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When Less is More: Approximating the Quantum Geometric Tensor with Block Structures

Ahmedeo Shokry, Alessandro Santini, Filippo Vicentini

TL;DR

The paper tackles the high cost of inverting the quantum geometric tensor (QGT) in natural-gradient optimization for neural quantum states. It introduces a block-diagonal QGT that partitions the metric by network modules, preserving intra-layer curvature while removing noisy cross-layer correlations. This approach improves conditioning and scalability, with empirical gains in faster convergence, lower energies, and greater stability on Heisenberg and frustrated $J_1$-$J_2$ models. The results suggest a practical, scalable alternative to full SR that can extend to other differentiable scientific simulators and modular neural architectures.

Abstract

The natural gradient is central in neural quantum states optimizations but it is limited by the cost of computing and inverting the quantum geometric tensor, the quantum analogue of the Fisher information matrix. We introduce a block-diagonal quantum geometric tensor that partitions the metric by network layers, analogous to block-structured Fisher methods such as K-FAC. This layer-wise approximation preserves essential curvature while removing noisy cross-layer correlations, improving conditioning and scalability. Experiments on Heisenberg and frustrated $J_1$-$J_2$ models show faster convergence, lower energy, and improved stability.

When Less is More: Approximating the Quantum Geometric Tensor with Block Structures

TL;DR

The paper tackles the high cost of inverting the quantum geometric tensor (QGT) in natural-gradient optimization for neural quantum states. It introduces a block-diagonal QGT that partitions the metric by network modules, preserving intra-layer curvature while removing noisy cross-layer correlations. This approach improves conditioning and scalability, with empirical gains in faster convergence, lower energies, and greater stability on Heisenberg and frustrated - models. The results suggest a practical, scalable alternative to full SR that can extend to other differentiable scientific simulators and modular neural architectures.

Abstract

The natural gradient is central in neural quantum states optimizations but it is limited by the cost of computing and inverting the quantum geometric tensor, the quantum analogue of the Fisher information matrix. We introduce a block-diagonal quantum geometric tensor that partitions the metric by network layers, analogous to block-structured Fisher methods such as K-FAC. This layer-wise approximation preserves essential curvature while removing noisy cross-layer correlations, improving conditioning and scalability. Experiments on Heisenberg and frustrated - models show faster convergence, lower energy, and improved stability.

Paper Structure

This paper contains 7 sections, 4 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Natural Gradient Descent for NQS and Block-Layer QGT
  3. Block-layer QGT.
  4. Numerical Results
  5. Heisenberg Chain.
  6. Frustrated $J_1\text{–}J_2$ model.
  7. Conclusions

Figures (2)

  • Figure 1: Comparison of block-layer and full QGT optimization for the $L=16$ Heisenberg chain. (a) Relative energy error; (b) infidelity to the exact ground state; (c) normalized QGT spectra at convergence; (d) correlation between approximate and exact QGT eigenvalues.
  • Figure 2: Comparison of block-layer and full QGT optimization in the $J_1\text{–}J_2$ model at $J_2/J_1=0.5$. (a): $6\times6$ lattice—energy error vs. GPU time. (b): $10\times10$ lattice—energy error vs. samples. Blue (red) shaded background, faster NTK (block-layer QGT) convergence.