In recent years, several experimental groups have claimed demonstrations of ``quantum supremacy'' or computational quantum advantage. A notable first claim by Google Quantum AI revolves around a metric called the Linear Cross Entropy Benchmarking (Linear XEB), which has been used in many quantum supremacy experiments since. The complexity-theoretic hardness of spoofing Linear XEB, however, depends on the Cross-Entropy Quantum Threshold (XQUATH) conjecture put forth by Aaronson and Gunn, which has been disproven for sublinear depth circuits. In the efforts on demonstrating quantum supremacy by quantum Hamiltonian simulation, a similar benchmarking metric called the System Linear Cross Entropy Score (sXES) holds firm in light of the aforementioned negative result due to its fundamental distinction with Linear XEB. Moreover, the complexity-theoretic hardness of spoofing sXES rests on the System Linear Cross-Entropy Quantum Threshold Assumption (sXQUATH), the formal relationship of which to XQUATH is unclear. Despite the promises offered by sXES for future demonstration of quantum supremacy, in this work we show that it can be classically simulated efficiently in certain regimes.

翻译：近年来，多个实验组宣称实现了“量子霸权”或计算量子优势。谷歌量子人工智能团队提出的首个著名主张围绕一种称为线性交叉熵基准测试的度量指标展开，该指标此后被广泛应用于众多量子霸权实验。然而，欺骗线性交叉熵基准测试在计算复杂性意义上的困难程度取决于Aaronson和Gunn提出的交叉熵量子阈值猜想，而该猜想对于亚线性深度电路已被证伪。在通过量子哈密顿量模拟证明量子霸权的努力中，一种称为系统线性交叉熵评分的类似基准测试指标，因其与线性交叉熵基准测试的本质区别，在面对上述否定性结果时仍保持稳固。此外，欺骗系统线性交叉熵评分的计算复杂性困难程度基于系统线性交叉熵量子阈值假设，该假设与交叉熵量子阈值猜想之间的形式化关系尚不明确。尽管系统线性交叉熵评分为未来证明量子霸权提供了希望，但本研究表明，在某些特定条件下该指标可以被经典计算机高效模拟。