Computing local properties in the trivial phase

TL;DR

This paper presents an efficient algorithm for computing local observable expectations in trivial phases of translation-invariant gapped Hamiltonians across different dimensions, without requiring knowledge of the connecting path.

Contribution

It introduces a polynomial-time algorithm for local property computation in trivial phases, applicable in finite and infinite systems, without prior path information.

Findings

Efficient polynomial-time computation in 1D systems.

Exponential efficiency in higher dimensions.

Applicable to finite and thermodynamic limit systems.

Abstract

A translation-invariant gapped local Hamiltonian is in the trivial phase if it can be connected to a completely decoupled Hamiltonian with a smooth path of translation-invariant gapped local Hamiltonians. For the ground state of such a Hamiltonian, we show that the expectation value of a local observable can be computed in time $\text{poly}(1/\delta)$ in one spatial dimension and $e^{\text{poly}\log(1/\delta)}$ in two and higher dimensions, where $\delta$ is the desired (additive) accuracy. The algorithm applies to systems of finite size and in the thermodynamic limit. It only assumes the existence but not any knowledge of the path.