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Fast Estimation of Outcome Probabilities for quantum Circuits

PRX quantum 2022年第2期3卷 020361-020361页

作者： Hakop Pashayan Oliver Reardon-Smith Kamil Korzekwa Stephen D. Bartlett Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo Ontario N2L 3G1 Canada Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada Faculty of Physics Astronomy and Applied Computer Science Jagiellonian University Kraków 30-348 Poland Centre for Engineered Quantum Systems School of Physics The University of Sydney Sydney New South Wales 2006 Australia

We present two classical algorithms for the simulation of universal quantum circuits on n qubits constructed from c instances of Clifford gates and t arbitrary-angle Z-rotation gates such as T gates. Our algorithms complement each other by performing best in different parameter regimes. The Estimate algorithm produces an additive precision estimate of the Born-rule probability of a chosen measurement outcome with the only source of run-time inefficiency being a linear dependence on the stabilizer extent (with scaling approximately equal to 1.17t for T gates). Our algorithm is state of the art for this task: as an example, in approximately 13 h (on a standard desktop computer), we estimate the Born-rule probability to within an additive error of 0.03, for a 50-qubit, 60 non-Clifford gate quantum circuit with more than 2000 Clifford gates. Our second algorithm, Compute, calculates the probability of a chosen measurement outcome to machine precision with run time O(2t−rt), where r is an efficiently computable, circuit-specific quantity. With high probability, r is very close to min{t,n−w} for random circuits with many Clifford gates, where w is the number of measured qubits. Compute can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+T circuits with n=55, w=5, c=105, and t=80T gates, and then compute the Born-rule probability with a run time consistently less than 10 min using a single core of a standard desktop computer. We provide a C+Python implementation of our algorithms and benchmark them using random circuits, the hidden-shift algorithm, and the quantum approximate optimization algorithm.

关键词： quantum simulation

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quantum State Reduction for Universal Measurement Based Computation

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Physical Review Letters 2010年第2期105卷 020502-020502页

作者： Xie Chen Runyao Duan Zhengfeng Ji Bei Zeng Department of Physics Massachusetts Institute of Technology Cambridge Massachusetts USA Centre for Quantum Computation and Intelligent Systems (QCIS) Faculty of Engineering and Information Technology University of Technology Sydney NSW Australia Department of Computer Science and Technology Tsinghua National Laboratory for Information Science and Technology Tsinghua University Beijing China Perimeter Institute for Theoretical Physics Waterloo Ontario Canada State Key Laboratory of Computer Science Institute of Software Chinese Academy of Sciences Beijing China Institute for Quantum Computing University of Waterloo Waterloo Ontario Canada Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario Canada

Measurement based quantum computation, which requires only single particle measurements on a universal resource state to achieve the full power of quantum computing, has been recognized as one of the most promising models for the physical realization of quantum computers. Despite considerable progress in the past decade, it remains a great challenge to search for new universal resource states with naturally occurring Hamiltonians and to better understand the entanglement structure of these kinds of states. Here we show that most of the resource states currently known can be reduced to the cluster state, the first known universal resource state, via adaptive local measurements at a constant cost. This new quantum state reduction scheme provides simpler proofs of universality of resource states and opens up plenty of space to the search of new resource states.

关键词： quantum computers

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Multi-qutrit exact synthesis

arXiv

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arXiv 2024年

作者： Kalra, Amolak Ratan Saikia, Manimugdha Valluri, Dinesh Winnick, Sam Yard, Jon Institute for Quantum Computing University of Waterloo WaterlooON Canada David R. Cheriton School of Computer Science University of Waterloo WaterlooON Canada Dept. of Combinatorics & Optimization University of Waterloo WaterlooON Canada Perimeter Institute for Theoretical Physics WaterlooON Canada Department of Mathematics University of Western Ontario LondonON Canada

We present an exact synthesis algorithm for qutrit unitaries in U3n([1/3, e2πi/3]) over the Clifford+T gate set with at most one ancilla. This extends the already known result of qutrit metaplectic gates being a subset of Clifford+T gate set with one ancilla. As an intermediary step, we construct an algorithm to convert 3-level unitaries into multiply-controlled gates, analogous to Gray codes converting 2-level unitaries into multiply-controlled gates. Finally, using catalytic embeddings, we present an algorithm to exactly synthesize unitaries U3n([1/3, e2πi/9]) over the Clifford+T gate set with at most 2 ancillas. This, in particular, gives an exact synthesis algorithm of single-qutrit Clifford+D over the multi-qutrit Clifford+T gate set with at most two ancillas. Copyright © 2024, The Authors. All rights reserved.

