Extremal eigenvalues of local Hamiltonians

Aram W Harrow,Ashley Montanaro

doi:10.22331/q-2017-04-25-6

Abstract

We apply classical algorithms for approximately solving constraint satisfaction problems to find bounds on extremal eigenvalues of local Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on the number of terms in which each spin participates, and find extensive bounds for the operator norm and ground-state energy of such Hamiltonians under this constraint. In each case the bound is achieved by a product state which can be found efficiently using a classical algorithm.

Highlights

The eigenvalue statistics of a local Hamiltonian are related to its structure
What about the large-scale features, such as the extremal eigenvalues? Do these scale differently for non-interacting or interacting systems? It is generally understood that interacting systems can be frustrated, meaning that all the local terms cannot simultaneously be in their ground state
We study the extremal eigenvalues of quantum Hamiltonians which are only weakly interacting, in the sense that they can be written as sums of terms where each term depends only on a few qubits, and each qubit is included in only a few terms

Summary

Introduction

The eigenvalue statistics of a local Hamiltonian are related to its structure. One example is the level spacings of chaotic vs integrable systems, which can be seen as the small-scale structure of the spectrum. Both results that make up Theorem 1 are based on the use of a correspondence between local quantum Hamiltonians and lowdegree polynomials, which allows us to apply classical approximation algorithms for constraint satisfaction problems. One other related work is [3], which showed that when k = 2 and the degree of the interaction graph is large, product states can provide a good approximation for any state, with respect to the metric given by averaging the trace distance over the pairs of systems acted on by the Hamiltonian In particular this means they can approximate the minimum and maximum eigenvalues.

The quantum-classical correspondence

Operator norm bounds

Bounds on extremal eigenvalues

Optimality