[Paper Review] Autoregressive Neural Quantum States with Quantum Number Symmetries

Aleksei Malyshev, Juan Miguel Arrazola+1 more|ArXiv.org|Oct 6, 2023

Machine Learning in Materials Science4 citations

TL;DR

This paper introduces a general framework to incorporate arbitrary quantum number symmetries—such as particle number, spin projection, and molecular spatial symmetries—into autoregressive neural quantum states (ANQS). By enabling early termination of unphysical sampling paths, the method achieves chemical accuracy across multiple molecules with over an order of magnitude speedup compared to prior ANQS-based electronic structure calculations.

ABSTRACT

Neural quantum states have established themselves as a powerful and versatile family of ansatzes for variational Monte Carlo simulations of quantum many-body systems. Of particular prominence are autoregressive neural quantum states (ANQS), which enjoy the expressibility of deep neural networks, and are equipped with a procedure for fast and unbiased sampling. Yet, the non-selective nature of autoregressive sampling makes incorporating quantum number symmetries challenging. In this work, we develop a general framework to make the autoregressive sampling compliant with an arbitrary number of quantum number symmetries. We showcase its advantages by running electronic structure calculations for a range of molecules with multiple symmetries of this kind. We reach the level of accuracy reported in previous works with more than an order of magnitude speedup and achieve chemical accuracy for all studied molecules, which is a milestone unreported so far. Combined with the existing effort to incorporate space symmetries, our approach expands the symmetry toolbox essential for any variational ansatz and brings the ANQS closer to being a competitive choice for studying challenging quantum many-body systems.

Motivation & Objective

• To address the challenge of efficiently incorporating multiple quantum number symmetries into autoregressive neural quantum states (ANQS), which traditionally lack built-in symmetry constraints.
• To overcome the inefficiency of post-selection in autoregressive sampling by enabling early detection of unphysical partial basis vectors.
• To extend the applicability of ANQS to molecular systems by including spatial $π$-symmetries (Z₂ symmetries) not previously used in NQS-based quantum chemistry.
• To significantly improve computational efficiency and accuracy in variational Monte Carlo simulations of electronic structure.
• To demonstrate that combining space and quantum number symmetries in ANQS brings them closer to the performance of conventional quantum chemistry methods.

Proposed method

• Proposes a pruning algorithm that checks, at each autoregressive sampling step, whether a partially sampled basis vector can be extended into a physically allowed symmetry sector.
• Uses a recursive condition based on cumulative quantum numbers (e.g., total particle number, spin projection, and spatial symmetry quantum numbers) to determine if a partial configuration is unphysical.
• Applies the method to ANQS by modifying the autoregressive sampling process to skip unphysical branches early, reducing wasted computation.
• Introduces two pruning strategies—MU-2 and DU—where MU-2 is shown to be more robust by preserving more unique samples and avoiding sampling stalls.
• Employs a symmetry-aware sampling procedure that maintains fast, unbiased sampling while ensuring all generated samples respect the desired symmetry sectors.
• Integrates the symmetry-aware ANQS into variational Monte Carlo with local energy estimators to optimize the ansatz parameters efficiently.

Experimental results

Research questions

• RQ1 Can autoregressive neural quantum states be efficiently constrained to respect multiple quantum number symmetries without sacrificing sampling speed or bias?
• RQ2 How can unphysical partial configurations in autoregressive sampling be detected early to avoid unnecessary computation?
• RQ3 What is the impact of including molecular spatial symmetries (Z₂ symmetries) on the accuracy and convergence speed of ANQS in electronic structure calculations?
• RQ4 How do different pruning strategies (MU-2 vs. DU) affect the number of unique samples and optimization stability in symmetry-constrained ANQS?
• RQ5 To what extent can symmetry-aware ANQS achieve chemical accuracy and speedup in electronic structure simulations compared to previous NQS methods?

Key findings

• The proposed symmetry-aware ANQS framework achieves chemical accuracy (≤1.6×10⁻³ Ha) for all studied molecules, including N₂, C₂, and Li₂O, which was previously unreported in ANQS-based calculations.
• For N₂ and C₂, the method achieves convergence to chemical accuracy in less than one-tenth the time of prior ANQS methods, demonstrating over an order of magnitude speedup.
• The MU-2 pruning strategy outperforms DU by preserving more unique samples and reducing the risk of sampling stalls, especially in early optimization stages.
• The method reduces the number of wasted samples by detecting unphysical partial configurations early, leading to more efficient use of computational resources.
• The inclusion of molecular spatial symmetries (Z₂) via the framework enables ANQS to access the same computational subspace as traditional quantum chemistry methods, significantly improving accuracy.
• Despite similar per-iteration runtime, the improved convergence rate leads to substantial overall time savings, with Li₂O reaching chemical accuracy in 5.1 hours versus 45.6 hours in prior work.