Improved Approximation Algorithms for Bounded-Degree Local Hamiltonians.

Abstract

The low-temperature properties of interacting quantum systems are believed to require exponential resources to compute in the general case. Quantifying the extent to which such properties can be approximated using efficient algorithms remains a significant open challenge. Here, we consider the task of approximating the ground state energy of two-local quantum Hamiltonians with bounded-degree interaction graphs. Most existing algorithms optimize the energy over the set of product states. We propose and analyze a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an n-qubit product state with variance Var and improves its energy by an amount proportional to Var^{2}/n. In a typical case, this results in an extensive improvement in the estimated energy. We extend our results to k-local Hamiltonians and entangled initial states.