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Quantum Natural Gradient (1909.02108v3)

Published 4 Sep 2019 in quant-ph, cs.LG, and stat.ML

Abstract: A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.

Citations (365)

Summary

  • The paper presents a novel optimization method that adapts natural gradient descent to quantum variational circuits using the Fubini-Study metric.
  • It proposes an efficient block-diagonal approximation of the quantum geometric tensor, significantly accelerating convergence on NISQ devices.
  • Numerical experiments demonstrate faster convergence compared to traditional methods like vanilla gradient descent and Adam, underscoring its practical impact.

Quantum Natural Gradient: An Overview

This article presents an innovative approach to optimizing variational quantum circuits via the Quantum Natural Gradient (QNG), offering a quantum adaptation of Natural Gradient Descent. The core of this method involves leveraging the Quantum Information Geometry, specifically the real part of the Quantum Geometric Tensor (QGT), also recognized as the Fubini-Study metric tensor. The authors introduce an efficient algorithm to approximate this tensor in a block-diagonal form, focusing on its application within the field of Noisy Intermediate-Scale Quantum (NISQ) devices.

Background and Motivation

Variational quantum circuits are integral for many quantum-classical hybrid algorithms, notably the Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and Quantum Neural Networks (QNNs). These circuits are particularly suited for NISQ devices due to their reliance on variational parameters and stochastic optimization methods. Traditional optimization techniques, including derivative-free methods and first-order gradient evaluations, have been limited by the challenges of parameter space dimensionality and hyperparameter tuning. In contrast, the QNG seeks to address these limitations by considering the intrinsic geometry of quantum states.

Contributions of the Paper

The paper makes several key contributions to quantum optimization:

  • Reparametrization Invariance: The method introduces a Riemannian metric tensor on quantum state space, ensuring invariance concerning arbitrary reparametrizations. The Fubini-Study metric helps define a descent direction independently of chosen parametrizations.
  • Efficient Computation of the QGT: The article proposes a quantum circuit methodology for approximating the QGT, focusing on a block-diagonal approach that outperforms traditional gradient descent strategies.
  • Numerical Validation: Through numerical experiments, the QNG is empirically shown to converge faster than methods like vanilla gradient descent and Adam, particularly in high-dimensional parameter spaces.

Implications and Future Directions

The implications of this work are substantial for both theory and practice in the field of quantum computing. The QNG aligns closely with the natural gradient descent known in the field of statistics, underscoring the foundational interplay between quantum and classical optimization frameworks. Practically, the algorithm shows promise for implementation on NISQ devices, improving the efficiency and scalability of quantum circuit optimization.

Future directions could explore further integration with non-local averaging methods, such as Adam, to enhance convergence properties. Additionally, extending the analysis to noise-affected quantum systems could involve examining geometries beyond pure states, like the Bures metric. This pivot could introduce robustness within noisy quantum environments, broadening the applicability of QNG.

In conclusion, the Quantum Natural Gradient provides a compelling framework for optimizing variational quantum circuits, leveraging quantum geometric properties to transcend conventional optimization limitations. The proposed method stands as a formidable technique with notable potential for advancing quantum computation, particularly within the operational scope of NISQ-era devices.

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