Approximation algorithms for quantum many-body problems

TL;DR

This paper presents classical approximation algorithms for the maximum eigenvalue of quantum many-body Hamiltonians, providing guarantees for separable and Gaussian states, and comparing their effectiveness.

Contribution

It introduces efficient algorithms for approximating maximum eigenvalues using separable and Gaussian states, with improved bounds and insights into their relative performance.

Findings

Algorithm achieves an $O(rac{ ext{max eigenvalue}}{ ext{log} n})$ approximation for qubit Hamiltonians.

Gaussian states outperform Slater determinants in approximating maximum eigenvalues.

Simplified proof of Lieb's theorem on separable states with high energy.

Abstract

We discuss classical algorithms for approximating the largest eigenvalue of quantum spin and fermionic Hamiltonians based on semidefinite programming relaxation methods. First, we consider traceless $2$-local Hamiltonians $H$ describing a system of $n$ qubits. We give an efficient algorithm that outputs a separable state whose energy is at least $\lambda_{\max}/O(\log n)$, where $\lambda_{\max}$ is the maximum eigenvalue of $H$. We also give a simplified proof of a theorem due to Lieb that establishes the existence of a separable state with energy at least $\lambda_{\max}/9$. Secondly, we consider a system of $n$ fermionic modes and traceless Hamiltonians composed of quadratic and quartic fermionic operators. We give an efficient algorithm that outputs a fermionic Gaussian state whose energy is at least $\lambda_{\max}/O(n\log n)$. Finally, we show that Gaussian states can vastly outperform Slater determinant states commonly used in the Hartree-Fock method. We give a simple family of Hamiltonians for which Gaussian states and Slater determinants approximate $\lambda_{\max}$ within a fraction $1-O(n^{-1})$ and $O(n^{-1})$ respectively.