002]. This error is defined in the equation below: U− n Y i=1 Ui < ε (2.1) where ∥A∥ ≡ max|ψ⟩|A|ψ⟩| ||ψ⟩| is the spectral norm of the matrix A, defined as the maximum absolute eigenvalue of Aand ||ψ⟩| =p⟨ψ|ψ⟩Horn and Johnson [1990]. In general, a universal set of quantum operations must include the ability to implement a general single-qubit operation, often represented by the U3gate (as shown in Equation 2.2), and a two-qubit gate such as the controlled-not (CNOT) gate (see Table 1)Barenco et al. [1995]. However, this is not the only universal set. For instance, the Clifford group (generated by the CNOT gate, Hadamard gate H, and Sgate) along with the Tgate (see Table 1) also forms a universal set Carignan-Dugas et al. [2015]. 18
U3(θ, ϕ, λ) = cos θ 2−eiλ sin θ 2 eiϕ sin θ 2ei(ϕ+λ)cos θ 2 (2.2) Since the focus of this work does not involve the specifics of how DQC implements the set of quantum gates, we will employ the commonly used set of quantum operations (see Table 1). Their names and matrix notations are presented below. Additionally, when constructing a quantum algorithm, a circuit notation is often used, where quantum operations are depicted as boxes arranged in horizontal lines. The sequence of operations proceeds from left to right, representing the order of operations over time Feynman [1985], Nielsen and Chuang [2002]. |ψ⟩ |+⟩ |i⟩ |0⟩ |1⟩ φ θ Figure 2: The figure illustrates a general pure state |ψ⟩defined by the azimuthal angle θand the polar angle φ. The azimuthal angle provides us with the amplitudes of the state |0⟩and |1⟩, while the polar angle represents the complex phase between the computational basis states |0⟩and |1⟩. The state |+⟩ corresponds to the eigenstate with a positive eigenvalue of the Pauli-Xoperator and can be described by θ=π 2and φ= 0. Similarly, the state |i⟩is the eigenstate with a positive eigenvalue of the Pauli-Y operator and is characterized by θ=π 2and φ=π 2. 19
Table 1: Common Quantum Gates and Their Representations. There are specific cases for the phase gate P(ϕ), namely: Pπ 2corresponds to the Sgate, and a phase of ϕ=π 4gives the Tgate. Gate Circuit Notation Matrix Notation Pauli-X (X)X 0 1 1 0 Pauli-Y (Y)Y 0−i i0 Pauli-Z (Z)Z 1 0 0−1 Hadamard (H)H1 √2 1 1 1−1 Phase (P(ϕ)) P(ϕ) 1 0 0eiϕ CNOT • 1000 0100 0001 0010 SWAP × × 1000 0010 0100 0001 Moreover, it is possible to control single-qubit operations by representing a qubit’s pure state in terms of the Bloch sphere Kok [2018]. A qubit’s state |ψ⟩can be depicted on the Bloch sphere, as illustrated in Figure 2, and is represented by |ψ⟩=cos θ 2|0⟩+eiφ sin θ 2|1⟩(2.3) Therefore, any single-qubit operation can be decomposed into single-qubit rotations along each axis. 20
This gives us the three possible rotations Rα(θ) = e−iθ 2σαfor α∈ {x, y, z}, where σαcorresponds to the Pauli-αoperator Nielsen and Chuang [2002]. These operations can be found in the table 2. Table 2: Single-Qubit Rotation Gates Rotation Circuit Notation Matrix Notation Rx(θ)Rx(θ) cos θ 2−isin θ 2 −isin θ 2cos θ 2 Ry(θ)Ry(θ) cos θ 2−sin θ 2 sin θ 2cos θ 2 Rz(θ)Rz(θ) e−iθ 20 0eiθ 2 2.3 Quantum Circuits and Techniques With the concept of quantum simulation and digital quantum computers introduced, this section now delves into the presentation of several cutting-edge quantum circuits and techniques, which are highly relevant to the focus of this dissertation on implementing quantum digital simulation. These quantum circuits hold significant importance for the context of this dissertation. 2.3.1 Lie-Trotter-Suzuki Formula The exponentiation of the sum of two operators, Aand B, that do not commute [A, B]= 0 cannot be exactly decomposed as the product of the individual exponentiations of these operators. However, this exponentiation can be decomposed using the following relation. e(A+B)=lim n→∞eA neB nn(2.4) This is a product formula named after Trotter [1959] and is widely used for approximating the exponentiation of operators. With this formula, it becomes possible to exponentiate an operator Hcomposed of sums of other operators, given by H=P j hj, using nproducts of individual exponentials ehj/n, where nis a finite integer. This approximation has a bounded error of O(1/n). Therefore, the first-order TrotterSuzuki formula for approximating the exponential eiHt, where t∈R, is provided below Trotter [1959], 21
1 √2n 2n−1 X k=0 ei2πkϕ |k⟩⊗|ψ⟩QF T † −−−→ 1 2n 2n−1 X k=0 2n−1 X j=0 exp −i2π 2nk(j−2nϕ)|j⟩⊗|ψ⟩ =1 2n 2n−1 X k=0 2n−1 X j=0 exp −i2π 2nk(j−φ)ei2πδk |j⟩⊗|ψ⟩ (2.19) Since ϕ∈[0,1], it is possible to approximate the value of 2nϕ=φ+2nδ, where φ∈Z+represents the nearest integer that approximates the eigenvalue of 2nϕ. The value of δrepresents the error in the approximation, and it is bounded by 0≤2n|δ| ≤ 1 2. This error arises when 2nϕcannot be exactly represented as a binary integer of nbits Cleve et al. [1998]. For example, consider ϕ=1 3. In its binary fraction representation, ϕis 0.0101 . . . , which cannot be exactly represented as a unique integer value for any n-bit string. Increasing the resolution of the ancillary register by augmenting the number of qubits can provide a more accurate approximation, but for this example, an exact solution is not possible. The probability of successfully measuring in the ancillary register the integer that best approximates the value of 2nϕ, denoted as φ, is given by P(φ): P(φ) = 1 2n 2n−1 X j=0 ei2πδk 2 = 1 2n 1−(ei2πδ)2n 1−ei2πδ 2 =1 2n sin πδ2n sin πδ 2 (2.20) ≥1 2n 2δ2n πδ 2 =4 π2 (2.21) Therefore, the probability of measuring |φ⟩in the ancillary register and estimating the eigenvalue ϕ is at least 4 π2for δ= 0 and ϕ≈φ 2n. On the other hand, when δ= 0, the probability P(φ)=1and ϕ=φ 2ncan be exactly obtained. The complete algorithm is summarized in Figure 7. For more information about this algorithm, please refer to the works by Cleve et al. [1998], Abrams and Lloyd [1999], Nielsen and Chuang [2002] 28
|0⟩⊗nH⊗n O(U) QFT† |φ⟩ |ψ⟩ Figure 7: The circuit represents the Quantum Phase Estimation (QPE) algorithm, where Uis the unitary operator of interest, |ψ⟩is the quantum eigenstate of Uwith eigenvalue ei2πϕ, and nis the number of qubits in the ancillary register. The box labeled with O(U)represents the oracles corresponding to the controlled operations. The j-th qubit controls the operation U2n−j. Upon measurement, the state |φ⟩is obtained, where φis the integer that provides the best estimate for 2nϕ. 2.3.6 Efficient Implementation of |W⟩States The |W⟩state, named after Wolfgang Dür, is an entangled quantum state involving multiple qubits that demonstrates a form of multipartite entanglement. The general form of the |W⟩Nstate for Nqubits is given in the equation below Dür et al. [2000]. |W⟩N=1 √N(|0. . . 01⟩+|0. . . 10⟩+···+|1. . . 00⟩)(2.22) Implement the |W⟩state can be achieved using an algorithm proposed by Cruz et al. [2019]. This algorithm enables the preparation of the desired state for a general number of qubits Nwith a logarithmic time complexity O(log2N), making it particularly useful for NISQ devices. This logarithmic complexity O(log2N)is attained by leveraging the parallel execution capability of quantum operations on different qubits. By carefully selecting the target and control qubits, multiple controlled operations can be executed simultaneously, resulting in a logarithmic-depth O(log2N)execution instead of the linear depth O(N) required by a sequential approach Cruz et al. [2019]. The algorithm relies on the application of unitary blocks B(p), which are two-qubit gates with the effect shown in Equation 2.23. These gates can be decomposed into a controlled G(p)(see Equation 2.24) operation and a standard CNOT gate, as illustrated in Figure 8. B(p)|00⟩=|00⟩, B(p)|10⟩=√p|10⟩+p1−p|01⟩(2.23) G(p) = √p−√1−p √1−p√p (2.24) In the sequential method, the unitary block is applied to pairs of qubits starting with qubits 1and 2, then 2and 3, and so on, up to the pair N−1and Nfor Nqubits. The corresponding values of pfor 29
• G(p)• Figure 8: Decomposition of the B(p)operator into a controlled G(p)operation and a inverted CNOT gate Cruz et al. [2019]. each block are determined by 1 N,1 N−1, up to 1 2. In contrast, the logarithmic approach involves using different qubits for each parallel execution block. The values of pfor each B(p)block in this approach are calculated differently. The method proposed by D. Cruz employs a dichotomy tree to construct the algorithm. A dichotomy for an integer nis defined as n=⌊n/2⌋+⌈n/2⌉, where ⌊n/2⌋and ⌈n/2⌉are the lower and upper integer parts of n/2, respectively.The main idea is to build a dichotomy tree starting from the root, which is given by the pair (⌊N/2⌋, N). The leaves for a given node (n, m)are obtained by recursively applying the dichotomy operation, resulting in (⌊n/2⌋,⌊m/2⌋)and (⌈n/2⌉,⌈m/2⌉). This process continues until the tree reaches the leaves (0,1),(1,1), or (1,2), at which point the construction of the tree stops Cruz et al. [2019]. To illustrate this concept, let’s consider the case of N= 5. The root of the dichotomy tree for N= 5 is (2,5), and the subsequent leaves can be seen in Figure 9a. In order to adapt the dichotomy tree concept for the |W⟩Nalgorithm implementation, a pruning step is necessary. This pruning involves swapping the leaves (1,1) with (1,2) and eliminating the leaves (0,1) and (1,1). This results in a pruned dichotomy tree, which is used to construct the quantum algorithm. Each node (n, m)in the pruned dichotomy tree corresponds to a block B(n/m)in the algorithm. The control qubit for the controlled G(p)operation is determined by the ”upper line” if the block corresponds to the ”upper child” (left node on the tree). Conversely, if the block corresponds to the ”lower child” (right node on the tree), the control qubit is determined by the ”lower line” of the parent node. The pruned tree for the given example with N= 5 can be seen in Figure 9b. The corresponding quantum circuit that implements the |W⟩5state can be found in Figure 10. The controlled operation G(p)can be constructed in various ways using decomposition rules and relations with available quantum operations. In this case, the relation presented in Figure 11 is used, which involves Ry(θp)(see Table 2) rotations and a CNOT gate, where θp=arcsin √p. 30
