Solving Gapped Hamiltonians Locally

TL;DR

This paper demonstrates that gapped short-range Hamiltonians can be decomposed into local operators with ground states approximable by matrix product states, facilitating quantum simulations.

Contribution

It introduces a method to express gapped Hamiltonians as sums of local operators, enabling efficient approximation of ground states with matrix product states.

Findings

Ground states are close to generalized matrix product states.

Range of operators scales with the square of the logarithm of system size.

Applications to many-body quantum simulation are discussed.

Abstract

We show that any short-range Hamiltonian with a gap between the ground and excited states can be written as a sum of local operators, such that the ground state is an approximate eigenvector of each operator separately. We then show that the ground state of any such Hamiltonian is close to a generalized matrix product state. The range of the given operators needed to obtain a good approximation to the ground state is proportional to the square of the logarithm of the system size times a characteristic "factorization length". Applications to many-body quantum simulation are discussed. We also consider density matrices of systems at non-zero temperature.