Digital-Analog Quantum Machine Learning

Digital-Analog Quantum Machine Learning Lucas Lamata Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Sevilla, Spain Correspondence: Lucas Lamata ([email protected]) Received: 17 November 2024 |Revised: 30 December 2024 |Accepted: 20 January 2025 Funding: MCIN/AEI, Grant/Award Number: PID2022-136228NB-C21PID2022-136228NB-C22; ERDF A way of making Europe; Ministry for Digital Transformation and of Civil Service of the Spanish Government; European Union Keywords: digital-analog quantum protocols |machine learning |quantum technologies ABSTRACT Machine learning algorithms are extensively used in an increasing number of systems, applications, technologies, and products, both in industry and in society as a whole. They enable computing devices to learn from previous experience and therefore improve their performance in a certain context or environment. In this way, many useful possibilities have been made accessible. However, dealing with an increasing amount of data poses difficulties for classical devices. Quantum systems may offer a way forward, possibly enabling to scale up machine learning calculations in certain contexts. On the contrary, quantum systems themselves are also hard to scale up, due to decoherence and the fragility of quantum superpositions. In the short and mid term, it has been evidenced that a quantum paradigm that combines evolution under large analog blocks with discrete quantum gates, may be fruitful to achieve new knowledge of classical and quantum systems with no need of having a fault-tolerant quantum computer. In this perspective, we review some recent works that employ this digital-analog quantum paradigm to carry out efficient machine learning calculations with current quantum devices. 1|Quantum Machine Learning Machine learning is a knowledge field, a tool, a computing paradigm, a technology, which is significantly impacting society at large, by enabling a plethora of possibilities, as well as challenges, as with any new and highly powerful technology. However, its full deployment is hampered by the fact that an increasing amount of data is hard to process with current classical computers, and consumes a significant amount of energy. A possible way forward to scale up machine learning calculations would be via the use of quantum devices employing genuine quantum properties, such as superposition and entanglement. In this way, it may be possible to take advantage of the speedup of quantum computers to carry out machine learning calculations in a more efficient way, at least in some instances. Even though the field of quantum machine learning, which connects machine learning calculations to a quantum hardware seeking for these advantages, is not absent of controversy and speculation, as it is really hard to prove speedup with respect to the best classical machine learning algorithms, this is a field that is rising much expectation and hope that with NISQ devices one may beat classical computers for useful tasks. For a recent review of the field, see ref. [1]. 2|Digital-Analog Quantum Paradigm Full-fledged, scalable, digital quantum computers are hard to achieve, as they normally require millions of physical qubits to encode fault-tolerance and error-correction protocols. On the contrary, analog quantum simulators are typically scalable but Dedicated to the founders of Quantum Mechanics. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). Advanced Intelligent Discovery published by Wiley-VCH GmbH. Advanced Intelligent Discovery, 2025; 0:e2400023 1of4 https://doi.org/10.1002/aidi.202400023 Advanced Intelligent Discovery PERSPECTIVE limited in the amount of problems they are able to solve, or reproduce in a quantum simulation. A possible combination of both approaches has emerged in the past few years as a strategy for the short and mid term, which may enable NISQ quantum devices to achieve useful tasks, for instance, in learning new properties of quantum systems. The focus on the field of digital-analog quantum protocols has been mostly on quantum simulations in the past few years, with theory proposals and experiments in a variety of quantum platforms. However, most recently several works that aim at carrying out machine learning calculations via digitalanalog quantum protocols have appeared in the literature. For an early review on the field of digital-analog quantum simulations see ref. [2]. In Figure 1, I plot a scheme of a generic extreme digital-analog quantum protocol (EDAQP). This consists of a combination of the largest analog blocks possible (a set of global native interactions, namely, global unitary gates on all qubits based on the quantum platform native Hamiltonian, U 1 (t 1 ), U 2 (t 2 ),…, U k (t k ),…, and the smallest digital gates possible, that is, singlequbit gates, R i,j , acting on qubit jat step i, which can be done with the best fidelities in most quantum platforms. 