Quantum natural gradient with thermal-state initialization

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.

arXiv commentv2: 10 pages of main text, 13 pages of appendices, 3 figures

References in corpus (54)

• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Barren plateaus in quantum neural network training landscapes
• The theory of variational hybrid quantum-classical algorithms
• Quantum Circuit Learning
• Quantum Algorithms Revisited
• Evaluating analytic gradients on quantum hardware
• An adaptive variational algorithm for exact molecular simulations on a quantum computer
• Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms
• From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
• Quantum Fisher information matrix and multiparameter estimation
• Efficient Learning for Deep Quantum Neural Networks
• Challenges and Opportunities in Quantum Machine Learning
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Toward the first quantum simulation with quantum speedup
• Variational ansatz-based quantum simulation of imaginary time evolution
• Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
• Quantum generative adversarial networks
• Quantum Natural Gradient
• Training variational quantum algorithms is NP-hard
• Robust data encodings for quantum classifiers
• The Expressive Power of Parameterized Quantum Circuits
• A Geometric Perspective on Quantum Parameter Estimation
• Exploring entanglement and optimization within the Hamiltonian Variational Ansatz
• Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps
• Barren Plateaus in Variational Quantum Computing
• Stochastic gradient descent for hybrid quantum-classical optimization
• Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer
• Metric adjusted skew information
• Fisher Information in Noisy Intermediate-Scale Quantum Applications
• Wigner-Yanase information on quantum state space:the geometric approach
• Expressibility of the alternating layered ansatz for quantum computation
• Classical Optimizers for Noisy Intermediate-Scale Quantum Devices
• A Comparison of Various Classical Optimizers for a Variational Quantum Linear Solver
• Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule
• Variational Quantum Algorithm for Estimating the Quantum Fisher Information
• Using models to improve optimizers for variational quantum algorithms
• Quantum Fisher information and symmetric logarithmic derivative via anti-commutators
• Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations
• Quantum natural gradient generalised to noisy and non-unitary circuits
• Optimal learning of quantum Hamiltonians from high-temperature Gibbs states
• Sub-Quantum Fisher Information
• On the Sample Complexity of Quantum Boltzmann Machine Learning
• Practical Black Box Hamiltonian Learning
• Variational Quantum Algorithms for Semidefinite Programming
• Geometric approach to quantum statistical inference
• Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes
• Single-component gradient rules for variational quantum algorithms
• Quantum computational advantage with constant-temperature Gibbs sampling
• Quantum Natural Policy Gradients: Towards Sample-Efficient Reinforcement Learning
• QSlack: A slack-variable approach for variational quantum semi-definite programming
• Estimation of Quantum Fisher Information via Stein's Identity in Variational Quantum Algorithms