(2,5) (1,3) (1,2) (0,1) (1,2) (a) Dichotomy tree. (0,2,5) (1,1,3) (3,1,2) . . . (0,1,2) (b) Pruned dichotomy tree. Figure 9: The Figure 9a shows the dichotomy tree for the case N= 5. The leaves on the right side of the tree correspond to the ”lower child,” while the leaves on the left side correspond to the ”upper child.” The Figure 9b displays the pruned tree obtained from the unpruned tree shown in the left figure. In the pruned tree, the node (0,1) has been removed. To clarify the control qubit for each leaf, the notation used for each node is (c, n, m), where crepresents the index of the control qubit (starting from 0), and nand m retain the same values as in the unpruned tree. 2.4 Spin Systems The focus of this dissertation work is to study quantum systems through quantum simulation using digital quantum computers. The systems of interest in this study are those that can be described utilizing the spin degree of freedom. Hence, this section’s objective is to introduce the key concepts related to such systems and introduce several pertinent models that will prove beneficial for the core discussions within this dissertation. 2.4.1 Spin To comprehend the fundamental concept of a spin system, it’s essential to begin with an exploration of spin itself. For this purpose, let’s delve into the Stern-Gerlach Experiment (SGE) Sakurai and Commins [1995], Kok [2018]. This experiment involves a particle source that emits neutral spin-1 2particles through an inhomogeneous magnetic field Bgenerated by two magnets aligned along the vertical axis, commonly referred to as the ˆzaxis. At the end of their trajectory, the particles are directed toward a screen where their positions are detected Sakurai and Commins [1995], Kok [2018]. Refer to Figure 12 for a schematic representation of this experimental arrangement. To elucidate the behavior of spin-1 2particles in this setup, let’s initially consider classical magnetic dipoles, represented as µ, traversing the magnetic field created by the magnets Sakurai and Commins 31
|1⟩• • |0⟩G(2 5)• • |0⟩G(1 2)• |0⟩G(1 3)• • |0⟩G(1 2)• Figure 10: Quantum circuit for implementing the |W⟩5state using the pruned dichotomy tree in 9b. Each block delimited by the barrier represents a parallel execution block, where, for instance, B(1 2)and B(1 3) are executed simultaneously. This reflects the logarithmic depth of the method, which in this case is ⌈log25⌉= 3. • G(p)=• Ry(θp)Ry(−θp) Figure 11: Controlled G(p)operator decomposition. [1995]. The resulting force experienced by these dipoles can be expressed as: F=∇(µ · B)(2.25) Consequently, the deflection of these dipoles and their resulting positions on the detector screen will depend on their orientation with respect to the magnetic field. This phenomenon is visually depicted in Figure 13a Sakurai and Commins [1995]. In this setup, the classical dipoles are distributed along the entire ˆzaxis, depending on the orientation of their magnetic momentum µ. However, when considering spin-1 2 particles, where the only degree of freedom under consideration for the Stern-Gerlach experiment is the spin, the results are depicted in Figure 13b. In contrast to classical magnetic dipoles, the quantum particles are deflected and hit the screen at only two positions along the ˆzaxis. These positions correspond to the two possible values of the particle’s spin Sz=±1 2. Hence, it is evident that the spin of a particle behaves like a magnetic dipole ˆ µ (see Equation 2.26) in the presence of an external magnetic field. However, it is quantized, and for the present example of spin S=1 2, it has two possible values for its ˆzcomponent Sakurai and Commins [1995], Kok [2018]. 32
Figure 12: The figure depicts the schematic representation of the Stern-Gerlach experiment. It illustrates the particle source, the magnetic monopoles creating an inhomogeneous magnetic field B, and the screen detector where particles are detected after passing through the magnetic field. ˆ µ =gµBˆ S(2.26) In this equation, grepresents the gyromagnetic ratio, which has a value of g= 2 for particles with intrinsic spin. µB=eℏ 2mdenotes the elementary unit of magnetic moment, known as the Bohr magneton, where estands for the electron charge, ℏrepresents the reduced Planck constant, and mdenotes the particle’s mass Sakurai and Commins [1995], Parkinson and Farnell [2010]. The operator ˆ Scorresponds to the spin vector operator associated with the particle. Indeed, since the experiment involved non-charged particles, the observed outcomes can only be explained by the existence of spin as an intrinsic property of particles. This property is not restricted to a specific type of particle; rather, all elementary particle possess this degree of freedom Merzbacher [1998]. For instance, electrons, neutrons, and protons are all spin-1 2particles. Photons have a spin of S= 1, and the Higgs boson has a spin of S= 0. Here, particles can be categorized into two groups: Fermions and Bosons, which obey Fermi-Dirac and Bose-Einstein statistics, respectively. In terms of spin, fermions are those with half-integer spins (1 2,3 2,. . . ), while bosons have integer spins Sakurai and Commins [1995], Merzbacher [1998]. 33
(a) Classical dipoles (b) Quantum spin-1 2particles Figure 13: Schematic representation of the Stern-Gerlach experiment results: 13a Using classical magnetic dipoles with different orientations of their magnetic moments µ relative to the ˆzaxis, the magnets deflect them based on their orientations, resulting in a distribution across the screen detector that spans the entire ˆzaxis. 13b In the quantum case, neutral spin-1 2particles with various initial states are considered. The diagram illustrates the deflection of these particles to only two distinct points on the ˆzaxis, determined by their ˆzspin components. 2.4.2 Spin Operators and Their Eigenstates Spin behaves similarly to orbital angular momentum, and as a result, the quantum mechanics of spin operators follows the same principles. Consequently, we can describe the behavior of the total spin operator S2and Szon their respective eigenstates. These eigenstates |S, Sz⟩are identified by their spin Svalue and their ˆzcomponent Sakurai and Commins [1995], Parkinson and Farnell [2010]. S2|S, Sz⟩=S(S+ 1)ℏ2|S, Sz⟩ Sz|S, Sz⟩=Szℏ|S, Sz⟩ (2.27) The vector operator Shas three components that correspond to the ˆx,ˆy, and ˆzcomponents of the spin. They are denoted as Sx,Sy, and Szrespectively. They follow the cyclic commutation relation [Si, Sj] = iℏεijkSk, where εijk is the Levi-Civita symbol Parkinson and Farnell [2010]. Moreover, two new operators, known as the raising and lowering operators, can be defined as S±=Sx±iSy. Their effects on the eigenstates are presented below Parkinson and Farnell [2010], Pires [2021]. S±|S, Sz⟩= 0, Sz=S S±|S, Sz⟩=ℏpS(S+ 1) −Sz(Sz±1) |S, Sz+ 1⟩, Sz< S (2.28) As the name suggests, these operators increase or decrease the ˆzcomponent of the spin while transitioning between the eigenstates |S, Sz⟩. For a particle with spin S, the allowed values for Szare shown in Equation 2.29, and similarly for the other components when they are measured. For the purpose 34
of this work, the spin particles that we are going to consider are S=1 2particles, such as electrons Sakurai and Commins [1995], Parkinson and Farnell [2010]. Sz∈ {−S, −S+ 1, . . . , S −1, S}(2.29) Therefore, the possible values that they can have for the ˆzcomponent are Sz∈ −1 2,1 2. Moreover, the spin operators for these particles are directly related to the Pauli operators (X,Yand Z). From now on, we will consider ℏ= 1. The dimensionless spin operators can be written as follows Sakurai and Commins [1995]. Sx=1 2X Sy=1 2Y Sz=1 2Z (2.30) A final aspect to consider regarding spin angular momentum is the fact that it is possible to have systems with many spin particles. Therefore, describing these systems in terms of their spin degree of freedom is essential. The common approach to finding a basis that describes the total angular momentum ST= S1+ S2+. . . of the system is to use the angular momentum addition theorem Sakurai and Commins [1995]. Using this theorem, it is straightforward to find the basis of this system of many particles. The eigenstates are calculated using the Clebsch-Gordan coefficients , which are commonly found in a table ( Appendix C.I of Messiah [2014]). To better understand this addition, let’s consider a system consisting of two spin-1 2particles. The basis states of the composite system resulting from the addition of the two individual particles, denoted as S1 2⊗S1 2 , can be described in terms of their total spin ST∈ {1,0}. This implies that the system forms two distinct subspaces, one with a total spin of S= 1 and the other with S= 0. The possible ˆzspin projections for each subspace follow the relationship shown in Equation 2.29. By utilizing the ClebschGordan coefficients, it becomes possible to determine how these eigenstates |ST, Sz⟩are represented in terms of tensor product of the individual particle states |S, Sz⟩. |1,−1⟩=1 2,−1 2⊗1 2,−1 2=|↓↓⟩ |1,0⟩=1 √21 2,1 2⊗1 2,−1 2+1 2,−1 2⊗1 2,1 2 =1 √2(|↑↓⟩+|↓↑⟩) |1,1⟩=1 2,1 2⊗1 2,1 2=|↑↑⟩ |0,0⟩=1 √21 2,1 2⊗1 2,−1 2−1 2,−1 2⊗1 2,1 2 =1 √2(|↑↓⟩−|↓↑⟩) (2.31) 35
2.4.3 Heisenberg Model The previous subsections have introduced the concept of spin and its description within quantum mechanics. This subsection and the following ones are dedicated to introducing various spin models that are relevant to this dissertation. As previously mentioned, a spin system comprises particles with spin, where their interactions with each other and the environment are primarily described using the spin degree of freedom. These interactions are encoded in spin model Hamiltonians, utilizing the quantum operators discussed earlier. First, let’s introduce the Heisenberg model. While this model was previously defined using the example of two spin-1 2particles in the first chapter, it can actually be generalized to encompass varying numbers of spin particles, diverse geometries, and different types of spin particles. Fundamentally, the Heisenberg model elucidates interactions between spin particles. Its general representation is depicted in the Hamiltonian equation below Pires [2021]. HHeis =1 2X ij Jij Si· Sj(2.32) In this equation, the interaction is represented by the dot product of the spin operators of particles i and j, where iand jencompass all particles within the system. The factor of 1 2is included to avoid double counting, and the exchange constant Jij =Jji quantifies the strength of the interaction between particles iand j. To simplify the Hamiltonian, it is often assumed that only first-neighbour ⟨i, j⟩interactions exist, all with the same strength Jij =−J. This simplification, depicted in Equation 2.33, is common Pires [2021]. Furthermore, for the purpose of this work, the considered geometry will be limited to 1Dchains with periodic boundary conditions, as illustrated in Figure 14. HHeis =−JX ⟨i,j⟩ Si· Sj(2.33) If J > 0, the model is ferromagnetic, favoring a ground state where all spins align in the same direction. Conversely, if J < 0, the interaction is antiferromagnetic, causing spins to tend to align in opposite directions Parkinson and Farnell [2010], Pires [2021]. 2.4.4 Spin Waves To introduce the concept of spin waves, let’s consider a Ferromagnetic Heisenberg model for Nparticles in a 1Ddimensional chain. The ground state of this system is straightforward to find, as it’s the state 36
