Motivated by notions of quantum heuristics and by average-case rather than worst-case algorithmic analysis, we define quantum computational advantage in terms of individual problem instances. Inspired by the classical notions of Kolmogorov complexity and instance complexity, we define their quantum versions. This allows us to define queasy instances of computational problems, like e.g. Satisfiability and Factoring, as those whose quantum instance complexity is significantly smaller than their classical instance complexity. These instances indicate quantum advantage: they are easy to solve on a quantum computer, but classical algorithms struggle (they feel queasy). Via a reduction from Factoring, we prove the existence of queasy Satisfiability instances; specifically, these instances are maximally queasy (under reasonable complexity-theoretic assumptions). Further, we show that there is exponential algorithmic utility in the queasiness of a quantum algorithm. This formal framework serves as a beacon that guides the hunt for quantum advantage in practice, and moreover, because its focus lies on single instances, it can lead to new ways of designing quantum algorithms.

翻译：受量子启发式算法以及平均情况而非最坏情况算法分析的启发，我们从单个问题实例的角度定义量子计算优势。借鉴经典柯尔莫哥洛夫复杂度和实例复杂度的概念，我们定义了它们的量子版本。这使得我们能够定义计算问题（例如可满足性问题和因数分解问题）的"眩晕实例"——即那些量子实例复杂度显著小于经典实例复杂度的实例。这些实例表明了量子优势：它们在量子计算机上易于求解，但经典算法却难以应对（它们感到"眩晕"）。通过从因数分解问题归约，我们证明了眩晕可满足性实例的存在性；具体而言，这些实例在合理的复杂性理论假设下是最大程度眩晕的。此外，我们证明了量子算法的眩晕性具有指数级的算法效用。这一形式化框架可作为在实践中探寻量子优势的指引，并且，由于其关注点在于单个实例，它能够为设计量子算法开辟新的途径。