Optimizing Strongly Interacting Fermionic Hamiltonians

TL;DR

This paper investigates the complexity of optimizing strongly interacting fermionic Hamiltonians, especially when Gaussian states are insufficient, providing efficient algorithms for certifying eigenvalues in the SYK model.

Contribution

It introduces efficient classical and quantum certification algorithms for eigenvalue bounds in the SYK model, advancing understanding of non-Gaussian fermionic Hamiltonian optimization.

Findings

Classical certification algorithm for upper-bounding eigenvalues in SYK model.

Quantum certification algorithm for lower-bounding eigenvalues in SYK model.

Both algorithms achieve constant-factor approximations with high probability.

Abstract

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.