Figure 14: The figure illustrates a schematic representation of a 1D Heisenberg chain with periodic boundary conditions. Here, arepresents the lattice constant, such that each spin particle is positioned at ri= (i−1)a, where i∈[1, N], along the 1D chain. The value Jsignifies the exchange constant, determining the strength of interactions. It’s important to note that the spin orientations shown are arbitrary, and the diagram is solely intended for didactic purposes. in which all the spins point in the same direction. In this case, for simplicity, let’s choose the positive ˆz orientation Parkinson and Farnell [2010], Pires [2021]. We’ll refer to this state as |ψ0⟩, as depicted below. |ψ0⟩=|↑↑ . . . ↑⟩ (2.34) The excited states of this simplified Hamiltonian would correspond to states where one or more spins are flipped to the |↓⟩state. We will label these new states as |ψi⟩=S− i|ψ0⟩when the i-th spin is flipped Parkinson and Farnell [2010]. |ψi⟩=|. . . ↑↓i↑. . .⟩(2.35) However, these states are not eigenstates of the Hamiltonian because there is no particular reason for a single i-th spin to be flipped while keeping the rest unchanged. Additionally, all the states that have a single spin flipped have the same energy. Therefore, the actual excited eigenstate, which is an eigenstate of the Hamiltonian, would be a uniform superposition of all these states, as shown below Parkinson and Farnell [2010]. |q⟩=1 √N N X n=1 eiqrnS− n|↑↑ . . . ↑⟩ (2.36) This state |q⟩represents a magnon, which is a quantized version of the propagation of the spin perturbation, also known as a spin-wave. The term spin-wave refers to a plane wave eiqrnthat traverses 37
quantum simulation of chiral spin states involves two main research directions: the synthesis of chiral logic gates to prepare and simulate the dynamics of chiral spin states Ma et al. [2017], Wang et al. [2019], and the development of measurement schemes to identify chiral spin states Mazurenko et al. [2023], Sotnikov et al. [2022]. Our work contributes to the latter aspect. While Sotnikov et al. [2022] proposed a hardware-efficient approach to identify phase transitions associated with changes in scalar spin chirality by measuring in two or more random single-qubit bases and coarse-graining the outcomes to probe inter-scale dissimilarities Sotnikov et al. [2021], here we follow a physically-motivated strategy in which we compute the expectation value of the scalar spin chirality. 2.7 Methodology This chapter serves as a guide to the methodologies employed throughout this dissertation to address its primary objectives. The methods chosen for this research encompass analytical calculations and computer-assisted simulations. 2.7.1 Python Python, a high-level programming language created by Guido van Rossum, plays a central role in this research. Python is renowned for its versatility and wide application in scientific and numerical programming Van Rossum et al. [1995]. In this work, Python, specifically version 3.10, along with its open-source libraries, serves as the primary toolset for executing all calculations and simulations. In the following sections, we will introduce the most relevant libraries used in this dissertation for analytical calculations and simulations. Additionally, we employed several Python libraries to streamline calculation and simulation programming and to visualize the results presented in this work. These libraries include NumPy, which specializes in multidimensional array operations for scientific computing Harris et al. [2020]; Matplotlib, utilized for visualizing results Hunter [2007]; and SciPy, a library encompassing a wide range of mathematical algorithms and tools Virtanen et al. [2020]. 2.7.2 Analytical Calculations This dissertation utilizes analytical calculations to assess the feasibility and validity of the developed quantum circuits. These calculations adhere to the quantum mechanics and quantum computing principles outlined in the previous chapter. They constitute a fundamental aspect of this research, supporting the theoretical investigation of the proposed chirality indirect measurement methods and validating the results 44
obtained in the exploration of diverse spin models. Furthermore, while most analytical calculations were performed manually, Python tools were employed to verify preliminary results and facilitate other calculations. For instance, the SymPy library was used for this purpose. SymPy is an open-source Python library designed for symbolic mathematics, a form of computation known as a computer algebra system (CAS) Meurer et al. [2017]. This library streamlines tasks such as diagonalizing quantum operators to find their exact eigenvalues and eigenstates, as well as determining various many-body quantum operators through their explicit expressions. 2.7.3 Computer-Assisted Simulations The computational simulations used in this research rely on two main libraries: Qiskit Qiskit contributors [2023] and QuSpin Weinberg and Bukov [2017,2019]. Qiskit aids in quantum computing simulations to test quantum circuits, while QuSpin is employed for classical simulations of the quantum systems used in this work. Qiskit Qiskit is an open-source software library powered by Python Qiskit contributors [2023]. It facilitates the construction of quantum programs at the level of quantum circuits. Developed by IBM (International Business Machines Corporation), this software facilitates the creation and simulation of quantum programs using provided quantum computer simulators running on classical computers. These simulators can operate either noiselessly or with noise, mimicking real quantum hardware. Furthermore, Qiskit, in conjunction with the IBM cloud, enables the execution of quantum circuits on actual IBM-provided quantum hardware Qiskit contributors [2023]. In this dissertation, we exclusively use the noiseless simulator, as the primary aim is to identify a well-suited Noisy Intermediate-Scale Quantum (NISQ) quantum circuit for chirality probing under ideal noiseless conditions. Moreover, all quantum circuits proposed for chirality measurement are tested using Qiskit. In Chapter 5, where chirality for various models will be tested, Qiskit simulations estimate chirality by emulating a noiseless quantum computer. QuSpin QuSpin is an open-source library employed for exact diagonalization and quantum dynamics of arbitrary boson, fermion, and spin many-body systems Weinberg and Bukov [2017,2019]. QuSpin enables the construction of diverse quantum systems and the determination of their properties through classical methods, including finding eigenstates and eigenvalues. It facilitates the modeling of spin systems and the explo45
ration of quantum observables, which can also be implemented Weinberg and Bukov [2017,2019]. In this dissertation, QuSpin serves as a tool to implement spin models and obtain their chirality. The results obtained using QuSpin are considered exact, against which estimations using Qiskit are compared. 46
Chapter 3 Chirality Eigenstate Preparation The objective of this dissertation is to propose quantum circuits for measuring the chirality of a given quantum state. To achieve this goal, it is necessary to start with states whose chirality is well-known, using them to test various measurement approaches. The straightforward strategy involves utilizing chirality eigenstates. Hence, this chapter will identify these eigenstates and present a quantum circuit capable of preparing them within a quantum computer. The chirality eigenstates can be easily obtained by diagonalizing the chirality operator, as referenced in Wen et al. [1989]. Since the chirality operator is Hermitian, its eigenvalues are real. Furthermore, the rotational symmetry of this operator results in the chirality operator sharing the same eigenstates as the total spin operator S2=ˆ S1+ˆ S2+ˆ S32and the Szoperator. This relationship can also be understood through the properties of the ˆχ2operator (refer to Equation 3.1). ˆχ2=−4ˆ S1+ˆ S2+ˆ S32 + 15 (3.1) According to the angular momentum addition theorem, for a system of three spin-1 2particles, a basis can be formed in which the total angular momentum is divided into a spin-3 2multiplet denoted as S=3 2, Sz(see Equation 3.2), and two spin-1 2multiplets denoted as S=1 2, Szλ(see Equation 3.3), where λcan take values of ±1Wen et al. [1989]. Equation 3.1 demonstrates that the multiplet with S=3 2has zero chirality, while the multiplet with S=1 2should have non-zero chirality values, specifically ±2√3. To construct the chirality operator’s eigenstates, the states with S=3 2are first generated using the angular momentum addition theorem. Then, the corresponding spin-1 2states with S=1 2are constructed to be orthogonal to the S=3 2, Sz states by adding certain phases ωand ω2, which are the cube roots of unity, satisfying the condition 1+ω+ω2= 0. In this context, ω=ei2π 3. Hence, the eigenstates of the chirality operator are presented below. 47
S=3 2, Sz=3 2=|↑↑↑⟩ S=3 2, Sz=1 2=1 √3(|↓↑↑⟩+|↑↓↑⟩+|↑↑↓⟩) (3.2) S=1 2, Sz=1 2+1 =1 √3(|↓↑↑⟩+ω|↑↓↑⟩+ω2|↑↑↓⟩) S=1 2, Sz=1 2−1=1 √3(|↓↑↑⟩+ω2|↑↓↑⟩+ω|↑↑↓⟩) (3.3) The eigenstates with Sz=−1 2and Sz=−3 2can be easily found by applying the spin flip operator |↑⟩ → |↓⟩ to all three spins. Therefore, for the purpose of our discussion, we can focus on these eigenstates based on their chirality eigenvalue χ. ˆχ|χ=λ, Sz⟩= 2√3λ|χ=λ, Sz⟩(3.4) The notation λ∈ {0,±1}represents the values of zero, positive and negative chirality, respectively. Therefore, the following section will cover the implementation of these states in a quantum computer. 3.1 Building Chirality Eigenstates In the context of our quantum simulation on quantum computers, it is necessary to map the previously discussed eigenstates to the quantum computational basis. This mapping can be easily accomplished by associating |↑⟩with |0⟩and |↓⟩with |1⟩. Furthermore, it’s important to note that in our earlier discussion, we adopted a ket notation order where the first particle is placed on the top left and the last particle on the top right. This arrangement is referred to as little-endian notation, indicating that the lowest index comes first. However, in contrast, Qiskit, which we will be using along with Python, employs the big-endian notation. In this notation, the last particle is positioned on the top left, and the first particle on the top right. For consistency, we will adhere to the ’big-endian’ scheme in our notation. Hence, in this section, the chirality eigenstates in the quantum computational notation are divided based on the difficulty of preparing them on a quantum computer. The trivial chirality eigenstates with χ= 0 are |000⟩and |111⟩. The non-trivial eigenstates can be found below. χ=λ, Sz=1 2=1 √3|001⟩+ωλ|010⟩+ω2 λ|100⟩(3.5) Here, λ∈ {0,±1}denotes the zero, positive, and negative chirality non-trivial eigenstates with Sz=1 2, and ωλ=ei2π 3λ. Similarly, the remaining eigenstates with Sz=−1 2are obtained by flipping the states |0⟩ → |1⟩and |1⟩ → |0⟩. 48
