Quantifying quantum speedups: improved classical simulation from tighter magic monotones
James R. Seddon, Bartosz Regula, Hakop Pashayan, Yingkai Ouyang, Earl T. Campbell
备注：
24+13 pages, 8 figures; final author copy. Since v1: restructured with additional discussion, proof sketches and examples. Since v3: minor revisions to improve clarity, additional acknowledgments
Consumption of magic states promotes the stabilizer model of computation to universal quantum computation. Here, we propose three different classical algorithms for simulating such universal quantum circuits, and characterize them by establishing precise connections with a family of magic monotones. Our first simulator introduces a new class of quasiprobability distributions and connects its runtime to a generalized notion of negativity. We prove that this algorithm has significantly improved exponential scaling compared to all prior quasiprobability simulators for qubits. Our second simulator is a new variant of the stabilizer-rank simulation algorithm, extended to work with mixed states and with significantly improved runtime bounds. Our third simulator trades precision for speed by discarding negative quasiprobabilities. We connect each algorithm's performance to a corresponding magic monotone and, by comprehensively characterizing the monotones, we obtain a precise understanding of the simulation runtime and error bounds. Our analysis reveals a deep connection between all three seemingly unrelated simulation techniques and their associated monotones. For tensor products of single-qubit states, we prove that our monotones are all equal to each other, multiplicative and efficiently computable, allowing us to make clear-cut comparisons of the simulators' performance scaling. Furthermore, our monotones establish several asymptotic and non-asymptotic bounds on state interconversion and distillation rates. Beyond the theory of magic states, our classical simulators can be adapted to other resource theories under certain axioms, which we demonstrate through an explicit application to the theory of quantum coherence.
The development of a framework for quantifying "non-stabiliserness" of quantum operations is motivated by the magic state model of fault-tolerant quantum computation, and by the need to estimate classical simulation cost for noisy intermediate-scale quantum (NISQ) devices. The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford+T gates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the "non-stabiliserness" of quantum operations r…
Stabilizer Ranks, Barnes Wall Lattices and Magic Monotones
Institute for Quantum Computing, University of Waterloo（量子计算研究所，滑铁卢大学）；David R. Cheriton School of Computer Science, University of Waterloo（大卫·R·切里顿计算机科学学院，滑铁卢大学）；Perimeter Institute for Theoretical Physics（基础物理前沿研究所）
Magic states can be used as a resource to circumvent the restrictions due to stabilizer-preserving operations, and magic-state conversion has not been studied in the single-copy regime thus far. Here we solve the question of whether a stabilizer-preserving quantum operation exists that can convert between two given magic states in the single-shot regime. We first phrase this question as a feasibility problem for a semi-definite program (SDP), which provides a procedure for constructing a stabilizer-preserving quantum operation (free channel) if it exists. Then we employ a variant of the Farkas…
Motivated by their necessity for most fault-tolerant quantum computation schemes, we formulate a resource theory for magic states. We first show that robustness of magic is a well-behaved magic monotone that operationally quantifies the classical simulation overhead for a Gottesman-Knill type scheme using ancillary magic states. Our framework subsequently finds immediate application in the task of synthesizing non-Clifford gates using magic states. When magic states are interspersed with Clifford gates, Pauli measurements and stabilizer ancillas - the most general synthesis scenario - then the…
Stabilizer Statistical Mechanics: A Framework for Efficient Quantification and Classification of Magic States
LG Electronics Toronto AI Lab（LG电子多伦多人工智能实验室）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654（MajuLab，CNRS-UNS-NUS-NTU国际联合研究单元，UMI 3654）；National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）
Recent results on the non-universality of fault-tolerant gate sets underline the critical role of resource states, such as magic states, to power scalable, universal quantum computation. Here we develop a resource theory, analogous to the theory of entanglement, for resources for stabilizer codes. We introduce two quantitative measures - monotones - for the amount of non-stabilizer resource. As an application we give absolute bounds on the efficiency of magic state distillation. One of these monotones is the sum of the negative entries of the discrete Wigner representation of a quantum state, …
Magic states are key ingredients in schemes to realize universal fault-tolerant quantum computation. Theories of magic states attempt to quantify this computational element via monotones and determine how these states may be efficiently transformed into useful forms. Here, we develop a statistical mechanical framework based on majorization to describe Wigner negative magic states for qudits of odd prime dimension processed under Clifford circuits. We show that majorization allows us to both quantify disorder in the Wigner representation and derive upper bounds for magic distillation. These bou…
Bayesian Monotone Metrics for Multiparameter Quantum Estimation
Graduate School of Informatics and Engineering, The University of Electro-Communications（电气通信大学信息工程学院）；Institute of Science and Engineering, Kanazawa University（金泽大学理工研究院）
Efficient witnessing and testing of magic in mixed quantum states
Technology Innovation Institute（技术创新研究所）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）
Kirkwood-Dirac Nonpositivity is a Necessary Resource for Quantum Computing
Univ. of Cambridge（剑桥大学）；Hitachi Cambridge Lab.（日立剑桥实验室）；Joint Center for Quantum Information and Computer Science, NIST and University of Maryland（美国国家标准与技术研究院与马里兰大学量子信息与计算科学联合中心）；Institute for Physical Science and Technology, University of Maryland（马里兰大学物理科学与技术研究所）；Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé（里尔大学、法国国家科学研究中心、法国研究计算机科学与应用数学研究所，保罗·潘勒瓦实验室（UMR 8524））
Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；International Institute of Physics, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）；Center for Macroscopic Quantum States bigQ, Department of Physics, Technical University of Denmark（丹麦技术大学宏观量子态中心bigQ）
Non-stabilizerness and U(1) symmetry in chaotic many-body quantum systems
Scuola Superiore Meridionale（南方高等学院）；INFN Sezione di Napoli（那不勒斯 INFN 分部）；SISSA（的里雅斯特国际高等研究学校）；INFN Sezione di Trieste（的里雅斯特 INFN 分部）；Dipartimento di Fisica ‘Ettore Pancini’, Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学物理系）
We establish lower-bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks, treating stabilizer operations as free. Our bounds apply to adaptive computations using measurements and an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational tasks. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that bo…
The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over de…
We investigate the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer product input states. First, we consider the case when the input state is mixed, and give an efficient classical algorithm to approximate the output probabilities, with respect to the $l_1$ norm, of a large fraction of Clifford circuits. The running time of our algorithm decreases as the inputs become more mixed. Second, we consider the case when the input state is a pure nonstabilizer product state, and show that a similar efficient algorithm exists to approximate the output probabilities…