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Quantifying quantum Speedups: Improved Classical Simulation From Tighter Magic Monotones

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PRX quantum 2021年第1期2卷 010345-010345页

作者： James R. Seddon Bartosz Regula Hakop Pashayan Yingkai Ouyang Earl T. Campbell Department of Physics and Astronomy University College London London United Kingdom School of Physical and Mathematical Sciences Nanyang Technological University 637371 Singapore Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo Ontario N2L 3G1 Canada Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada Centre for Engineered Quantum Systems School of Physics The University of Sydney Sydney New South Wales 2006 Australia Department of Physics & Astronomy University of Sheffield Sheffield S3 7RH United Kingdom AWS Center for Quantum Computing Pasadena California 91125 USA

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 nonasymptotic 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.

关键词： quantum algorithms quantum computation quantum simulation Resource theories

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Approximate Randomization of quantum States With Fewer Bits of Key

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AIP Conference Proceedings 2006年第1期864卷 18-36页

作者： Paul A. Dickinson Ashwin Nayak *Department of Combinatorics and Optimization University of Waterloo 200 University Ave. W. Waterloo Ontario N2L 3G1 Canada **Institute for Quantum Computing University of Waterloo 200 University Ave. W. Waterloo Ontario N2L 3G1 Canada †Perimeter Institute for Theoretical Physics Waterloo Ontario Canada

Randomization of quantum states is the quantum analogue of the classical one‐time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/ε2) Pauli operators to map any d‐dimensional state to a state that is within trace distance ε of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter, and Ambainis and ***, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d‐dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small ε.

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Achieving fault tolerance on capped color codes with few ancillas

arXiv

引用

arXiv 2021年

作者： Tansuwannont, Theerapat Leung, Debbie Institute for Quantum Computing Department of Physics and Astronomy University of Waterloo WaterlooONN2L 3G1 Canada Department of Electrical and Computer Engineering Duke University DurhamNC27708 United States Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo WaterlooONN2L 3G1 Canada Perimeter Institute for Theoretical Physics WaterlooONN2L 2Y5 Canada

Attaining fault tolerance while maintaining low overhead is one of the main challenges in a practical implementation of quantum circuits. One major technique that can overcome this problem is the flag technique, in which high-weight errors arising from a few faults can be detected by a few ancillas and distinguished using subsequent syndrome measurements. The technique can be further improved using the fact that for some families of codes, errors of any weight are logically equivalent if they have the same syndrome and weight parity, as previously shown in [1]. In this work, we develop a notion of distinguishable fault set which captures both concepts of flags and weight parities, and extend the use of weight parities in error correction from [1] to families of capped and recursive capped color codes. We also develop fault-tolerant protocols for error correction, measurement, state preparation, and logical T gate implementation via code switching, which are sufficient for performing fault-tolerant Clifford computation on a capped color code, and performing fault-tolerant universal quantum computation on a recursive capped color code. Our protocols for a capped or a recursive capped color code of any distance require only 2 ancillas, assuming that the ancillas can be reused. The concept of distinguishable fault set also leads to a generalization of the definitions of fault-tolerant gadgets proposed by Aliferis, Gottesman, and Preskill. Copyright © 2021, The Authors. All rights reserved.

关键词： Error correction

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Preparing High Purity Initial States for Nuclear Magnetic Resonance quantum computing

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Physical Review Letters 2004年第4期93卷 040501-040501页

作者： M. S. Anwar D. Blazina H. A. Carteret S. B. Duckett T. K. Halstead J. A. Jones C. M. Kozak R. J. K. Taylor Centre for Quantum Computation Clarendon Laboratory University of Oxford Parks Road Oxford OX1 3PU United Kingdom Department of Chemistry University of York Heslington York YO10 5DD United Kingdom Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario Canada N2L 3G1 Physics Department Imperial College Prince Consort Road London SW7 2BZ United Kingdom

Here we demonstrate how parahydrogen can be used to prepare a two-spin system in an almost pure state which is suitable for implementing nuclear magnetic resonance quantum computation. A 12 ns laser pulse is used to initiate a chemical reaction involving pure parahydrogen (the nuclear spin singlet of H2). The product, formed on the μs time scale, contains a hydrogen-derived two-spin system with an effective spin-state purity of 0.916. To achieve a comparable result by direct cooling would require an unmanageable (in the liquid state) temperature of 6.4 mK or an impractical magnetic field of 0.45 MT at room temperature. The resulting spin state has an entanglement of formation of 0.822 and cannot be described by local hidden variable models.

