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arXiv:2008.05177 · Statistical Science

Statistical Aspects of the Quantum Supremacy Demonstration

Yosef Rinott, Tomer Shoham, Gil Kalai

Published Jun 22, 2022• 11 citations

Abstract

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.

DOI: 10.1214/21-sts836

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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.

In October 2019, Nature published a paper [6] describing an experimental work that was performed at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. Since then we have been involved in a long-term project to study various statistical aspects of the Google experiment. In [30] we studied Google's statistical framework that we found to be very sound and offered some technical improvements. This document describes three main concerns (based on statistical analysis) about the Google 2019 experiment. The first concern is that the data do not agree with Google's noise model (or any other specific model). The second concern is that a crucial simple formula for a priori estimation of the fidelity seems to involve an unexpected independence assumption, and yet it gives very accurate predictions. The third concern is about statistical properties of the calibration process.

In October 2019, Nature published a paper describing an experiment that took place at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. Since September 2019 we have been involved in a long-term project to study various statistical aspects of the Google experiment. We have been trying to gather the relevant data and information in order to reconstruct and verify those parts of the Google experiment that are based on classical computations (except when the required computation is too heavy), and to perform a statistical analysis on the data. This document describes the available data and information for the Google experiment, some main questions in the evaluation of the experiment, and some of our results and plans.

My 2018 lecture at the ICA workshop in Singapore dealt with quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics. We described a computational complexity argument against the feasibility of quantum computers: we identified a very low-level complexity class of probability distributions described by noisy intermediate-scale quantum computers, and explained why it would allow neither good-quality quantum error-correction nor a demonstration of "quantum supremacy," namely, the ability of quantum computers to make computations that are impossible or extremely hard for classical computers. We went on to describe general predictions arising from the argument and proposed general laws that manifest the failure of quantum computers. In October 2019, "Nature" published a paper describing an experimental work that took place at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer, thus clearly challenging my theory. In this paper, I will explain and discuss my work in the perspective of Google's supremacy claims.

As Moore's law reaches its limits, quantum computers are emerging with the promise of dramatically outperforming classical computers. We have witnessed the advent of quantum processors with over

50

quantum bits (qubits), which are expected to be beyond the reach of classical simulation. Quantum supremacy is the event at which the old Extended Church-Turing Thesis is overturned: A quantum computer performs a task that is practically impossible for any classical (super)computer. The demonstration requires both a solid theoretical guarantee and an experimental realization. The lead candidate is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of random quantum circuits. Google recently announced a

53-

qubit experimental demonstration of RCS. Soon after, classical algorithms appeared that challenge the supremacy of random circuits by estimating their outputs. How hard is it to classically simulate the output of random quantum circuits? We prove that estimating the output probabilities of random quantum circuits is formidably hard (

\#P

-Hard) for any classical computer. This makes RCS the strongest candidate for demonstrating quantum supremacy relative to all other proposals. The robustness to the estimation error that we prove may serve as a new hardness criterion for the performance of classical algorithms. To achieve this, we introduce the Cayley path interpolation between any two gates of a quantum computation and convolve recent advances in quantum complexity and information with probability and random matrices. Furthermore, we apply algebraic geometry to generalize the well-known Berlekamp-Welch algorithm that is widely used in coding theory and cryptography. Our results imply that there is an exponential hardness barrier for the classical simulation of most quantum circuits.

Quantum computation promises to execute certain computational tasks on time scales much faster than any known algorithm on an existing classical computer, for example calculating the prime factors of large integers. Recently a research team from Google claimed to have carried out such a task with a quantum computer, demonstrating in practice a case of this so-called quantum supremacy. Here we argue that this claim was not justified. Unlike other comments, our criticism is concerned with the missing verification of the output data of the quantum computation.

Recently, a team of scientists from Google claims to have carried a computation on their noisy, intermediate-scale quantum (NISQ) computer which no regular computer can achieve. A feat that is sometimes referred as quantum supremacy. In the first part of this work, we explain their approach, their randomised circuit construction and the consequences of their work. Having achieved this milestone, the natural question becomes: what else can we do with a quantum computer? We answer this question in the second part of this work and give an overview of the most widely used quantum primitives. We will see how one can implement them using quantum circuits and how these primitive circuits are composed to create fast algorithms capable of solving commercially relevant problems like simulation of complicated quantum systems for chemistry and material science, Monte Carlo simulation, solving large systems of linear equations, or breaking widely used cryptography like RSA.