|0⟩X• |0⟩G(1 3)• • |0⟩G(1 2)• Figure 20: This figure depicts the quantum circuit |W⟩3⟨0|⊗3that prepares the |W⟩3state. These non-trivial eigenstates are entangled states that belong to the Wclass, with both zero chirality or non-zero chirality. This is why the preparation of the |W⟩states was discussed in the previous chapter 2.3.6. Consequently, the entangled state with zero chirality and Sz=1 2can be implemented straightforwardly by following the implementation of |W⟩states for N= 3 qubits (see Figure 20). |W⟩3=1 √3(|001⟩+|010⟩+|100⟩)(3.6) The state with Sz=−1 2has the opposite spin orientation, and the quantum operations required to transform the simple |W⟩3state into this state involve flipping each individual qubit. The gate that performs this operation is the Pauli X. Hence, at the end of the encoding of the |W⟩3state, it is only necessary to apply the X⊗3operator. These relations are summarized in Equation 3.7. χ= 0, Sz=1 2=|W⟩3 χ= 0, Sz=−1 2=X⊗3|W⟩3 (3.7) The eigenstates with non-zero chirality can be prepared by adding certain phases to the |W⟩3states and flipping the qubits in the case of Sz=−1 2. The operator P(ϕ)applies the desired phase ϕto the qubits. For instance, consider the states |χ=−1, Sz⟩. The construction of these states is depicted by Equation 3.8. χ=−1, Sz=1 2=P2π 3⊗P4π 3⊗1|W⟩3 χ=−1, Sz=−1 2=X⊗3χ=−1, Sz=1 2(3.8) The states with positive chirality can be constructed using the same approach, with the only difference being the change in phases from ωto ω2, as shown in Equation 3.9. The circuit that summarizes the preparation of these states can be seen in Figure 21. Note that this circuit uses the computational basis notation, specifically the Big-endian scheme, where the higher index is positioned on the left side of the ket notation. In terms of the quantum circuit representation, the higher index corresponds to the bottom line of the circuit. 49
χ= 1, Sz=1 2=P4π 3⊗P2π 3⊗1|W⟩3 χ= 1, Sz=−1 2=X⊗3χ= 1, Sz=1 2(3.9) |0⟩ |W⟩3⟨0|⊗3(X⊗3)a |0⟩P(θ1) |0⟩P(θ2) χ P(θ1)P(θ2) 01 1 2√3ω ω2 −2√3ω2ω Figure 21: The left Figure represents the quantum algorithm in the form of a circuit, with the corresponding values for the phases provided in the table on the right. The values of these phases depend on the desired state to be prepared. For states with zero chirality, no phase is applied. For states with positive chirality, the phases used are ωand ω2, while for states with negative chirality, the phases are ω2and ω, as indicated in the table and explained in the text. The value of ain the last operator changes between states with Sz=1 2when a= 0, and Sz=−1 2when a= 1. 50
Chapter 4 Chirality Measurement In the field of quantum simulation, considerable efforts have been made to develop algorithms that can prepare specific quantum states. However, there is a noticeable lack of quantum algorithms focused on extracting information from a system through measurements. The most commonly employed method for measuring the expectation values of operators is quantum tomography. Unfortunately, this approach has the drawback of requiring measurements on all the qubits that encode the states, considering all possible combinations of Pauli’s strings that form the operator. Therefore, the objective of this dissertation project is to propose alternative approaches that minimize the number of measurements needed. This chapter will present the developed methods, offering promising solutions to this challenge. These approaches are based on different existing methods such as the Linear Combination of Unitaries (LCU), Quantum Phase Estimation (QPE), Hadamard Test, and combinations thereof. In the subsequent sections, these methods will be discussed in more detail. The chirality eigenstates, which were previously discussed, assist in testing the proposed approaches since their chirality is already known. This chapter exclusively employs chirality eigenstates for evaluating and comparing the methods via simulations implemented using Qiskit. Towards the end of the chapter, a thorough comparison will be carried out to discern the unique strengths exhibited by each algorithm. This comparative analysis will enable the identification of the strengths and weaknesses of each algorithm, facilitating the selection of the most suitable approach based on specific applications. 4.1 Characterizing Chirality through Permutation Operations The chirality operator is defined as the box product of spin vector operators for three particles. However, as outlined in the theoretical background chapter, the concept of chirality is intimately linked to order parameters associated with the violation of parity (P) and time-reversal (T) symmetry. This connection has inspired an alternative definition of the chirality operator, proposed by Subrahmanyam [1994], in terms of 51
spin permutation operators. To define the chirality operator as proposed by Subrahmanyam [1994], we start by introducing the operator Θi. This operator is defined for a general number of Nspin-1/2 particles and is obtained by taking the product of N−1 2permutations Pij. These permutation operators swap the states between sites iand j. In the context of spin operators, the permutation operator can be described as Pij = 2 Si· Sj+1 2. In quantum computing, the equivalent operation to the permutation operator is a SWAP operation that exchanges qubits iand j. Θi=⌊N−1 2⌋ Y j=1 P(i+j)(N+i−j) = 2⌊N−1 2⌋⌊N−1 2⌋ Y j=1 Si+j· SN+i−j+1 2 (4.1) Hence, the definition proposed by Subrahmanyam [1994] can be observed in Equation 4.2 for the case of N= 3. In his paper, this definition is generalized for an arbitrary number of spin-1/2 particles. The issue with this definition is that it is specific to a particular type of system governed by the Heisenberg Hamiltonian, unlike the conventional definition based on the box product, which is problem-independent. ˆχ= 2i[Θi,Θi+1](4.2) Nevertheless, it is possible to observe that although this definition cannot be assumed as the ”correct” one for generalized chirality in the case of an arbitrary number of particles, the scenario presented here for N= 3 spin-1/2 particles is indeed equivalent to the scalar spin chirality defined by the box product. This correspondence can be demonstrated by employing the commutation relations of Pauli operators and utilizing properties of the cross product. Therefore, to explicitly illustrate this relation, let us consider Θi= Θ1=P23 and Θ2=P31. By doing so, it becomes apparent that the traditional definition is recovered, as depicted below. ˆχ= 2i[P23, P31] =i 2[σ2·σ3, σ3·σ1] =σ1·σ2×σ3 (4.3) This representation will prove useful in the upcoming sections since, in the realm of quantum operations, the chirality operator is defined as the commutation between two SWAP operations. This definition offers a simpler approach compared to the complete description involving the expansion of the box product, 52
which entails considering six Pauli strings. 4.2 Employing the Linear Combination of Unitaries (LCU) method for Chirality Operator Implementation The chirality operator, as described previously, is a Hermitian operator defined as the sum of six Pauli strings or, alternatively, in terms of permutation operators, as the difference of two permutation products. However, quantum computers, by their inherent nature, can only implement unitary operations, which is not the case for the chirality operator. Therefore, directly implementing the chirality operator in a quantum computer is not possible; alternative techniques must be employed to enable its implementation. To address this challenge, a method known as the Linear Combination of Unitaries (LCU) (see Subsection 2.3.2), proposed by Childs and Wiebe [2012], can be employed. 4.2.1 Chirality Operator using LCU The chirality operator, being Hermitian, cannot be implemented directly on a quantum computer. However, by expressing it as a combination of unitary operators, either through Pauli strings or permutation operators, the LCU method offers a viable solution. The permutation description of the chirality operator stands out for its simplicity compared to the Pauli string representation, which involves six terms. The permutation description requires only the difference of two terms, making it more resource-efficient. By utilizing the permutation description, the chirality operator can be expressed as the difference of two terms, as shown in Equation 4.4 with R=P23P31. ˆχ= 2iR−R†(4.4) Given that the chirality operator can be described using two unitary operators, we can employ the previously discussed example illustrated in Figure 3. By setting U=Rand V=R†. Since the desired operation is a difference, the phase to be applied in this case is φ=π. The ancillary register should be initialized by applying a rotation Op=Ry(θ), where cos2θ=sin2θ=1 2, which corresponds to the Hadamard operator. Therefore, Op=H. Figure 22 summarizes the LCU method for applying the difference of Rand R†. The probability of success can be obtained using the result derived in Equation 2.8, and it can be expressed in terms of the chirality operator as follows. 53