关键词： quantum theory

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Approximation algorithms for quantum many-body problems

arXiv

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arXiv 2018年

作者： Bravyi, Sergey Gosset, David König, Robert Temme, Kristan IBM T.J. Watson Research Center Yorktown HeightsNY10598 United States Department of Combinatorics & Optimization and Institute for Quantum Computing University of Waterloo WaterlooONN2L 3G1 Canada Institute for Advanced Study & Zentrum Mathematik Technical University of Munich Garching85748 Germany

We discuss classical algorithms for approximating the largest eigenvalue of quantum spin and fermionic Hamiltonians based on semidefinite programming relaxation methods. First, we consider traceless 2-local Hamiltonians H describing a system of n qubits. We give an efficient algorithm that outputs a separable state whose energy is at least λmax/O(log n), where λmax is the maximum eigenvalue of H. We also give a simplified proof of a theorem due to Lieb that establishes the existence of a separable state with energy at least λmax/9. Secondly, we consider a system of n fermionic modes and traceless Hamiltonians composed of quadratic and quartic fermionic operators. We give an efficient algorithm that outputs a fermionic Gaussian state whose energy is at least λmax/O(n log n). Finally, we show that Gaussian states can vastly outperform Slater determinant states commonly used in the Hartree-Fock method. We give a simple family of Hamiltonians for which Gaussian states and Slater determinants approximate λmax within a fraction 1 − O(n−1) and O(n−1) respectively. Copyright © 2018, The Authors. All rights reserved.

关键词： Hamiltonians

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Neural ensemble decoding for topological quantum error-correcting codes

arXiv

引用

arXiv 2019年

作者： Sheth, Milap Jafarzadeh, Sara Zafar Gheorghiu, Vlad Institute for Quantum Computing University of Waterloo WaterlooON Canada Department of Combinatorics and Optimization University of Waterloo WaterlooON Canada David R. Cheriton School of Computer Science University of Waterloo WaterlooON Canada Department of Computer Science and Operations Research Université de Montreal MontrealQC Canada softwareQ Inc. KitchenerON Canada

Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We use machine learning techniques to assign a given error syndrome to the decoder which is likely to decode it correctly. We apply our framework to an ensemble of Minimum-Weight Perfect Matching (MWPM) and Hard-Decision Re-normalization Group (HDRG) decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudo-threshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability. Copyright © 2019, The Authors. All rights reserved.

关键词： Decoding

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Tight Bell inequalities with no quantum violation from qubit unextendible product bases

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Physical Review A 2012年第4期85卷 042113-042113页

作者： R. Augusiak T. Fritz Ma. Kotowski Mi. Kotowski M. Pawłowski M. Lewenstein A. Acín ICFO–Institut de Ciències Fotòniques 08860 Castelldefels (Barcelona) Spain Department of Combinatorics and Optimization and Institute for Quantum Computing University of Waterloo Waterloo Ontario Canada N2L 3G1 Department of Mathematics University of Bristol Bristol BS8 1TW United Kingdom ICREA–Institució Catalana de Recerca i Estudis Avançats 08010 Barcelona Spain

We investigate the relation between unextendible product bases (UPB) and Bell inequalities found recently in R. Augusiak et al. [Phys. Rev. Lett. 107, 070401 (2011)]. We, first, review the procedure introduced there that associates to any set of mutually orthogonal product vectors in a many-qubit Hilbert space a Bell inequality. We then show that if a set of mutually orthogonal product vectors can be completed to a full basis, then the associated Bell inequality is trivial, in the sense of not being violated by any nonsignalling correlations. This implies that the relevant Bell inequalities that arise from the construction all come from UPBs, which adds additional weight to the significance of UPBs for Bell inequalities. Then, we provide additional examples of tight Bell inequalities with no quantum violation constructed from UPBs in this way. Finally, it is proven that the Bell inequalities with no quantum violation introduced recently in M. Almeida et al. [Phys. Rev. Lett. 104, 230404 (2010)] are tight for any odd number of parties.

关键词： Bells

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