August 2022 Statistical Aspects of the Quantum Supremacy Demonstration
Yosef Rinott, Tomer Shoham, Gil Kalai
Author Affiliations +
Statist. Sci. 37(3): 322-347 (August 2022). DOI: 10.1214/21-STS836


In quantum computing, a demonstration of quantum supremacy (or quantum advantage) consists of presenting a task, possibly of no practical value, whose computation is feasible on a quantum device, but cannot be performed by classical computers in any feasible amount of time. The notable claim of quantum supremacy presented by Google’s team in 2019 consists of demonstrating the ability of a quantum circuit to generate, albeit with considerable noise, bitstrings from a distribution that is considered hard to simulate on classical computers. Very recently, in 2020, a quantum supremacy claim was presented by a group from the University of Science and Technology of China, using a different technology and generating a different distribution, but sharing some statistical principles with Google’s demonstration.

Verifying that the generated data is indeed from the claimed distribution and assessing the circuit’s noise level and its fidelity is a statistical undertaking. The objective of this paper is to explain the relations between quantum computing and some of the statistical aspects involved in demonstrating quantum supremacy in terms that are accessible to statisticians, computer scientists, and mathematicians. Starting with the statistical modeling and analysis in Google’s demonstration, which we explain, we study various estimators of the fidelity, and different approaches to testing the distributions generated by the quantum computer. We propose different noise models, and discuss their implications. A preliminary study of the Google data, focusing mostly on circuits of 12 and 14 qubits is given in different parts of the paper.

Funding Statement

The second and third author were supported by ERC advanced grant 834735.


The numerical work in this paper was done by the R Package and Wolfram Mathematica. We are grateful to Larry Goldstein for insightful comments and questions, to Alexander Vlasov for helpful comments on the Google data and other matters, and to Carsten Voelkmann for two very careful readings of the manuscript and many invaluable corrections. We are also thankful to members of the Google team and especially to John Martinis and Sergio Boixo for helpful discussions. Two referees, an associate editor, and the Editor provided very useful comments on the content and organization of the paper.


Download Citation

Yosef Rinott. Tomer Shoham. Gil Kalai. "Statistical Aspects of the Quantum Supremacy Demonstration." Statist. Sci. 37 (3) 322 - 347, August 2022.


Published: August 2022
First available in Project Euclid: 21 June 2022

MathSciNet: MR4444370
zbMATH: 07569964
Digital Object Identifier: 10.1214/21-STS836

Keywords: estimation of sampling weights , Google’s quantum computer , random distributions , size bias

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.37 • No. 3 • August 2022
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