[Paper Review] Group Convolutional Neural Networks Improve Quantum State Accuracy

Christopher G. Roth, A. H. MacDonald|arXiv (Cornell University)|Apr 11, 2021

Quantum many-body systems39 references26 citations

TL;DR

The paper shows that group equivariant convolutional neural networks (G-CNNs) improve ground-state energy accuracy for frustrated Heisenberg models on square and triangular lattices, achieving state-of-the-art VMC energies on the triangular lattice without extra memory overhead.

ABSTRACT

Neural networks are a promising tool for simulating quantum many body systems. Recently, it has been shown that neural network-based models describe quantum many body systems more accurately when they are constrained to have the correct symmetry properties. In this paper, we show how to create maximally expressive models for quantum states with specific symmetry properties by drawing on literature from the machine learning community. We implement group equivariant convolutional networks (G-CNN) \cite{cohen2016group}, and demonstrate that performance improvements can be achieved without increasing memory use. We show that G-CNNs achieve very good accuracy for Heisenberg quantum spin models in both ordered and spin liquid regimes, and improve the ground state accuracy on the triangular lattice over other variational Monte-Carlo methods.

Motivation & Objective

• Motivate the use of symmetry-respecting neural network states to improve accuracy in quantum many-body simulations.
• Develop maximally expressive wavefunction models by leveraging group equivariant convolutions over discrete symmetry groups (wallpaper group).
• Demonstrate that G-CNNs outperform symmetry-averaged models and previous neural-network quantum states on J1-J2 Heisenberg models.

Proposed method

• Adopt group equivariant convolutional networks (G-CNNs) with equivariant convolutions over the full wallpaper group (e.g., p6m for triangular lattices).
• Construct wavefunctions by feeding lattice spin configurations through stacked equivariant layers and nonlinearities, then combine per-group embeddings with symmetry characters to form the final state amplitude.
• Use SELU activations on real/imaginary parts to stabilize training and optimize via gradient-based methods (Adam).
• Compare full G-CNNs against symmetry-averaged (diagonal/masked) variants to quantify the benefit of full group connectivity.
• Demonstrate mapping between G-CNNs and symmetry-averaged linear models by filter masking to illustrate model expressiveness differences.

Experimental results

Research questions

• RQ1 Can group equivariant convolutions over discrete symmetry groups yield more accurate quantum state approximations than symmetry-averaged models?
• RQ2 How does incorporating full wallpaper group symmetry (translations, rotations, reflections) affect ground-state energies of frustrated Heisenberg models on square and triangular lattices?
• RQ3 What is the impact of off-diagonal vs. diagonal filter connectivity on model performance and training efficiency?
• RQ4 Do G-CNNs achieve state-of-the-art variational Monte Carlo energies for challenging spin-liquid regimes?

Key findings

• G-CNNs achieve very good accuracy for Heisenberg models in both ordered and spin-liquid regimes.
• On the 6x6 triangular lattice, G-CNNs attain approximately -0.55922 (J2=0) and -0.51365 (J2=J1/8), improving over several VMC methods and approaching exact/DMRG references.
• For J2=J1/8 on the triangular lattice, G-CNN energies surpass those of many VMC variants and are competitive with or better than several baselines without extra Lanczos steps.
• Full connectivity (against masking) yields better energies than symmetry-averaged variants, and translational connectivity has notably strong impact due to parameter scaling.
• On the square lattice, G-CNNs show competitive to superior ground-state energies across tested regimes, with favorable memory-to-parameter ratios compared to some competing methods.