[Paper Review] Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage

Xun Gao, M. W. Kalinowski+4 more|arXiv (Cornell University)|Dec 3, 2021

Quantum Computing Algorithms and Architecture87 references28 citations

TL;DR

The paper critically assesses Linear Cross-Entropy Benchmark (XEB) as a proxy for quantum fidelity and shows an efficient classical spoofing algorithm that achieves high XEB values, revealing fundamental limitations of XEB as a stand-alone benchmark for quantum advantage.

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 class of highly entangled many-body wavefunctions. It has been suggested that this approach can be certified with the Linear Cross-Entropy Benchmark (XEB). We critically examine this notion. First, in a "benign" setting where an honest implementation of noisy quantum circuits is assumed, we characterize the conditions under which the XEB approximates the fidelity. Second, in an "adversarial" setting where all possible classical algorithms are considered for comparison, we show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. We present an efficient classical algorithm that, with 1 GPU within 2s, yields high XEB values, namely 2-12% of those obtained in experiments. By identifying and exploiting several vulnerabilities of the XEB, we achieve high XEB values without full simulation of quantum circuits. Remarkably, our algorithm features better scaling with the system size than noisy quantum devices for commonly studied random circuit ensembles. To quantitatively explain the success of our algorithm and the limitations of the XEB, we use a theoretical framework in which the average XEB and fidelity are mapped to statistical models. We illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different gate choices, and in the presence of noise. Our results show that XEB's utility as a proxy for fidelity hinges on several conditions, which must be checked in the benign setting but cannot be assumed in the adversarial setting. Thus, the XEB alone has limited utility as a benchmark for quantum advantage. We discuss ways to overcome these limitations.

Motivation & Objective

• Evaluate under what conditions XEB approximates quantum fidelity in benign noisy-quantum circuit settings.
• Assess whether high XEB values imply faithful quantum dynamics in adversarial classical settings.
• Develop a theoretical framework linking XEB and fidelity via classical statistical mechanics models.
• Demonstrate a classical algorithm that achieves high XEB values comparable to state-of-the-art experiments.
• Discuss how to mitigate XEB vulnerabilities and improve quantum advantage certification.

Proposed method

• Define and explain linear cross-entropy benchmark (XEB) and its relation to fidelity.
• Analyze XEB-fidelity relation under benign, noisy-circuit assumptions and various architectures/gate sets.
• Construct a classical spoofing algorithm that omits or modifies a few gates to split circuits into smaller sub-circuits for efficient simulation.
• Quantitatively compare spoofed XEB values to those from Google/Sycamore and USTC experiments across circuit architectures.
• Map circuit dynamics to diffusion-reaction and Ising-type statistical mechanics models to interpret XEB and fidelity behavior.
• Provide a gate-set optimization (e.g., introducing fSim* gate) to minimize XEB-fidelity discrepancy.

Experimental results

Research questions

• RQ1 Under what conditions does XEB reliably approximate quantum fidelity in benign settings?
• RQ2 Can a classical algorithm achieve high XEB values without simulating full quantum dynamics, thereby spoofing quantum advantage?
• RQ3 How do circuit architecture and gate choices affect the relationship between XEB and fidelity?
• RQ4 What theoretical framework can describe XEB and fidelity dynamics across architectures and noise regimes?
• RQ5 What practical measures could mitigate XEB vulnerabilities to reliably certify quantum advantage?

Key findings

• An efficient classical algorithm can spoof XEB, achieving high values (2-12% of state-of-the-art experiments) using a single GPU within seconds.
• XEB may outpace fidelity under adversarial settings, especially when errors are correlated or when circuit architectures enable boundary- versus bulk-dominated effects.
• XEB and fidelity scale differently with system size; fidelity multiplies across disjoint subsystems while XEB adds, enabling spoofing advantages at larger scales.
• A diffusion-reaction statistical mechanics framework explains how XEB and fidelity relate under different gate sets and noise, and why homogeneous error assumptions are crucial for XEB as a fidelity proxy.
• The commonly used Google fSim gate is not optimal for minimizing XEB-fidelity discrepancy; a proposed fSim* gate can further reduce this gap.
• XEB’s utility as a standalone benchmark is limited; independent checks beyond XEB are necessary to certify quantum advantage.