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quantum algorithms for powering stable Hermitian matrices

Physical Review A 2021年第6期103卷 062420-062420页

作者： Guillermo González Rahul Trivedi J. Ignacio Cirac Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Strasse 1 85748 Garching Germany and Munich Center for Quantum Science and Technology (MCQST) Schellingstrasse 4 D-80799 Munich Germany

Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of discrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms: (i) a fast-forwarding algorithm that builds on construction of Apers and Sarlette [quantum Inf. Comput. 19, 181 (2019)] and (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.

关键词： quantum algorithms

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Optimization of Robot-Trajectory Planning with Nature-Inspired and Hybrid quantum algorithms

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Physical Review Applied 2022年第5期18卷 054045-054045页

作者： Martin J.A. Schuetz J. Kyle Brubaker Henry Montagu Yannick van Dijk Johannes Klepsch Philipp Ross Andre Luckow Mauricio G.C. Resende Helmut G. Katzgraber Amazon Quantum Solutions Lab Seattle Washington 98170 USA AWS Intelligent and Advanced Compute Technologies Professional Services Seattle Washington 98170 USA AWS Center for Quantum Computing Pasadena California 91125 USA BMW Group Munich Germany *** Inc. Bellevue Washington 98004 USA University of Washington Seattle Washington 98195 USA

We solve robot-trajectory planning problems at industry-relevant scales. Our end-to-end solution integrates highly versatile random-key algorithms with model stacking and ensemble techniques, as well as path relinking for solution refinement. The core optimization module consists of a biased random-key genetic algorithm. Through a distinct separation of problem-independent and problem-dependent modules, we achieve an efficient problem representation, with a native encoding of constraints. We show that generalizations to alternative algorithmic paradigms such as simulated annealing are straightforward. We provide numerical benchmark results for industry-scale data sets. Our approach is found to consistently outperform greedy baseline results. To assess the capabilities of today’s quantum hardware, we complement the classical approach with results obtained on quantum annealing hardware, using qbsolv on Amazon Braket. Finally, we show how the latter can be integrated into our larger pipeline, providing a quantum-ready hybrid solution to the problem.

关键词： Classical mechanics Optimization problems quantum algorithms quantum benchmarking Genetic algorithm Ising model Simulated annealing

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quantum algorithms with local particle-number conservation: Noise effects and error correction

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Physical Review A 2021年第4期103卷 042412-042412页

作者： Michael Streif Martin Leib Filip Wudarski Eleanor Rieffel Zhihui Wang Quantum Artificial Intelligence Laboratory (QuAIL) NASA Ames Research Center Moffett Field California 94035 USA USRA Research Institute for Advanced Computer Science (RIACS) Mountain View California 94043 USA Data:Lab Volkswagen Group Ungererstrasse 69 80805 München Germany University Erlangen-Nürnberg (FAU) Institute of Theoretical Physics Staudtstrasse 7 91058 Erlangen Germany

quantum circuits with local particle-number conservation restrict the quantum computation to a subspace of the Hilbert space of the qubit register. In a noiseless or fault-tolerant quantum computation, such quantities are preserved. In the presence of noise, however, the evolution's symmetry could be broken and nonvalid states could be sampled at the end of the computation. On the other hand, the restriction to a subspace in the ideal case suggests the possibility of more resource-efficient error-mitigation techniques for circuits preserving symmetries that are not possible for general circuits. Here, we analyze the probability of staying in such symmetry-preserved subspaces under noise, providing an exact formula for local depolarizing noise. We apply our findings to benchmark, under depolarizing noise, the symmetry robustness of the XY quantum alternating operator Ansatz, which has local particle-number conserving symmetries and is a special case of the quantum alternating operator Ansatz. We also analyze the influence of the choice of encoding the problem on the symmetry robustness of the algorithm and discuss a simple adaption of the bit-flip code to correct for symmetry-breaking errors with reduced resources.

关键词： quantum algorithms quantum computation

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quantum algorithms for hedging and the learning of Ising models

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Physical Review A 2021年第1期103卷 012418-012418页

作者： Patrick Rebentrost Yassine Hamoudi Maharshi Ray Xin Wang Siyi Yang Miklos Santha Université de Paris IRIF CNRS F-75006 Paris France and Centre for Quantum Technologies and MajuLab UMI 3654 National University of Singapore Singapore 117543 Singapore

A paradigmatic algorithm for online learning is the Hedge algorithm by Freund and Schapire. An allocation into different strategies is chosen for multiple rounds and each round incurs corresponding losses for each strategy. The algorithm obtains a favorable guarantee for the total losses even in an adversarial situation. This work presents quantum algorithms for such online learning in an oracular setting. For T time steps and N strategies, we exhibit run times of about Opoly(T)N for estimating the losses and for betting on individual strategies by sampling. In addition, we discuss a quantum analog of the Sparsitron, a machine learning algorithm based on the Hedge algorithm. The quantum algorithm inherits the provable learning guarantees from the classical algorithm and exhibits polynomial speedups. The speedups may find relevance in finance, for example, for hedging risks, and machine learning, for example, for learning generalized linear models or Ising models.