Therefore, by setting the time t=2π 3, it becomes apparent that the time evolution operator for chirality simplifies to the cyclical permutation operator R. Uχ2π 3=R(4.16) The relationship between Uχ(t)and the specific time t=2π 3allows for a more straightforward implementation of this operator on quantum computers. Since the operator Ris simpler to implement compared to the Trotterization of Uχ(t), the upcoming algorithms will utilize this implementation. Indeed, it is important to emphasize that the main objective is to measure the chirality of a given state, rather than implementing the operator for arbitrary times. In this context, the simplified implementation using the operator Rbecomes highly valuable and practical. 4.5 Using Quantum Phase Estimation (QPE) for Chirality Determination As discussed in the previous section, a straightforward method was presented for implementing the chirality time evolution operator Uχ(t)by wisely selecting the time interval as t=2π 3. Consequently, the eigenvalues of this operator would be given by e−i2π 3λ, where the corresponding chirality eigenstates are labeled as |χ=λ⟩with λ= 0,±1. To estimate the phase of this unitary operation, a commonly used approach is the Quantum Phase Estimation (QPE) algorithm (see Subsection 2.3.5)Abrams and Lloyd [1999]. This algorithm utilizes an ancillary register consisting of nqubits to estimate the phase ϕof a given eigenvalue ei2πϕ. The best estimation of 2nϕ, denoted as φ∈Z+, is encoded in the state |φ⟩obtained after measuring the ancillary register. The probability of measuring the best estimation φis lower bounded by P(φ)≥4 π2Cleve et al. [1998]. Since the eigenvalues of the chirality operator are also encoded in exponential form e−i2π 3λ, the QPE approach can be employed to estimate these values. By comparing the Uχ(2π 3)eigenvalues e−i2π 3λwith the exponential form ei2πϕ estimated in the QPE algorithm, we can determine the value to be estimated using this approach, which is ϕ=−λ 3∈0,±1 3. It is important to note that when λ=−1, we have ϕ=1 3. As shown in the previous example in the QPE Subsection 2.3.5, this value cannot be exactly measured for any arbitrary number of nqubits in the ancillary register because it does not have a finite binary fraction representation. Increasing the number of qubits in the ancillary register only increases the resolution of the approximation φ≈2n·1 3. On the other hand, when λ= 1, we have ϕ=−1 3, which is a negative value. However, the QPE method only 60
estimates positive integer values of φ≈2nϕ. Therefore, the value that would be obtained through QPE by utilizing the periodicity of the complex exponential is ei(2π−2π 3λ)=e−i2π 3λ. Consequently, when λ= 1, the value to be estimated is ϕ= 1 −1 3=2 3. Once again, this value cannot be exactly represented by a binary integer φ. As observed, there is no specific number of qubits, denoted as n, for which the values of λcan be exactly obtained using the Quantum Phase Estimation (QPE) method. However, the objective of utilizing the QPE algorithm is to have a reliable method to distinguish between the three values of λ, namely λ∈ {0,±1}. To achieve this objective, it is crucial to determine the minimum number of qubits needed in the ancillary register to effectively discriminate between the eigenvalues. To effectively distinguish between the three possible outcomes of the QPE algorithm, namely ϕ∈ {0,1 3,2 3}, it is crucial that the separation between the outcomes must be significant enough to allow for accurate discrimination. Therefore, the number of qubits nrequired can be determined using the following expression: log2 1 2n=log21 3 n=⌈log23⌉= 2 (4.17) To distinguish between the three possible outcomes of φ, a total of n= 2 qubits is required. With this configuration, we can proceed to calculate the probability of accurately estimating the value of ϕ=1 3 when λ=−1using the Equation 2.20. First, let’s determine the nearest integer to the correct solution and calculate its deviation δusing the following equation: 2nϕ= 221 3 Binary −−→ 220.0101 ··· = 01 + 0.0101 . . . =φ+ 22δ (4.18) Therefore, the nearest integer to the correct solution is φ= 1, which in binary representation is 01. The deviation δcan be represented as δ=1 12 , which in binary fraction is 0.0001010 . . . . Consequently, we can calculate the probability of obtaining the correct solution as follows: P(φ= 1) = 1 4 sin π 3 sin π 12 2 ≈0.70 (4.19) The probability of correctly measuring the state |01⟩, which corresponds to the value that best approximates the actual value of ϕ=1 3, is approximately P(φ= 1) ≈0.70. Similarly, for measuring ϕ=2 3, the nearest integer is the value of φ= 3, represented in binary as 11, with a deviation of δ=1 12 such that 61
|00i → 0|01i → 1 4|10i → 1 2|11i → 3 4 Basis state 0.0 0.2 0.4 0.6 0.8 1.0 Probability 0.06 0.7 0.19 0.05 Input state: |χ=−1i (a) |00i → 0|01i → 1 4|10i → 1 2|11i → 3 4 Basis state 0.0 0.2 0.4 0.6 0.8 1.0 Probability 1.0 0.0 0.0 0.0 Input state: |χ= 0i (b) |00i → 0|01i → 1 4|10i → 1 2|11i → 3 4 Basis state 0.0 0.2 0.4 0.6 0.8 1.0 Probability 0.07 0.05 0.19 0.7 Input state: |χ= 1i (c) Figure 24: The figure displays the probabilities of measuring each basis state of n= 2 qubits using the QPE algorithm, obtained through a noiseless simulation using Qiskit. The simulation consisted of Nshoots = 2,500 runs to estimate the probabilities with an error of ε= 0.01, represented in the figure by error bars. Figure 24a shows the probability estimation using the chirality eigenstate |χ=−1⟩as the input, figure 24b corresponds to |χ= 0⟩, and figure 24c corresponds to |χ= 1⟩. The x-axis of each figure also represents the corresponding values of φ 22. The chirality eigenvalue was estimated by considering the state with the highest probability (see Table 5). 22ϕ=φ−22δ. The probability of correctly measuring this value is approximately P(φ= 3) ≈0.70. Conversely, since 2nϕ= 0 can be exactly represented with n= 2 qubits as φ= 0, the probability of correctly measuring this value is P(φ= 0) = 1. Therefore, the error in the estimation of ϕis bounded by δmax =1 2n+1 = 0.125. Consequently, the estimation of χ=λis subject to an error bounded by ϵ= 3δmax = 0.375. To validate these results through a noiseless Qiskit simulation, we conducted Nshoots = 2,500 runs to obtain an estimation error on the probability P(φ)of ε=1 2√Nshoots = 0.01. The input states used in the simulation were the chirality eigenstates discussed in section 3.1 and previously employed to test the other methods. The measured probabilities for each input state are depicted in Figure 24, and the estimated eigenvalues are presented in Table 5. The obtained results are consistent with the expected outcomes. The previous results showed the minimum resources required to distinguish between the three possible eigenstates χ∈0,±1. To obtain a more precise estimation using the Quantum Phase Estimation (QPE) algorithm, let’s add more qubits to the ancillary register. With n= 5 qubits, the maximum error in the estimated value of ϕis δmax =1 2n+1 ≈0.02, and the error in the chirality estimation is ϵ= 3δmax ≈0.05. The results are summarized in Table 6. 62
Table 5: Estimated eigenvalues χ=λusing QPE. The estimation error is limited by the precision determined by the number of qubits. With n= 2 qubits, the binary fraction error is bounded by δmax =1 2n+1 = 0.125. Consequently, the chirality estimation error is ϵ= 3δmax = 0.375. The measured value of φ 22, providing the best approximation of the binary fraction representing ϕ=−λ 3, was obtained by selecting the result with the highest probability from simulations with different input states (see Figure 24). Input State Measured φ 22Estimated χ |χ=−1⟩0.2±0.1−0.8±0.4 |χ= 0⟩0.0±0.1 0 ±0.4 |χ= 1⟩0.8±0.1 0.8±0.4 Table 6: Estimated eigenvalues χ=λusing QPE with n= 5 qubits. The estimation error is limited by the precision determined by the number of qubits. With n= 5 qubits in the ancillary register, the maximum error in the estimated value of ϕis δmax =1 2n+1 ≈0.02. Consequently, the error in the chirality estimation is ϵ= 3δmax ≈0.05. The table summarizes the results obtained for the estimated eigenvalues using the enhanced QPE setup. Input State Measured φ 25Estimated χ |χ=−1⟩0.34 ±0.02 −1.03 ±0.05 |χ= 0⟩0.00 ±0.02 0.00 ±0.05 |χ= 1⟩0.65 ±0.02 1.03 ±0.05 4.5.1 QPE with Qutrits Another alternative to address the issue of encoding the values ϕ=1 3(χ=−1) and ϕ=2 3(χ= 1) in binary fractions is to utilize a qutrit, which is a quantum system with 3-level quantum states, in the ancillary register. By employing a qutrit, it becomes possible to precisely encode the potential outcomes since they can be represented exactly in terms of ternary fractions, with 31ϕ∈ {0,1,2}. While our focus for this problem is on qutrits, it is important to note that this concept can be extended to the use of qudits, which are quantum systems with d-level quantum states, either in the ancillary register or the main register. This extension has been discussed by Tonchev and Vitanov [2016]. One notable change when transitioning from QPE with qubits to qutrits is the preparation of the ancillary register. In the case of qubits, Hadamard gates are used for this purpose. However, for qutrits, the 63
equivalent operation is the application of a one-qutrit Quantum Fourier Transform (QF T3), as depicted in the equation below. Similarly, the final transformation is accomplished using the adjoint of the QFT3, denoted as QFT† 3. QFT3=1 √3 1 1 1 1ω ω2 1ω2ω QFT† 3=1 √3 1 1 1 1ω2ω 1ω ω2 , ω =ei2π 3(4.20) In the case of using qutrits, the controlled unitary operation Uχ2π 3, also known as oracles O(Uχ), involves a qutrit as the control. The state |k⟩, k ∈ {0,1,2}is used to control the application of the corresponding power of Uk χto the main register, which encodes the input state |χ⟩using qubits. O(Uχ)|k⟩⊗|χ⟩=e−i2kπ 3χ|k⟩⊗|χ⟩(4.21) The final state of the Quantum Phase Estimation (QPE) algorithm with the ancillary qutrit, when applied to a chirality eigenstate |χ⟩as input, can be described by the following equations. Please note that the label |⟩dindicates that the state is encoded using a qudit. |0⟩3⊗|χ⟩2 QF T3 −−−→ 1 √3(|0⟩3+|1⟩3+|2⟩3)⊗|χ⟩2 O(Uχ) −−−→ 1 √3|0⟩3+e−i2π 3χ|1⟩3+e−i4π 3χ|2⟩3⊗|χ⟩2 QF T † 3 −−−→ |0⟩3⊗|χ⟩2if χ= 0 |1⟩3⊗|χ⟩2if χ=−1 |2⟩3⊗|χ⟩2if χ= 1 (4.22) Indeed, the result can be easily observed since ωand ω2are the cube roots of unity and satisfy the relation 1 + ω+ω2= 0. This relationship leads to a circuit modification in the QPE that enables the discrimination of the chirality eigenvalue with a single measurement. In this modified version, the probability of successfully measuring the correct value is always P(φ)=1for all possible chirality eigenstates as input. This demonstrates the effectiveness of utilizing a qutrit-based encoding and the resulting circuit structure in achieving accurate and deterministic measurements of the chirality eigenvalue. On the other hand, if the input state is a general state |ψ⟩, this circuit allows for the estimation of the chirality expectation value ⟨ψ|ˆχ|ψ⟩by computing the probabilities of measuring each qutrit basis state. The final state of the ancillary register is directly related to the chirality distribution of the state |ψ⟩. Specifically, the probability of measuring |0⟩in the ancillary register is directly related to the probability 64