3|Digital-Analog Quantum Machine Learning In this perspective, I give a non-exhaustive overview of the field of quantum machine learning, when digital-analog quantum protocols are employed for its deployment. I propose to use the name “Digital-analog quantum machine learning”(DAQML) for this research avenue. I will cite several papers in this area as well as briefly describe each of them. In ref. [3], the authors propose to use a digital-analog quantum protocol to implement a variational quantum eigensolver for estimating ground state energies of molecules. A subsequent work [4], analyzed the deployment of a quantum genetic algorithm via the Qadence digital-analog coding language for Rydberg atom arrays [5], to address also calculations of ground state energies of molecules, in a scalable way. This instance of digital-analog quantum protocol was of the EDAQP as shown in Figure 1. Ref. [6] studies the use of a digital-analog quantum approximate optimization algorithm for solving a series of tasks with NISQ devices. In ref. [7], the authors propose to use a digital-analog quantum paradigm for quantum kernel implementation in the context of quantum machine learning. In ref. [8], a quantum Fourier transform implementation is proposed via the use of digital-analog quantum protocols, showing possible gains in resources with respect to purely digital ones, and enabling in this way the use of this primitive in a variety of protocols, which may also include quantum machine learning ones, in particular. This is for instance analyzed in ref. [9] with respect to the HarrowHassidim–Lloyd (HHL) protocol. In ref. [10], the authors propose digital-analog quantum learning algorithms for Rydberg atom systems, which they claim can combine the usefulness of quantum machine learning protocols in the near term, with the efficient scalability that is recently being achieved with Rydberg atoms. Most previous works, which are first instances of the combination of quantum machine learning algorithms with the digitalanalog quantum paradigm, show evidence that this combination can be fruitful, and perhaps, by enabling a better scalability of quantum machine learning algorithms, may provide a quantum advantage with respect to classical machine learning calculations. 4|Implementations of the Digital-Analog Quantum Paradigm Several experiments in quantum platforms have reached a significantly large amount of qubits and/or Hilbert space dimension, by combining state-of-the-art developments in these platforms, and the use of digital-analog or similar quantum techniques. Some of the quantum platforms employed are trapped ions, superconducting circuits, and Rydberg atoms. This kind of approach has been considered in parallel by several groups, which employ large analog blocks combined with discrete gates, although some of these techniques are also referred to in the literature as “programmable analog quantum simulations”. For a recent review that describes analog, digital, and digitalanalog quantum paradigms, in the context of quantum simulation experiments, see ref. [11]. Even though these pioneering experiments have mainly been focused on quantum simulations, similar setups may address quantum machine learning tasks as the ones described above, with possible gains with respect to classical computers at least in some favorable situations [See, e.g., work done in startups PASQAL [12] and QUERA [13] in this sense]. 5|Outlook The previous account of recent literature combining digitalanalog quantum protocols with quantum machine learning algorithms, which I name “Digital-Analog Quantum Machine Learning”(DAQML), is just an appetizer for what may come in the future. With the increasing efficiency of quantum platforms such as trapped ions, superconducting circuits, and Rydberg atoms, in achieving devices with tens, or even, hundreds of quantum bits, the possibility to combine large analog blocks with digital steps, in the paradigm that we created more than FIGURE 1 |scheme of a generic EDAQP. This consists of a combination of the largest analog blocks possible, that is, a set of global native interactions, namely, global unitary gates on all qubits based on the quantum platform native Hamiltonian, U 1 (t 1 ), U 2 (t 2 ),…, U k (t k ),…, and the smallest digital gates possible, that is, single-qubit gates, which can be done with the best fidelities in most quantum platforms. 2of4 Advanced Intelligent Discovery, 2025 29439981, 0, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/aidi.202400023 by Readcube (Labtiva Inc.), Wiley Online Library on [14/02/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 10 years ago in the context of quantum simulations, may represent a significant step forward in the field of quantum technologies. Perhaps one of the first approaches that could possibly impact the industry for carrying out useful tasks might be the one I describe here, of combining our digital-analog quantum paradigm with the fledgling field of quantum machine learning. This could, in turn, impact subsequently society at large, as well as the scientific enterprise, via enhancing scientific discovery, as this very new journal, Advanced Intelligent Discovery, aims to achieve. In general terms, considering analog