Improved algorithms for learning quantum Hamiltonians, via flat polynomials

TL;DR

This paper presents an improved algorithm for learning quantum Hamiltonians from Gibbs states, achieving lower sample complexity and runtime by introducing a new flat polynomial approximation of the exponential function.

Contribution

The authors develop a new flat polynomial approximation that reduces the degree and improves the efficiency of learning algorithms for quantum Hamiltonians.

Findings

Reduced sample complexity to singly exponential in inverse temperature

Lowered runtime dependence compared to previous methods

Introduced a novel flat polynomial approximation for the exponential function

Abstract

We give an improved algorithm for learning a quantum Hamiltonian given copies of its Gibbs state, that can succeed at any temperature. Specifically, we improve over the work of Bakshi, Liu, Moitra, and Tang [BLMT24], by reducing the sample complexity and runtime dependence to singly exponential in the inverse-temperature parameter, as opposed to doubly exponential. Our main technical contribution is a new flat polynomial approximation to the exponential function, with significantly lower degree than the flat polynomial approximation used in [BLMT24].