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Quantum natural gradient without monotonicity

Toi Sasaki, Hideyuki Miyahara

TL;DR

The paper addresses the limitation that conventional quantum natural gradient (QNG) relies on monotone SLD-based geometry. By introducing non-monotone quantum Fisher metrics via the Petz framework and the rescaled sandwiched Rényi divergence, it demonstrates both analytically and numerically that non-monotone QNG can accelerate convergence compared to SLD-based QNG. The work also shows that diagonal approximations to the quantum Fisher metric retain these benefits and analyzes how metric ordering governs optimization speed. The results challenge the view that monotonicity is a necessary constraint for geometry in physics, with practical implications for faster parameter estimation on quantum devices and related Monte Carlo methods.

Abstract

Natural gradient (NG) is an information-geometric optimization method that plays a crucial role, especially in the estimation of parameters for machine learning models like neural networks. To apply NG to quantum systems, the quantum natural gradient (QNG) was introduced and utilized for noisy intermediate-scale devices. Additionally, a mathematically equivalent approach to QNG, known as the stochastic reconfiguration method, has been implemented to enhance the performance of quantum Monte Carlo methods. It is worth noting that these methods are based on the symmetric logarithmic derivative (SLD) metric, which is one of the monotone metrics. So far, monotonicity has been believed to be a guiding principle to construct a geometry in physics. In this paper, we propose generalized QNG by removing the condition of monotonicity. Initially, we demonstrate that monotonicity is a crucial condition for conventional QNG to be optimal. Subsequently, we provide analytical and numerical evidence showing that non-monotone QNG outperforms conventional QNG based on the SLD metric in terms of convergence speed.

Quantum natural gradient without monotonicity

TL;DR

The paper addresses the limitation that conventional quantum natural gradient (QNG) relies on monotone SLD-based geometry. By introducing non-monotone quantum Fisher metrics via the Petz framework and the rescaled sandwiched Rényi divergence, it demonstrates both analytically and numerically that non-monotone QNG can accelerate convergence compared to SLD-based QNG. The work also shows that diagonal approximations to the quantum Fisher metric retain these benefits and analyzes how metric ordering governs optimization speed. The results challenge the view that monotonicity is a necessary constraint for geometry in physics, with practical implications for faster parameter estimation on quantum devices and related Monte Carlo methods.

Abstract

Natural gradient (NG) is an information-geometric optimization method that plays a crucial role, especially in the estimation of parameters for machine learning models like neural networks. To apply NG to quantum systems, the quantum natural gradient (QNG) was introduced and utilized for noisy intermediate-scale devices. Additionally, a mathematically equivalent approach to QNG, known as the stochastic reconfiguration method, has been implemented to enhance the performance of quantum Monte Carlo methods. It is worth noting that these methods are based on the symmetric logarithmic derivative (SLD) metric, which is one of the monotone metrics. So far, monotonicity has been believed to be a guiding principle to construct a geometry in physics. In this paper, we propose generalized QNG by removing the condition of monotonicity. Initially, we demonstrate that monotonicity is a crucial condition for conventional QNG to be optimal. Subsequently, we provide analytical and numerical evidence showing that non-monotone QNG outperforms conventional QNG based on the SLD metric in terms of convergence speed.
Paper Structure (8 sections, 5 theorems, 30 equations, 3 figures)

This paper contains 8 sections, 5 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Natural gradient
  3. Quantum natural gradient
  4. Monotonicity
  5. Fisher metrics and speed of QNG
  6. Numerical simulations
  7. Conclusions
  8. Proof of Thm. \ref{['main_theorem_relation_Petz_function_quantum_Fisher_metrics_001_001']}

Key Result

Theorem 1

$f_\mathrm{SLD} (\cdot)$ and $f_\mathrm{rRLD} (\cdot)$ are the maximum and the minimum elements with respect to Eq. main_eq_def_order_operator_monotone_functions_001_001 under the condition of monotonicity, Eq. main_eq_def_operator_monotone_function_001_001, $f (1) = 1$, and $f (t) = t f (t^{-1})$. where $f_\mathrm{SLD} (\cdot)$ and $f_\mathrm{rRLD} (\cdot)$ are the Petz functions for the SLD and

Figures (3)

  • Figure 1: $f_\alpha (t)$, Eq. \ref{['main_eq_def_f_alpha_001_001']}, with $\alpha = 0.1, 0.3, 0.5, 100.0, -100.0, -1.0, -0.3, -0.1$. Note that $\alpha = 0.5$ and $\alpha = -1.0$ yield the SLD and rRLD metrics, respectively. The regime of the monotone Petz functions is highlighted by light cyan.
  • Figure 2: Cost functions for several $\alpha$ in the case of Eq. \ref{['main_eq_update_theta_001_001']}. We set $\epsilon = 1.0 \times 10^{-8}$, $\xi = 1.0 \times 10^{-3}$, and $\delta = 1.0 \times 10^{-3}$. The regime of the monotone metrics is highlighted by light cyan.
  • Figure 3: Cost functions for several $\alpha$ in the case of Eq. \ref{['main_eq_update_theta_001_002']}. We set $\eta = 5.0 \times 10^{-4}$, $\xi = 1.0 \times 10^{-3}$, and $\delta = 1.0 \times 10^{-3}$. The regime of the monotone metrics is highlighted by light cyan.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof