Learning many-body Hamiltonians with Heisenberg-limited scaling

TL;DR

This paper introduces a new quantum algorithm that efficiently learns many-body Hamiltonians with Heisenberg-limited precision, requiring minimal total evolution time and experiments, outperforming previous methods significantly.

Contribution

The paper presents the first algorithm achieving Heisenberg-limited scaling for learning many-body Hamiltonians, using a quantum divide-and-conquer approach with proven optimality.

Findings

Achieves Heisenberg-limited scaling in Hamiltonian learning

Requires only polylogarithmic experiments in inverse error

Proven to be asymptotically optimal

Abstract

Learning a many-body Hamiltonian from its dynamics is a fundamental problem in physics. In this work, we propose the first algorithm to achieve the Heisenberg limit for learning an interacting $N$-qubit local Hamiltonian. After a total evolution time of $\mathcal{O}(\epsilon^{-1})$, the proposed algorithm can efficiently estimate any parameter in the $N$-qubit Hamiltonian to $\epsilon$-error with high probability. The proposed algorithm is robust against state preparation and measurement error, does not require eigenstates or thermal states, and only uses $\mathrm{polylog}(\epsilon^{-1})$ experiments. In contrast, the best previous algorithms, such as recent works using gradient-based optimization or polynomial interpolation, require a total evolution time of $\mathcal{O}(\epsilon^{-2})$ and $\mathcal{O}(\epsilon^{-2})$ experiments. Our algorithm uses ideas from quantum simulation to decouple the unknown $N$-qubit Hamiltonian $H$ into noninteracting patches, and learns $H$ using a quantum-enhanced divide-and-conquer approach. We prove a matching lower bound to establish the asymptotic optimality of our algorithm.