Ansatz-free Hamiltonian learning with Heisenberg-limited scaling

TL;DR

This paper introduces a quantum algorithm for learning arbitrary sparse Hamiltonians without prior structural assumptions, achieving Heisenberg-limited scaling and robustness to errors, advancing quantum system characterization.

Contribution

The work presents the first ansatz-free method for Heisenberg-limited Hamiltonian learning applicable to arbitrary sparse Hamiltonians using black-box queries.

Findings

Achieves Heisenberg-limited scaling in Hamiltonian estimation.

Demonstrates robustness to state-preparation-and-measurement errors.

Establishes a fundamental trade-off between evolution time and controllability.

Abstract

Learning the unknown interactions that govern a quantum system is crucial for quantum information processing, device benchmarking, and quantum sensing. The problem, known as Hamiltonian learning, is well understood under the assumption that interactions are local, but this assumption may not hold for arbitrary Hamiltonians. Previous methods all require high-order inverse polynomial dependency with precision, unable to surpass the standard quantum limit and reach the gold standard Heisenberg-limited scaling. Whether Heisenberg-limited Hamiltonian learning is possible without prior assumptions about the interaction structures, a challenge we term \emph{ansatz-free Hamiltonian learning}, remains an open question. In this work, we present a quantum algorithm to learn arbitrary sparse Hamiltonians without any structure constraints using only black-box queries of the system's real-time evolution and minimal digital controls to attain Heisenberg-limited scaling in estimation error. Our method is also resilient to state-preparation-and-measurement errors, enhancing its practical feasibility. We numerically demonstrate our ansatz-free protocol for learning physical Hamiltonians and validating analog quantum simulations, benchmarking our performance against the state-of-the-art Heisenberg-limited learning approach. Moreover, we establish a fundamental trade-off between total evolution time and quantum control on learning arbitrary interactions, revealing the intrinsic interplay between controllability and total evolution time complexity for any learning algorithm. These results pave the way for further exploration into Heisenberg-limited Hamiltonian learning in complex quantum systems under minimal assumptions, potentially enabling new benchmarking and verification protocols.