Quantum natural gradient with thermal-state initialization

TL;DR

This paper develops exact quantum algorithms to compute Fisher information matrices for thermal-state initialized PQCs, enabling improved natural gradient optimization in variational quantum algorithms with mixed states.

Contribution

It introduces exact methods for estimating three quantum Fisher information matrices for thermal-state PQCs, extending quantum natural gradient techniques to mixed states.

Findings

Exact algorithms for Fisher-Bures, Wigner-Yanase, Kubo-Mori matrices.

Enabling quantum natural gradient descent for mixed-state PQCs.

Fundamental limits on parameter estimation for unknown PQCs.

Abstract

Parameterized quantum circuits (PQCs) are central to variational quantum algorithms (VQAs), yet their performance is hindered by complex loss landscapes that make their trainability challenging. Quantum natural gradient descent, which leverages the geometry of the parameterized space through quantum generalizations of the Fisher information matrix, offers a promising solution but has been largely limited to pure-state scenarios, with only approximate methods available for mixed-state settings. This paper addresses this question, originally posed in [Stokes et al., Quantum 4, 269 (2020)], by providing exact methods to compute three quantum generalizations of the Fisher information matrix-the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrices-for PQCs initialized with thermal states. We prove that these matrix elements can be estimated using quantum algorithms combining the Hadamard test, classical random sampling, and Hamiltonian simulation. By broadening the set of quantum generalizations of Fisher information and realizing their unbiased estimation, our results enable the implementation of quantum natural gradient descent algorithms for mixed-state PQCs, thereby enhancing the flexibility of optimization when using VQAs. Another immediate consequence of our findings is to establish fundamental limitations on the ability to estimate the parameters of a state generated by an unknown PQC, when given sample access to such a state.