Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage

Authors:
raw_author_name: Xun Gao
raw_affiliation_string: Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA; JILA and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
raw_author_name: Marcin Kalinowski
raw_affiliation_string: Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
raw_author_name: Chi-Ning Chou
raw_affiliation_string: School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
raw_author_name: Mikhail D. Lukin
raw_affiliation_string: Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
raw_author_name: Soonwon Choi
raw_affiliation_string: Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Abstract: Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a certain class of highly entangled many-body wave functions. It has been suggested that such a quantum advantage can be certified with the linear cross-entropy benchmark (XEB). We critically examine this notion. First, we consider a “benign” setting, where an honest implementation of a noisy quantum circuit is assumed, and characterize the conditions under which the XEB approximates the fidelity of quantum dynamics. Second, we assume an “adversarial” setting, where all possible classical algorithms are considered for comparisons, and show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. Specifically, we present an efficient classical algorithm that achieves high XEB values, namely 5–12% of those obtained in the state-of-the-art experiments, within just a few seconds using a single GPU machine. This is made possible by identifying and exploiting several vulnerabilities of the XEB, which allows us to achieve high XEB values without simulating a full quantum circuit. Remarkably, our algorithm features better scaling with the system size than a noisy quantum device for commonly studied random circuit ensembles in various architecture. We quantitatively explain the success of our algorithm and the limitations of the XEB by using a theoretical framework, in which the dynamics of the average XEB and fidelity are mapped to classical statistical mechanics models. Using this framework, we illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different choices of gate sets, and in the presence of noise. Taken together, our results demonstrate that XEB’s utility as a proxy for fidelity hinges on several conditions, which should be independently checked in the benign setting, but cannot be assumed in the general adversarial setting. Therefore, the XEB on its own has a limited utility as a benchmark for quantum advantage. We briefly discuss potential ways to overcome these limitations.6 MoreReceived 28 January 2022Revised 24 July 2023Accepted 20 November 2023DOI:https://doi.org/10.1103/PRXQuantum.5.010334Published 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 SocietyPhysics Subject Headings (PhySH)Research AreasQuantum benchmarkingQuantum chaosQuantum computationQuantum protocolsQuantum verificationQuantum Information, Science & Technology