Abstract
We present two classical algorithms for the simulation of universal quantum circuits on qubits constructed from instances of Clifford gates and arbitrary-angle -rotation gates such as 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 for gates). Our algorithm is state of the art for this task: as an example, in approximately h (on a standard desktop computer), we estimate the Born-rule probability to within an additive error of , for a -qubit, non-Clifford gate quantum circuit with more than Clifford gates. Our second algorithm, Compute, calculates the probability of a chosen measurement outcome to machine precision with run time , where is an efficiently computable, circuit-specific quantity. With high probability, is very close to for random circuits with many Clifford gates, where is the number of measured qubits. Compute can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+ circuits with , , , and gates, and then compute the Born-rule probability with a run time consistently less than 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.
4 More- 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)
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.