of measuring the state |ψ⟩with χ= 0, while the probabilities of measuring |1⟩and |2⟩correspond to the chirality eigenvalues χ=−1and χ= 1, respectively. The chirality expectation value can then be calculated as follows: ⟨ψ|ˆχ|ψ⟩=P(φ= 2) −P(φ= 1) (4.23) Moreover, if the input state |ψ⟩is a linear combination of states with different chirality, applying the QPE with the qutrit ancillary register will result in the collapse of the wavefunction into a single chirality eigenstate, with the specific chirality indicated by the measurement outcome. Therefore, this circuit provides a means to initialize states with well-defined chirality, similar to recently proposed algorithms for preparing valence-bond-solid states Murta et al. [2023] and the Gutzwiller wave function Murta and Fernández-Rossier [2021]. 4.6 Assessing Chirality using the Cycle Test This section will discuss a final method to probe chirality over a given input state |ψ⟩. First, according to the result provided by equation 4.14,Uχ(t)can be expressed in terms of a real and imaginary part. Additionally, let’s recall the definition of the time evolution operator Uχ(t=2π 3) = Rfor the specific time t=2π 3(see Section 4.4). This operator is straightforward to implement in a quantum computer since it is composed of just two qubit permutations R=P23P31. Therefore, the equation below shows the combination of both definitions. Uχt=2π 3= 1 −3 2ˆχ2−i√3 2ˆχ=R(4.24) The equation above shows that the operator Ris directly related to the chirality operator, as its real part is formed by ˆχ2and its imaginary part by ˆχ. Consequently, the expectation value of this operator for an arbitrary state can be expressed as equation 4.25. In this dissertation work, a circuit has already been presented to measure either the real or imaginary part of a given unitary operator, known as the Hadamard test. Therefore, it is possible to perform the Hadamard test on the operator Rto obtain the chirality of a given input state |ψ⟩. The circuit can be seen in Figure 25. ⟨ψ|R|ψ⟩= 1 −3 2⟨ψ|ˆχ2|ψ⟩−i√3 2⟨ψ|ˆχ|ψ⟩(4.25) The algorithm used in the Hadamard test, which involves a cyclical permutation operator, is referred to as the cycle test. It was recently proposed by Oszmaniec et al. [2021] as a method for measuring the 65
|0⟩H• • (S†)bH × |ψ⟩ × × × _ _ _ _ _ _ Figure 25: The figure depicts the Hadamard test for the unitary operator Uχ2π 3=R, indicated by the dashed box. As mentioned in the main text (see Subsection 2.3.4), when measured in the Ybasis (b= 1), the result corresponds to the imaginary part of the unitary operator R, which in this case gives ⟨ψ|ˆχ|ψ⟩(see Equation 4.25). On the other hand, the measurement in the Xbasis (b= 0) captures the real part of R, which is associated with ⟨ψ|ˆχ2|ψ⟩(see Equation 4.25). unitary invariant properties of a set of states, which characterize their relative geometrical orientation. The relationship between these properties will be further explored in the next chapter. By measuring the ancilla in the Y(X) basis with b= 1 (b= 0), it becomes possible to obtain ⟨ψ|ˆχ|ψ⟩(⟨ψ|ˆχ2|ψ⟩) as shown below. ⟨ψ|ˆχ|ψ⟩=−2 √3⟨Yancilla⟩ ⟨ψ|ˆχ2|ψ⟩=2 3(1 −⟨Xancilla⟩) (4.26) Similarly to the previous methods, this method was also tested using Qiskit, with the input being the chirality eigenstates we have been using. The noiseless simulation involved Nshots = 10,000 trials to obtain an estimation in the Pauli expectation value with an error bound of ε=1 √Nshots . Consequently, the bounded errors for ⟨ˆχ⟩and ⟨ˆχ2⟩are ϵχ=2 √3εand ϵχ2=2 3ε, respectively. The results of these simulations are presented in Table 7, and as expected, the correct outcomes were obtained. Table 7: The table shows the results of the noiseless simulation of the cycle test (see Figure 25), for the chirality eigenstates as input. The estimation was obtained with Nshots = 10,000 trials. The error bounds for the chirality and chirality squared are ϵχ=2 √3Nshots ≈0.01 and ϵχ2=2 3 1 √Nshots ≈0.007, respectively. |ψ⟩χ(estimated) χ2(estimated) |χ=−1⟩ −1.01 ±0.01 0.993 ±0.007 |χ= 0⟩ −0.01 ±0.01 0.000 ±0.007 |χ= 1⟩0.99 ±0.01 0.999 ±0.007 66
4.7 Selecting the Best Approach This chapter was dedicated to presenting all the methods proposed for estimating the chirality of a given input state |ψ⟩using alternatives to conventional quantum tomography. All of these methods were tested using the chirality eigenstates prepared in the previous chapter (see Chapter 3), and as expected, they all enabled the determination of chirality. However, not all of the proposed approaches are suitable for NISQ devices due to the required resources. In this section, we aim to compare the methods based on the number of gates, depth, ancillary qubits required, and the number of trials needed to achieve a certain precision εin the estimation. This comparison will help us choose the most suitable approach for the chirality test and also describe the advantages and limitations of each method. The comparison in terms of the number of gates for both one-qubit and two-qubit operations, specifically the count of CNOT gates for the two-qubit operations, as well as the circuit depth was conducted as follows: Firstly, the circuits built in Qiskit were decomposed into the basic gates that can be handled by the Python tool PyZx Kissinger and van de Wetering [2020]. Then, the quantum circuit instructions were transformed into the Open Quantum Assembly Language (QASM) Cross et al. [2017], which can be understood by the PyZx module. This procedure was performed to utilize the PyZx library to optimize the circuits. By using the basic_optimization function provided by the PyZx module, the circuits were optimized. This ensures that the comparison is made in a fair manner. The statistics of each circuit were then obtained for this comparison (see Table 8). Table 8: The table presented below provides an overview of the resources needed for performing the chirality estimation. It compares the number of ancillary qubits, circuit depth, and the total number of gates. Specifically, the count of CNOT gates is explicitly displayed. Methods Number of Ancillas Depth Number of Gates CNOT count LCU 1 50 66 27 HT + LCU 2 123 142 66 QPE (n= 2)2 82 116 47 QPE (n= 5)5 793 1115 460 Cycle Test 1 26 36 14 The application of the chirality operator using the LCU method (see Section 4.2), which we will simply refer to as LCU for simplicity, is intended to implement non-unitary operators using a probabilistic approach. However, it has been observed that by repeating the process, the squared chirality operator 67
can be estimated. It should be noted that while the goal of this chapter is to develop a circuit that estimates chirality, this approach does not achieve that objective. As a result, this method is excluded from the comparison. Nonetheless, statistics using PyZx were computed for the LCU method. Although the LCU method cannot be used to measure chirality directly, it does provide a means to apply the chirality operator to a given state, as utilized in the combined approach of the Hadamard test and LCU (HT+LCU) (see Section 4.3). The other methods discussed in this chapter enable chirality estimation, although not all of them can estimate the chirality of an arbitrary state. For example, the Quantum Phase Estimation (QPE) method (see Section 4.5) only provides an estimation with a certain probability of success for a given chirality input state, and the precision of the estimation depends on the number of ancillary qubits. The main advantage of this method lies in using a qutrit as an ancilla (see Subsection 4.5.1), which transforms the circuit into a qutrit Hadamard test functioning as a QPE. As demonstrated earlier, this approach allows for either the deterministic single-shot determination of chirality when the input state is a chirality eigenstate, the preparation of states with known chirality, or the estimation of chirality by estimating the probability of each outcome. However, since our goal is to design a circuit that utilizes only qubits to operate on digital NISQ devices, the qutrit QPE is not the optimal approach. Moreover, the other comparison mentioned at the beginning of this section is related to the precision of the estimation. It aims to determine which algorithm achieves a better precision with fewer resources. This comparison will consider the number of ancillary qubits and the number of trials required to obtain the chirality estimation with a bounded error of ε. The HT + LCU method requires a constant number of two ancillary qubits, and the number of shots, denoted by Nshoots, scales with O1 ε2in order to achieve the desired precision. On the other hand, the QPE methods only estimate the chirality for the chirality eigenstates with a certain probability, but not for general input states. The precision of the QPE method depends on the number of ancillary qubits, denoted by n, and scales with Olog21 ε. Lastly, the cycle test requires a constant number of ancillary qubits, and the number of runs needed scales with O1 ε2. The exact number of resources required to estimate the chirality with a desired precision of ε= 0.01 is summarized in Table 9. Indeed, the cycle test approach appears to be the best method for probing chirality, as it achieves the goal with fewer resources, including the number of ancillary qubits, circuit depth, and CNOT count. The reduction in the number of CNOT gates is particularly important as multi-qubit operations can introduce errors due to the physical interactions between qubits Möttönen et al. [2004]. The cycle test method achieves the desired goal with a circuit depth of 26 and a CNOT count of 14, achieved by decomposing 7 68
Table 9: The table below compares the number of ancillary qubits and the number of shots Nshots required to obtain the chirality estimation with a bounded error of ε= 0.01. For the QPE method, the number of shots is associated with a probability of obtaining the correct result in a single shot, and this probability is explicitly stated in the table. Methods Number of Ancillas Nshots HT + LCU 24 ε2= 160,000 QPE log21 ε−1≈4 1 with a probability of success of 70% Cycle Test 14 3ε2≈13,334 CNOT gates for each Fredkin gate, as described by Cruz and Murta [2023]. Furthermore, the cycle test method allows for the estimation of the chirality expectation value for a general input state with better precision and fewer trials (Nshots) and ancillas compared to other methods. Therefore, considering both the reduction in resources and the improved precision, it is evident that the cycle test algorithm is the most suitable approach for probing chirality. 69
