Certified algorithms for quantum Hamiltonian learning via energy-entropy inequalities

TL;DR

This paper develops certified algorithms for quantum Hamiltonian learning using energy-entropy inequalities, providing provable bounds and convergence guarantees, especially for commuting Hamiltonians, bridging practical scalability and theoretical correctness.

Contribution

It extends existing scalable algorithms to include certified bounds and proves convergence for commuting Hamiltonians, enhancing reliability in quantum Hamiltonian learning.

Findings

Provides a posteriori bounds on Hamiltonian parameters.

Proves a priori convergence for commuting Hamiltonians.

Scales well into 100-qubit systems with certification.

Abstract

We consider the problem of learning the Hamiltonian of a quantum system from estimates of Gibbs-state expectation values. Various methods for achieving this task were proposed recently, both from a practical and theoretical point of view. On the one hand, some practical algorithms have been implemented and used to analyze experimental data but these algorithms often lack correctness guarantees or fail to scale to large systems. On the other hand, theoretical algorithms with provable asymptotic efficiency guarantees have been proposed, but they seem challenging to implement. Recently, a semidefinite family of Hamiltonian learning algorithms was proposed which was numerically demonstrated to scale well into the 100-qubit regime, but provided no provable accuracy guarantees. We build on this work in two ways, by extending it to provide certified a posteriori lower and upper bounds on the parameters to be learned, and by proving a priori convergence in the special case where the Hamiltonian is commuting.