关键词： quantum algorithms quantum computation

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Efficient Measure for the Expressivity of Variational quantum algorithms

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Physical Review Letters 2022年第8期128卷 080506-080506页

作者： Yuxuan Du Zhuozhuo Tu Xiao Yuan Dacheng Tao JD Explore Academy Beijing 101111 China School of Computer Science Faculty of Engineering The University of Sydney Darlington NSW 2008 Australia Center on Frontiers of Computing Studies Department of Computer Science Peking University Beijing 100871 China

The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.

关键词： Learning theory Machine learning quantum algorithms quantum computation Resource theories

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quantum algorithms for Machine Learning and Optimization

Quantum Algorithms for Machine Learning and Optimization

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作者： Li, Tongyang University of Maryland College Park

学位级别：Ph.D.

The theories of optimization and machine learning answer foundational questions in computer science and lead to new algorithms for practical applications. While these topics have been extensively studied in the context of classical computing, their quantum counterparts are far from well-understood. In this thesis, we explore algorithms that bridge the gap between the fields of quantum computing and machine learning. First, we consider general optimization problems with only function evaluations. For two core problems, namely general convex optimization and volume estimation of convex bodies, we give quantum algorithms as well as quantum lower bounds that constitute the quantum speedups of both problems to be polynomial compared to their classical counterparts. We then consider machine learning and optimization problems with input data stored explicitly as matrices. We first look at semidefinite programs and provide quantum algorithms with polynomial speedup compared to the classical state-of-the-art. We then move to machine learning and give the optimal quantum algorithms for linear and kernel-based classifications. To complement with our quantum algorithms, we also introduce a framework for quantum-inspired classical algorithms, showing that for low-rank matrix arithmetics there can only be polynomial quantum speedup. Finally, we study statistical problems on quantum computers, with the focus on testing properties of probability distributions. We show that for testing various properties including L1-distance, L2-distance, Shannon and Renyi entropies, etc., there are polynomial quantum speedups compared to their classical counterparts. We also extend these results to testing properties of quantum states.

关键词： Computer science quantum physics Machine learning Optimization theory quantum algorithms quantum computing quantum machine learning Theoretical computer science

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Thermodynamic optimization of quantum algorithms: On-the-go erasure of qubit registers

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Physical Review A 2022年第6期106卷 062426-062426页

作者： Florian Meier Lídia del Rio Institute for Theoretical Physics ETH Zurich 8093 Zurich Switzerland Atominstitut TU Wien 1020 Vienna Austria

We consider two bottlenecks in quantum computing: limited memory size and noise caused by heat dissipation. Trying to optimize both, we investigate “on-the-go erasure” of quantum registers that are no longer needed for a given algorithm: freeing up auxiliary qubits as they stop being useful would facilitate the parallelization of computations. We study the minimal thermodynamic cost of erasure in these scenarios, applying results on the Landauer erasure of entangled quantum registers. For the class of algorithms solving the Abelian hidden subgroup problem, we find optimal on-the-go erasure protocols. We conclude that there is a trade-off: if we have enough partial information about a problem to build efficient on-the-go erasure, we can use it to instead simplify the algorithm, so that fewer qubits are needed to run the computation in the first place. We provide explicit protocols for these two approaches.

关键词： quantum algorithms quantum entanglement quantum thermodynamics Resource theories Thermodynamics of computation

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Noisy intermediate-scale quantum algorithms

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Reviews of Modern Physics 2022年第1期94卷 015004-015004页

作者： Kishor Bharti Alba Cervera-Lierta Thi Ha Kyaw Tobias Haug Sumner Alperin-Lea Abhinav Anand Matthias Degroote Hermanni Heimonen Jakob S. Kottmann Tim Menke Wai-Keong Mok Sukin Sim Leong-Chuan Kwek Alán Aspuru-Guzik Department of Computer Science University of Toronto Toronto Ontario M5S 2E4 Canada Chemical Physics Theory Group Department of Chemistry University of Toronto Toronto Ontario M5G 1Z8 Canada Vector Institute for Artificial Intelligence Toronto Ontario M5S 1M1 Canada and Canadian Institute forthat Advanced Research Toronto Ontario M5G 1Z8 Canada