(see Equation 5.12), where niis the unit vector as shown in Figure 26a. Here, all spins have the same azimuthal angle θi=θ, which is a reasonable simplification since the system is symmetric under rotation around the axis parallel to D. However, they have different polar angles φi, which characterize the spin spiral. Si=S(cos φisin θˆx+sin φisin θˆy+cos θˆz)(5.12) Hence, considering a one-dimensional system with Nspin particles and periodic boundary conditions, the classical energy Eof the system can be rewritten as follows: E=−JX ⟨i,j⟩ S2cos (φj−φi)sin2θ+cos2θ+DX ⟨i,j⟩ S2sin (φj−φi)sin2θ(5.13) Here, it is possible to write each polar angle in terms of the spiral wave vector qas φi=qri, where ri=a(i−1) is the position vector of each particle i∈[1, N]in the ring with a lattice constant a= 1. Therefore, since the interactions are considered only for first neighbors, the energy of the system depends on the wave number of the spiral. E(q) = NS2(Dsin q−Jcos q)sin2θ−NJS2cos2θ(5.14) Therefore, the first thing to find is the condition for the wave number qmin that minimizes the energy of the system, which will correspond to its ground state. This can be achieved by computing the partial derivative of Ewith respect to q, as shown below. ∂E ∂q =NS2(Dcos q+Jsin q)sin2θ= 0 (5.15) tan qmin =−D J(5.16) By replacing the value of Dobtained from the condition found above, we can see that the value of the azimuthal angle θthat minimizes the ground energy is given by θ=π 2. This means that in the ground state, the spins in the classical spin spiral are aligned perpendicular to the Dvector, confirming the result described initially. E(qmin) = NS2Dsin qmin −NS2Jcos qmin (5.17) 76
5.3.2 Quantum Ground State and Broken Symmetry The previous subsection discussed the classical treatment of the problem involving Ferromagnetic and Dzyaloshinskii-Moriya (DM) interactions, whose ground state corresponds to a spin spiral residing in the plane perpendicular to the Dvector and wave number qmin. In this subsection, we will qualitatively explore the quantum version of this problem and confirm the expected results through simulations using QuSpin, a Python package that offers open-source tools for exact diagonalization and quantum dynamics simulations of various many-body systems, including spins Weinberg and Bukov [2017,2019]. Before delving into the quantum spin spiral, it is essential to note that the system exhibits invariance to global rotations of the spins. Consequently, all spin spirals with the same wave number qmin but different global phase ϕpossess the same energy. As a result, there exists an infinite set of spin spirals that are ground states of the system. Each classical spin vector Si,ϕ is depicted below. Si,ϕ =S[cos (qminri+ϕ) ˆx+sin (qminri+ϕ) ˆy](5.18) This degeneracy leads to a quantum ground state that is a superposition of all these spin spirals defined by the global phase ϕ. Analytically computing this result can be challenging and is not the primary focus of this work. Instead, we will compare the classical analysis with simulations using QuSpin. In these simulations, we will measure the spin vector Siobservables and correlators ˆ Sα 1ˆ Sα iwith α∈ {x, y, z} and i∈[2, N]. In the classical treatment, if we consider a spin spiral with, for example, ϕ= 0, the measurement of Si will be in the form of Si, 0. In the case of the correlators, only Sx 1Sx i=S2cos qminriis non-zero; the others are zero. On the other hand, if we assume that the ground state is a superposition of all the infinite possibilities of ϕ, the classical result for Siand the correlators would be an average over all values of ϕ, as shown below. Si=1 2πZ2π 0 dϕ Si,ϕ = 0 (5.19) Sα 1Sα i=1 2πZ2π 0 dϕSα 1,ϕSα i,ϕ =S2 2cos qminri, α ∈ {x, y}(5.20) The first step in the simulation is to choose the spin spiral wave number for the ground state, which should be qmin =2π Nfor the system with N= 10 particles and the ferromagnetic coupling constant J= 1. Following the classical ground state condition found in Equation 5.16, we have D=−Jtan qmin. This ground state should be a superposition of the infinite possible spin spirals with global phase ϕ, and 77
02468 ri −0.4 −0.2 0.0 0.2 0.4 Bx= 0 hˆ Sx ii hˆ Sy ii hˆ Sz ii h~ Sii(classical) (a) 02468 ri −0.4 −0.2 0.0 0.2 0.4 Bx=−1 hˆ Sx ii hˆ Sy ii hˆ Sz ii hˆ Sx ii(classical) hˆ Sy ii(classical) (b) Figure 28: The figures display the simulation results obtained using QuSpin for a ring with N= 10 particles, J= 1, and D=−Jtan 2π N. In these simulations, the operator Siwas measured for each particle i∈[1, N]in two systems: one without a local magnetic field (Bx= 0,28a) and the other with a local magnetic field (Bx=−1,28b) to achieve a broken-symmetry ground state. The dashed lines in the figures represent the analytical solutions obtained through classical analysis. the simulation results using QuSpin should be consistent with the classical analysis. However, to obtain a unique spin spiral, we need to add a local magnetic field Bx=−1located only on the first spin. This will lead to a broken-symmetry ground state that chooses the spin spiral with ϕ= 0. Again, the simulation results compared with the classical analysis will allow us to confirm that the assumptions were correct. By comparing the simulation results with the classical analysis for both cases (superposition of all ϕand broken-symmetry ground state with ϕ= 0), we can verify the expected behavior of the quantum ground state and its agreement with the classical treatment. The Figures 28 and 29 display the simulation results and their comparison with the classical analysis. Figures 28a and 29a demonstrate the concordance with the case without a local magnetic field (Bx= 0), in which the ground state is a superposition of all infinite spin spirals with a phase ϕ. On the other hand, figures 28b and 29b illustrate the case of broken symmetry achieved by adding a local magnetic field (Bx=−1), which selects the spin spiral with a global phase ϕ= 0. In both cases, the classical results obtained previously are shown by dashed lines. Indeed, these results and their agreement with the classical analysis strongly support the conclusion that the quantum ground state is a degenerate state. In this ground state, the system exists in a superposition of all spin spirals, obtained through global rotations of the spins. 78
2468 ri −0.4 −0.2 0.0 0.2 0.4 Bx= 0 hˆ Sx 1ˆ Sx ii hˆ Sy 1ˆ Sy ii hˆ Sz 1ˆ Sz ii hˆ Sα 1ˆ Sα ii(classical) (a) 2468 ri −0.4 −0.2 0.0 0.2 0.4 Bx=−1 hˆ Sx 1ˆ Sx ii hˆ Sy 1ˆ Sy ii hˆ Sz 1ˆ Sz ii hˆ Sx 1ˆ Sx ii(classical) (b) Figure 29: The figures display the simulation results obtained using QuSpin for a ring with N= 10 particles, J= 1, and D=−Jtan 2π N. In these simulations, the correlator Sα 1Sα i, α ∈ {x, y, z} was measured for each particle i∈[2, N]in two systems: one without a local magnetic field (Bx= 0, 29a) and the other with a local magnetic field (Bx=−1,29b) to achieve a broken-symmetry ground state. The dashed lines in the figures represent the analytical solutions obtained through classical analysis. In subfigure 29a, the dashed lines represent the analytical classical solutions for ⟨Sα 1Sα i⟩, where α∈ {x, y}. 5.3.3 Classical and Quantum Chirality The goal of studying the chirality of spin spirals is to explore the coexistence of classical and quantum chirality in the broken-symmetry case. When the Zeeman component of the Hamiltonian B= 0, the spins tend to align with this magnetic field, resulting in a non-coplanar spin structure in the classical description. This non-coplanarity leads to a state with non-zero classical chirality. Moreover, due to the nature of the interactions, the quantum version of the system also exhibits non-zero chirality. The quantum chirality arises from the entanglement of quantum states and is a unique feature of quantum systems. First, let’s calculate the analytical expression for the classical chirality of spin spirals’ trios ijk using the definition in terms of the box product of the unit vector ni, as shown below: χcla ijk =1 2√3ni·nj×nk(5.21) where ni,nj, and nkare the unit vectors corresponding to the spins at sites i,j, and kin the spin spiral, respectively. Thus, by using the expression for the classical spins in Equation 5.12 as ni= Si S, the classical chirality can then be written in terms of the azimuthal angle θ, the wave number qmin, and the difference between the sites ∆αβ ≡rα−rβas follows: 79
χcla ijk =cos θsin2θ[sin (qmin∆ji) + sin (qmin∆kj) + sin (qmin∆ik)] 2√3(5.22) Note that for the case in which the ground state is a superposition of all the spin spirals, the classical chirality is always zero. This result is obtained based on the conclusions from the previous subsection, as shown in Equation 5.19. Therefore, to obtain a broken-symmetry ground state with classical and quantum chirality, a local magnetic field Bx=−0.1is applied to the first spin. This state corresponds to a single spin spiral with a global phase ϕ= 0 in the absence of an external magnetic field (B= 0). In this study, a sweep was conducted over the values of Bin the range [0,1], and the classical and quantum chirality were computed for a system with N= 10 particles, a ferromagnetic coupling constant J= 1, and D=Jtan(qmin), where qmin =2π Nensures that a period of the spin spirals is completed upon covering all Nsites. To obtain the classical ground state of the system, numerical simulation was employed. The classical energy of the system E(θ, q)(see Equation 5.23) was minimized to find the optimal values for the wave number of the spin spiral qmin and the azimuthal angle θ, which depends on the external magnetic field B. Using the analytical expression found in Equation 5.22, the classical chirality was then computed. E(θ, q) = NS2Dsin qsin2θ−Jcos qsin2θ−Jcos2θ+BxSsin θ+NSB cos θ(5.23) The quantum ground state of the system was obtained using QuSpin through exact diagonalization. The quantum operator ˆχwas utilized to obtain the exact quantum chirality for each value of B, and the results were validated by applying the cycle test via noiseless simulation with Qiskit. Furthermore, the classical version of this quantum ground state was obtained by computing the expectation value of each spin component. The classical spin vector is given by si= (⟨Sx i⟩,⟨Sy i⟩,⟨Sz i⟩)for each particle i, and the box product using these vectors was computed to obtain the classical chirality χcla ijkof the quantum ground state |ψ0⟩. ⟨ψ0|χcla ijk|ψ0⟩=4 √3si·sj×sk(5.24) Hence, Figure 30 shows the results of the simulation where the chirality for the sites 1,4, and 9was computed for the cases in which Bx= 0 and Bx=−0.1. The broken-symmetry case exhibits non-zero classical chirality, as expected. Furthermore, the quantum chirality is larger than the classical chirality due to entanglement, consistent with previous discussions. In the Bx= 0 case, the system’s chirality is entirely quantum and appears to be discretized along the values of B, representing a fully quantum 80
