Optimal Hamiltonian recognition of unknown quantum dynamics

TL;DR

This paper introduces an optimal quantum algorithm for recognizing unknown Hamiltonians from limited dynamical data, combining quantum hypothesis testing and metrology, with proven success probabilities and experimental validation.

Contribution

It develops a novel quantum algorithm for Hamiltonian recognition using quantum signal processing, achieving optimal success probabilities and demonstrating practical implementation.

Findings

Achieves $O(1/k)$ decay in success probability with queries.

Validates protocol on superconducting quantum processor.

Provides numerical evidence for multi-qubit Hamiltonian recognition.

Abstract

Identifying unknown Hamiltonians from their quantum dynamics is a pivotal challenge in quantum technologies. In this paper, we introduce Hamiltonian recognition, a framework that bridges quantum hypothesis testing and quantum metrology, aiming to identify the Hamiltonian governing quantum dynamics from a known set of Hamiltonians. To identify $H$ for an unknown qubit quantum evolution $\exp(-iH\theta)$ with unknown $\theta$, from two or three orthogonal Hamiltonians, we develop a quantum algorithm for coherent function simulation, built on two quantum signal processing (QSP) structures. It can simultaneously realize a target polynomial based on measurement results regardless of the chosen signal unitary for the QSP. Utilizing semidefinite optimization and group representation theory, we prove that our methods achieve the optimal average success probability, taken over possible Hamiltonians $H$ and parameters $\theta$, decays as $O(1/k)$ with $k$ queries of the unknown unitary transformation. Furthermore, we demonstrate the validity of our protocol on a superconducting quantum processor. We also investigate a physically motivated recognition task for Heisenberg Hamiltonians, providing numerical evidence for effective multi-qubit quantum system recognition. This work presents an efficient method to recognize Hamiltonians from limited queries of the dynamics, opening new avenues in composite channel discrimination and quantum metrology.