Heisenberg-limited Hamiltonian learning continuous variable systems via engineered dissipation

TL;DR

This paper develops a rigorous framework for learning Hamiltonians in continuous-variable quantum systems, achieving Heisenberg-limited precision with logarithmic scaling in evolution time, and discusses potential experimental realizations.

Contribution

It introduces a novel analytic approach to Hamiltonian learning in bosonic systems, enabling Heisenberg-limited algorithms with efficient scaling and new adiabatic approximation estimates for unbounded Lindbladian evolutions.

Findings

Heisenberg-limited Hamiltonian learning algorithms with logarithmic time scaling.

A new quantitative adiabatic approximation for Lindbladian evolutions.

Framework supports experimental implementation discussions.

Abstract

Discrete and continuous variables oftentimes require different treatments in many learning tasks. Identifying the Hamiltonian governing the evolution of a quantum system is a fundamental task in quantum learning theory. While previous works mostly focused on quantum spin systems, where quantum states can be seen as superpositions of discrete bit-strings, relatively little is known about Hamiltonian learning for continuous-variable quantum systems. In this work we focus on learning the Hamiltonian of a bosonic quantum system, a common type of continuous-variable quantum system. This learning task involves an infinite-dimensional Hilbert space and unbounded operators, making mathematically rigorous treatments challenging. We introduce an analytic framework to study the effects of strong dissipation in such systems, enabling a rigorous analysis of cat qubit stabilization via engineered dissipation. This framework also supports the development of Heisenberg-limited algorithms for learning general bosonic Hamiltonians with higher-order terms of the creation and annihilation operators. Notably, our scheme requires a total Hamiltonian evolution time that scales only logarithmically with the number of modes and inversely with the precision of the reconstructed coefficients. On a theoretical level, we derive a new quantitative adiabatic approximation estimate for general Lindbladian evolutions with unbounded generators. Finally, we discuss possible experimental implementations.