Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.
Classical simulation of noisy random circuits from exponential decay of correlation
Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普里茨克分子工程学院）；Department of Computer Science, University of Chicago（芝加哥大学计算机科学系）；Department of Physics, Korea Advanced Institute of Science and Technology（韩国科学技术院物理系）；AWS Center for Quantum Computing（亚马逊云科技量子计算中心）
Approximation Methods for Simulation and Equivalence Checking of Noisy Quantum Circuits
University of Chinese Academy of Sciences（中国科学院大学）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；Centre for Quantum Software and Information, University of Technology Sydney（悉尼科技大学量子软件与信息研究中心）
Enhancing classical simulation with noisy quantum devices
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Department of Mathematics, Tsinghua University（清华大学数学系）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
Parallel classical simulation of noisy shallow circuits: no quantum advantage in 1D
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；National University of Singapore（新加坡国立大学）；Centre for Quantum Technologies（量子技术中心）
Limitations of Noisy Geometrically Local Quantum Circuits
University of Maryland（马里兰大学）；Joint Center for Quantum Information & Computer Science, University of Maryland and NIST（马里兰大学与美国国家标准与技术研究院联合量子信息与计算中心）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Classical simulation of a quantum circuit with noisy magic inputs
Graduate School of Quantum Science and Technology, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院量子科学与技术研究生院）；Department of Physics, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院物理系）
Sampling (noisy) quantum circuits through randomized rounding
IBM France（法国国际商业机器公司）；Univ Lyon, ENS Lyon, UCBL, Inria, LIP（里昂大学、巴黎高等师范学校里昂分校、里昂第一大学、法国国家信息与自动化研究所、计算与图像实验室）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）
Pilot-Wave Simulator: Exact Classical Sampling from Ideal and Noisy Quantum Circuits up to Hundreds of Qubits
Lomonosov Moscow State University（莫斯科国立大学）；University of Science and Technology of China（中国科学技术大学）；Peking University（北京大学）；Southern University of Science and Technology（南方科技大学）
A polynomial-time classical sampler for noisy quantum circuits from statistical mechanics
University of Maryland, College Park（马里兰大学帕克分校）；Electrical & Computer Engineering Department, UCLA（加州大学洛杉矶分校电气与计算机工程系）；Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气与计算机工程系）；QuEra Computing Inc.（QuEra计算公司）；Department of Physics, Stanford University（斯坦福大学物理系）
Estimates of loss function concentration in noisy parametrized quantum circuits
University of Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare, Trieste Section（意大利国家核物理研究所，的里雅斯特分部）；European Organization for Nuclear Research (CERN)（欧洲核子研究组织（CERN））
Measurement-induced entanglement in noisy 2D random circuits
Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（联合量子研究所和联合量子信息与计算科学中心，美国国家标准与技术研究院/马里兰大学）；Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）
Shallow instantaneous quantum polynomial-time circuits for generative modeling on noisy intermediate-scale quantum hardware
Eurecat, Centre Tecnològic de Catalunya（Eurecat，加泰罗尼亚技术中心）；Centre de Visió per Computador (CVC)（计算机视觉中心）；Universitat Autònoma de Barcelona (UAB)（巴塞罗那自治大学）
