• Open Access

Fast Estimation of Outcome Probabilities for Quantum Circuits

Hakop Pashayan, Oliver Reardon-Smith, Kamil Korzekwa, and Stephen D. Bartlett
PRX Quantum 3, 020361 – Published 23 June 2022
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Abstract

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(2trt), where r is an efficiently computable, circuit-specific quantity. With high probability, r is very close to min{t,nw} 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.

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  • Received 26 June 2021
  • Revised 24 March 2022
  • Accepted 9 May 2022

DOI:https://doi.org/10.1103/PRXQuantum.3.020361

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
Quantum Information, Science & Technology

Authors & Affiliations

Hakop Pashayan1,2,*, Oliver Reardon-Smith3, Kamil Korzekwa3, and Stephen D. Bartlett4

  • 1Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L 3G1, Canada
  • 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • 3Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Kraków 30-348, Poland
  • 4Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia

  • *hakopsadventures@gmail.com

Popular Summary

All known methods of simulating quantum mechanics using classical computers require exponential resources. It is widely believed that this difference is a fundamental one, and that quantum computers can efficiently solve problems for which no efficient classical algorithm exists. However, classical computers are cheaper, faster, more accessible, and more reliable than modern quantum computers, and so classical simulation algorithms continue to play a significant role in assessing and benchmarking the performance of quantum devices. In this paper we provide state-of-the-art classical algorithms for estimating the outcome probabilities that characterize the output of a quantum computer.

Recent works on classical simulations of quantum computers have determined what appears to be a fundamental limit on the cost of running such algorithms, scaling exponentially not in the number of qubits, but in a quantity called “magic,” which describes how far a particular operation is from a classical one. We present a classical simulation algorithm that in certain previously inaccessible parameter regimes, permits practical simulation run times for “typical quantum circuits.” For the cases where this result does not apply, e.g., for an adversarial choice of quantum circuit, we develop novel tools that allow us to achieve orders of magnitude improvements in simulation run time for practically relevant parameter regimes.

It is increasingly important to have methods for verifying and validating the outputs of quantum devices and assessing proposals for applications of near-term quantum devices using trusted classical methods. We expect our algorithms to be useful in this setting.

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Vol. 3, Iss. 2 — June - August 2022

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It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

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