Information geometry of nonmonotonic quantum natural gradient

TL;DR

This work extends quantum natural gradient (QNG) beyond the conventional symmetric logarithmic derivative (SLD) metric by employing a family of Petz functions that generate both monotone and nonmonotone quantum Fisher metrics. It establishes that monotonicity makes the SLD metric locally optimal for convergence speed, while nonmonotone metrics can yield faster QNG convergence, particularly in non-full-rank or pure-state regimes via the Petz framework. The authors derive bra-ket formulations for full- and non-full-rank density operators, analyze the role of Petz functions in metric design, and demonstrate, through numerical simulations in quantum circuit learning, that nonmonotone QNG can outperform SLD-based QNG. The study also connects QNG to stochastic reconfiguration and discusses design principles for geometry through linear combinations of Petz functions, offering practical pathways to accelerate quantum optimization tasks.

Abstract

Natural gradient is an advanced optimization method based on information geometry, where the Fisher metric plays a crucial role. Its quantum counterpart, known as quantum natural gradient (QNG), employs the symmetric logarithmic derivative (SLD) metric, one of the quantum Fisher metrics. While quantization in physics is typically well-defined via the canonical commutation relations, the quantization of information-theoretic quantities introduces inherent arbitrariness. To resolve this ambiguity, monotonicity has been used as a guiding principle for constructing geometries in physics, as it aligns with physical intuition. Recently, a variant of QNG, which we refer to as nonmonotonic QNG in this paper, was proposed by relaxing the monotonicity condition. It was shown to achieve faster convergence compared to conventional QNG. In this paper, we investigate the properties of nonmonotonic QNG. To ensure the paper is self-contained, we first demonstrate that the SLD metric is locally optimal under the monotonicity condition and that non-monotone quantum Fisher metrics can lead to faster convergence in QNG. Previous studies primarily relied on a specific type of quantum divergence and assumed that density operators are full-rank. Here, we explicitly consider an alternative quantum divergence and extend the analysis to non-full-rank cases. Additionally, we explore how geometries can be designed using Petz functions, given that quantum Fisher metrics are characterized through them. Finally, we present numerical simulations comparing different quantum Fisher metrics in the context of parameter estimation problems in quantum circuit learning.

Figures (21)

• Figure 1: $f_\mathrm{SLD} (t) = \frac{1 + t}{2}$, $f_{(1/2)} (t) = t^\frac{1}{2}$, $f_\mathrm{BKM} (t) = \frac{t - 1}{\ln t}$, $f_\mathrm{rRLD} (t) = \frac{2 t}{1 + t}$. The regimes of the monotone Petz functions and the Petz functions associated with the rescaled sandwiched quantum Rényi divergence are highlighted by light cyan and light yellow, respectively.
• Figure 2: $f_\alpha^\mathrm{st} (t)$, Eq. \ref{['main_eq_def_tilde_f_alpha_001_001']}, with $\alpha = 4.0, 2.0, 1.010, 0.50, -0.010, -1.0, -3.0$. The regimes of the monotone Petz functions and the Petz functions associated with the rescaled sandwiched quantum Rényi divergence are highlighted by light cyan and light yellow, respectively.
• Figure 3: $f_\alpha^\mathrm{sw} (t)$, 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 regimes of the monotone Petz functions and the Petz functions associated with the rescaled sandwiched quantum Rényi divergence are highlighted by light cyan and light yellow, respectively.
• Figure 4: $f_\alpha^\mathrm{sw} (t)$, Eq. \ref{['main_eq_def_f_alpha_001_001']}, with $\alpha = 0+, 0.5, \pm \infty, -1.0, -0.3, 0-$. Note that $\alpha = 0.5$ and $\alpha = -1.0$ yield the SLD and rRLD metrics, respectively. The regimes of the monotone Petz functions and the Petz functions associated with the rescaled sandwiched quantum Rényi divergence are highlighted by light cyan and light yellow, respectively.
• Figure 5: Value of $f_\alpha^\mathrm{sw} (t)$, Eq. \ref{['main_eq_def_f_alpha_001_001']}, for small $t$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• proof
• Lemma 7