The fundamental problem in much of physics and quantum chemistry is to optimize a low-degree polynomial in certain anticommuting variables. Being a quantum mechanical problem, in many cases we do not know an efficient classical witness to the optimum, or even to an approximation of the optimum. One prominent exception is when the optimum is described by a so-called "Gaussian state", also called a free fermion state. In this work we are interested in the complexity of this optimization problem when no good Gaussian state exists. Our primary testbed is the Sachdev--Ye--Kitaev (SYK) model of random degree-$q$ polynomials, a model of great current interest in condensed matter physics and string theory, and one which has remarkable properties from a computational complexity standpoint. Among other results, we give an efficient classical certification algorithm for upper-bounding the largest eigenvalue in the $q=4$ SYK model, and an efficient quantum certification algorithm for lower-bounding this largest eigenvalue; both algorithms achieve constant-factor approximations with high probability.

翻译：物理学和量子化学中的基本问题，本质上是优化某些反对易变量中的低次多项式。作为一个量子力学问题，在许多情况下我们并不知道优化结果（甚至近似优化结果）的高效经典验证方法。一个显著的例外是当优化结果由所谓的"高斯态"（也称为自由费米子态）描述时。本文的研究重点在于，当不存在良好高斯态时该优化问题的复杂度。我们采用随机$q$次多项式构成的Sachdev-Ye-Kitaev (SYK)模型作为主要测试平台——该模型在凝聚态物理和弦理论中具有重要研究价值，且从计算复杂性角度展现出非凡特性。在诸多成果中，我们给出了：针对$q=4$ SYK模型最大特征值的上界高效经典验证算法，以及该最大特征值的下界高效量子验证算法；两种算法均能以高概率实现常数因子近似。