Certified algorithms for equilibrium states of local quantum Hamiltonians

TL;DR

This paper introduces certified algorithms that provide rigorous bounds for computing expectation values of local observables in equilibrium states of quantum Hamiltonians, addressing challenges in the thermodynamic limit and showing practical convergence.

Contribution

The work develops certified algorithms with convergence guarantees for equilibrium states of local quantum Hamiltonians, including at finite temperature and in the thermodynamic limit.

Findings

Algorithms produce rigorous bounds on expectation values.

Finite-time approximation of local observables is possible.

Fast convergence proven for commuting Hamiltonians at high temperature.

Abstract

Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision $\varepsilon$, local observables can be approximated by a convex optimization program of quasi-polynomial size in $1/\varepsilon$.