A universal fault-tolerant quantum computer that can efficiently solve problems such as integer factorization and unstructured database search requires millions of qubits with low error rates and long coherence times. While the experimental advancement toward realizing such devices will potentially take decades of research, noisy intermediate-scale quantum (NISQ) computers already exist. These computers are composed of hundreds of noisy qubits, i.e., qubits that are not error corrected, and therefore perform imperfect operations within a limited coherence time. In the search for achieving quantum advantage with these devices, algorithms have been proposed for applications in various disciplines spanning physics, machine learning, quantum chemistry, and combinatorial optimization. The overarching goal of such algorithms is to leverage the limited available resources to perform classically challenging tasks. In this review, a thorough summary of NISQ computational paradigms and algorithms is provided. The key structure of these algorithms and their limitations and advantages are discussed. A comprehensive overview of various benchmarking and software tools useful for programming and testing NISQ devices is additionally provided.

关键词： Adiabatic quantum optimization Machine learning Nonequilibrium statistical mechanics Optimization problems quantum algorithms quantum benchmarking quantum computation quantum information processing

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On the Robustness of quantum algorithms for Blockchain Consensus

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SENSORS 2022年第7期22卷 2716-2716页

作者： Ullah, Muhammad Asad Setiawan, Jason William Rehman, Junaid Ur Shin, Hyundong Kyung Hee Univ Dept Elect & Informat Convergence Engn Yongin 17104 Gyeonggi Do South Korea

Blockchain has revolutionized many fields, such as distributed sensor networks, finance, and cryptocurrency. Consensus between distributed network nodes is at the core of such blockchain technologies. The three primary performance measures for any consensus algorithm are scalability, security, and decentralization. This paper evaluates the usefulness and practicality of quantum consensus algorithms for blockchain-enhanced sensor, and computing networks and evaluates them against the aforementioned performance measures. In particular, we investigate their noise robustness against quantum decoherence in quantum processors and over fiber-optic channels. We observe that the quantum noise generally increases the error rate in the list distribution. However, the effect is variable on different quantum consensus schemes. For example, the entanglement-free scheme is more affected than entanglement-based schemes for the local noise cases, while in the case of noisy optical fiber links, the effect is prominent on all quantum consensus schemes. We infer that the current quantum protocols with noisy intermediate-scale quantum devices and noisy quantum communication can only be employed for modular units in intraenterprise-level blockchain, such as Zilliqa, for sensor, and computing networks.

关键词： blockchain consensus quantum algorithms quantum noise distributed sensor networks modular computing units

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Error mitigation for variational quantum algorithms through mid-circuit measurements

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Physical Review A 2022年第2期105卷 022441-022441页

作者： Ludmila Botelho Adam Glos Akash Kundu Jarosław Adam Miszczak Özlem Salehi Zoltán Zimborás Institute of Theoretical and Applied Informatics Polish Academy of Sciences Bałtycka 5 44-100 Gliwice Poland Joint Doctoral School Silesian University of Technology Akademicka 2a 44-100 Gliwice Poland QWorld Association† Kesklinna linnaosa Tartu mnt 67/1-13b 10115 Tallinn Estonia Wigner Research Centre for Physics H-1525 P.O. Box 49 Budapest Hungary BME-MTA Lendület Quantum Information Theory Research Group 1111 Budapest Hungary

Noisy intermediate-scale quantum algorithms require novel paradigms of error mitigation. To obtain noise-robust quantum computers, each logical qubit is equipped with hundreds or thousands of physical qubits. However, it is not possible to use memory-consuming techniques for current quantum devices having at most hundreds or thousands of physical qubits on their own. For specific problems, valid quantum states have a unique structure as in the case of Fock states and W states where the Hamming weight is fixed, and the evolution takes place in a smaller subspace of the full Hilbert space. With this preknowledge, some errors can be detected during the evolution of the circuit, by filtering the states not obeying the pattern through postselection. In this paper, we present mid-circuit postselection schemes for frequently used encodings such as one-hot, binary, gray, and domain-wall encoding. For the particular subspace of one-hot states, we propose a method that works by compressing the full Hilbert space to a smaller subspace, allowing projecting to the desired subspace without using any ancilla qubits. We demonstrate the effectiveness of the approach for the quantum alternating operator ansatz algorithm. Our method is particularly suitable for the currently available hardware, where measuring and resetting are possible, but classical conditional operators are not.

关键词： quantum algorithms quantum computation

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