phenomenon with no classical explanation. 0.0 0.2 0.4 0.6 0.8 1.0 B 0.00 0.05 0.10 0.15 0.20 0.25 0.30 hψ0|ˆ χ149 |ψ0i Exact Estimated (a) 0.0 0.2 0.4 0.6 0.8 1.0 B 0.00 0.05 0.10 0.15 0.20 0.25 hψ0|ˆ χ149 |ψ0i Exact Classical (QGS) Classical (CGS) Estimated (b) Figure 30: Simulation results for the chirality measurement of the Hamiltonian in Equation 5.11 for Bx= 0 (30a) and Bx=−0.1(30b) with varying B∈[0,1]. The Quantum Ground State (QGS) was obtained through exact diagonalization using QuSpin. The Classical Ground State (CGS) was obtained by minimizing the classical energy E(θ, q)(Equation 5.23). In the case where Bx= 0, the ground state is a superposition of an infinite set of spin spirals, resulting in a classical chirality of zero. However, in the case with Bx=−0.1, the system leads to a broken-symmetry ground state that allows the existence of classical chirality. Black curve: Exact quantum chirality ⟨ˆχ149⟩. Blue dots: Estimated quantum chirality using the cycle test circuit in Qiskit. Green: Classical chirality χcla 149computed from the box product of classical spin vectors si= (⟨Sx i⟩,⟨Sy i⟩,⟨Sz i⟩)over the QGS (see Equation 5.24). Red: Classical chirality χcla 149 calculated using Equation 5.22 with values of θand qmin from the CGS. The broken-symmetry system with non-zero classical chirality allows us to create a classical representation of the spin vectors and visually show the non-coplanarity achieved due to the Zeeman component. In Figure 31, we present schematic representations for three different values of B. 1. For B= 0, the classical configuration is coplanar in the ˆxˆyplane. 2. For the chiral case with B= 0.17, the non-coplanarity is evident. 3. Lastly, for B= 1, the spins are aligned with the external magnetic field, and therefore, the chirality vanishes. 81
Figure 31: Classical spin vectors for three different values of B. The left figure corresponds to B= 0, and its spin configuration is a coplanar spin spiral with zero chirality. In the middle figure, for B= 0.17, the maximum quantum chirality is observed, and the non-zero classical chirality is apparent due to the non-coplanar spin configuration. The right figure, for B= 1, shows zero chirality, with the spins aligned with the external magnetic field. The chirality values were presented in Figure 30. 82
Chapter 6 Conclusions and Future Work 6.1 Conclusions In summary, this dissertation proposes quantum circuits for probing the scalar spin chirality of trios of spin-1 2particles in a wave function defined on an arbitrary lattice. Initially, we studied how to implement chiral states with well-known chirality values, identifying them as chirality eigenstates (Chapter 3). These states were employed to validate various approaches within our research. The initial approach to address this challenge involved employing the Linear Combination of Unitaries (LCU) method to map the non-Unitary chirality operator ˆχonto a digital quantum computer (Section 4.2). Through this approach, we established a connection between the probabilistic application of ˆχand the ancilla measurement, yielding the expectation value ⟨ˆχ2⟩. Subsequently, by utilizing the Hadamard Test and the LCU technique, we successfully measured the desired chirality expectation value of an input system (Section 4.3). Another approach to address this problem involved exponentiating the normalized chirality operator to obtain a unitary operator Uχ(t) = e−iˆχt (Section 4.4). The implementation of this unitary operator facilitated the use of various techniques, including Quantum Phase Estimation and the Hadamard test, to investigate the chirality of a given system. However, it was demonstrated that implementing this unitary operator for an arbitrary time twould require a substantial number of quantum operations, typically achieved through Trotterization. Nevertheless, through a clever selection of time, specifically t=2π 3, this unitary operator simplifies to a simple cyclic permutation. This simplified version can be readily implemented in a quantum computer using SW AP gates. Hence, employing this approach in conjunction with other quantum circuits enables the measurement of the chirality operator. The quantum circuit utilizing the Hadamard test with Uχmaps the expectation value of either ˆχor ˆχ2to the average value of Yor Xfor the ancillary qubit. When t=2π 3, the algorithm becomes equivalent to the cycle test (Section 4.6)Oszmaniec et al. [2021]. 83
Furthermore, when the Hadamard test is implemented with a qutrit as the ancilla, it is equivalent to quantum phase estimation (QPE) and enables the single-shot readout of the eigenvalues of ˆχwhen the main register is prepared in a corresponding eigenstate (Section 4.5). Importantly, if the main register is initialized in a linear superposition of states with different scalar spin chirality, executing the QPE algorithm causes the wave function to collapse onto the components indicated by the ancilla readout. This approach can be employed as a strategy to prepare states with well-defined scalar spin chirality in selected triads of spins, similar to recently proposed algorithms for preparing valence-bond-solid states Murta et al. [2023] and the Gutzwiller wave function Murta and Fernández-Rossier [2021]. All of the proposed algorithms were assessed in terms of the quantum resources needed to achieve a specific level of precision (Section 4.7). This allowed for the selection of the most efficient approach, which was found to be the cycle test quantum circuit. Subsequently, the cycle test quantum circuit was employed for the remainder of the dissertation’s work and was tested in various scenarios. We have demonstrated the applicability of this method (Chapter 5) to classical magnetism using spin states (Section 5.1). Our results indicate that the cycle test method is effective for both separable and entangled states. Moreover, in the context of classical magnetism, we have established a direct correlation between chirality and the imaginary component of the Bargmann invariant for three particles. This correlation aligns with existing descriptions of chirality, which consider the disparity in Berry Phases Wen et al. [1989]. The application of this method to one-magnon states (Section 5.2) has provided valuable insights into the results proposed by the authors. Specifically, it was demonstrated that for certain chirality values, the system serves as a witness for genuine tripartite entanglement (GTE) Tsomokos et al. [2008]. Consequently, chirality measurements were conducted and compared with theoretical predictions made within this work. Notably, it was revealed that only the N= 3 magnon state exhibited a significantly high chirality, confirming the presence of GTE. However, as the values were compared with the corresponding concurrence fill Xie and Eberly [2021], it became apparent that for larger values of N, there still existed a certain degree of GTE, which correlated with the limited presence of chirality. In the thermodynamic limit as N→ ∞, this GTE phenomenon gradually diminishes, which stands in contrast to the well-documented macroscopic bipartite entanglement observed in magnons in the thermodynamic limit as reported in existing literature Morimae et al. [2005], Zou et al. [2020]. The final application of the cycle test involved the ground state of a simplified model representing a chiral ferromagnet (Section 5.3). This analysis included a classical treatment to examine the behavior of the ground state’s spin spiral. Utilizing QuSpin, the quantum ground state was obtained and subsequently compared 84
with the results of the earlier classical analysis. To explore this system further, the chirality of the ground state was measured under various external magnetic field strengths, denoted as B. The measurements were conducted using QuSpin to determine the theoretical chirality, while the cycle test, implemented with Qiskit for a noiseless simulation, was employed to estimate the chirality. This project served as an invaluable learning experience, encompassing the acquisition of fundamental concepts within the field of quantum magnetism and its associated formalism, which underpins this dissertation. Furthermore, it involved an educational journey in quantum simulation, encompassing a solid foundation in quantum simulation principles. The project also entailed a comprehensive study of various quantum algorithms, each with significant applications in different facets of quantum simulation, allowing for the simulation of a wide range of systems. Consequently, the knowledge gained throughout this endeavor not only proved pivotal in achieving the primary objectives of the dissertation but also serves as a valuable resource for addressing future challenges within the domain of quantum simulation. In conclusion, our findings represent a significant step forward in the realm of efficient digital quantum simulation of magnetic materials with chiral characteristics. Furthermore, our quantum methodologies establish a crucial bridge between the scalar spin chirality formula, a staple in classical magnetism descriptions often formulated in terms of product states, and its quantum counterpart, which extends its applicability to entangled states. 6.2 Future Work Future work will address the extension of our results to the determination of the scalar spin chirality of spin systems with S > 1 2. This will involve investigating how our quantum circuit methodologies can serve as motivation to adapt the description of quantum chirality to handle higher spin states. Additionally, we will focus on the generalization of the chirality concept to sets of more than three spins. One promising avenue for this generalization is building upon the relation we discovered between the chirality and Bargmann invariants, as well as the self-definition of chirality in terms of Berry Phases. Therefore, the generalization will likely emerge from a broader concept of chirality related to cyclical permutations. Exploring the behavior of chirality in larger spin systems will provide valuable insights into the complex interactions that arise in such systems. Overall, these future research directions aim to expand the applicability of our findings and contribute to a deeper understanding of the theory of chirality, motivated by the quantum computing approach developed in this dissertation. 85
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