blocks in a quantum protocol allows one for making a much larger number of quantum systems to interact, and become entangled. For example, for long chains of 20–60 trapped ions, or 50 superconducting qubits, or tens to hundreds of Rydberg atoms, decomposing global entangling operations onto single and two-qubit gates is in general highly inefficient, due to digital errors introduced as well as errors produced in each two-qubit gate, such as turning pulses on and off, etc. Often, one may carry out quantum protocols for which it suffices to combine large analog blocks, with a native interaction coupling all qubits, with single-qubit gates, perhaps with a small number of additional two-qubit gates, and in this situation most often fidelities will be better than decomposing every multiqubit gate onto elementary gates. Of course, this will strongly depend on each particular case. Nevertheless, the digital-analog quantum paradigm includes the digital one and the analog one as particular cases, such that, given that the digital one is proven to be universal for quantum computing, digital-analog quantum protocols are trivially universal as the digital case is a particular case of the latter (e.g., for analog blocks with identity gates, having the native interaction with negligible coupling for arbitrary time). The main difficulty in deploying these kinds of protocols is elucidating the optimal multiqubit, single, and (perhaps) two-qubit gate decomposition, but in general this significantly enlarges the amount of choices available in the experimental toolbox for optimizing the quantum protocols. DAQML protocols will possibly perform better when one would need to act on many qubits collectively, producing global entanglement, and whenever single-qubit gates would suffice for convergence to the optimal solution, together with the analog global blocks (possibly by adding a few two-qubit gates, when needed). A promising scenario for this, as shown in ref. [4], is the quantum genetic algorithm realm, where universality is not so much needed at the level of elementary gates, but convergence is achieved via recombination and selection of the fittest, while some amount of diversity in terms of single-qubit gates and timedependent global analog blocks is introduced in each iteration in a rather robust and random way, where gate imperfections do not matter much. In this concrete scenario of quantum genetic algorithms, it may well be that a known problem in quantum machine learning protocols based on gradient descent convergence, namely, barren plateaus, may be avoided as the genetic protocol does not rely on gradients. Regarding potential future research directions in DAQML and, given that this kind of toolbox strongly depends on the developments of each quantum platform, it will be convenient to adapt the protocols to technological advances in each quantum implementation, namely, ions, superconducting circuits, Rydberg atoms, quantum photonics, etc. In general terms, it can be considered that large analog blocks and single-qubit gates may be robust in quantum machine learning given that in many protocols one is iterating until convergence, and, as far as one is approaching the optimal, specific imperfections, or timings, in the analog blocks, or the individual gates, do not matter so much. A similar consideration may be done for the digital-analog quantum paradigm for quantum simulations, given that real, large quantum systems typically have imperfections similar to those of the analog blocks of quantum platforms. Therefore, it is in general not detrimental to have imperfect analog blocks, as far as one can enrich them with individual gates, to produce, for example, a hard-to-reach Heisenberg model from an easily implementable Ising model. In summary, the digital-analog quantum paradigm is highly scalable and error-robust, both for the DAQML and the quantum simulation applications. We hope that the experimental quantum community finds useful ways to employ it for a new generation of highly complex, possibly useful, quantum experiments. Acknowledgments The author acknowledges the support from grants PID2022-136228NBC21 and PID2022-136228NB-C22 funded by MCIN/AEI/10.13039/ 50110001103 and “ERDF A way of making Europe”. This work has also been financially supported by the Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call—Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan—NextGenerationEU within the framework of the “Digital Spain 2026 Agenda”. Data Availability Statement The data that support the findings of this study are available from the corresponding author upon reasonable request. References 1. Y. Wang and J. Liu, “A Comprehensive Review of Quantum Machine Learning: From NISQ to Fault Tolerance,”Reports on Progress in Physics 87 (2024): 116402. 2. L. Lamata, A. Parra-Rodriguez, M. Sanz, and E. Solano, “Digital-Analog Quantum Simulations with Superconducting Circuits,”Advances in Physics 3, no. 1 (2018): 1457981. 3. A. Michel, S. Grijalva, L. Henriet, C. Domain, and A. 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