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arXiv:2601.20782

Neural Quantum States in Mixed Precision

Massimo Solinas, Agnes Valenti, Nawaf Bou-Rabee, Roeland Wiersema

quant-phcond-mat.str-elcs.LGSubmitted Jan 28, 2026• Announced Jan 29, 2026• 0 citations
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Abstract

Scientific computing has long relied on double precision (64-bit floating point) arithmetic to guarantee accuracy in simulations of real-world phenomena. However, the growing availability of hardware accelerators such as Graphics Processing Units (GPUs) has made low-precision formats attractive due to their superior performance, reduced memory footprint, and improved energy efficiency. In this work, we investigate the role of mixed-precision arithmetic in neural-network based Variational Monte Carlo (VMC), a widely used method for solving computationally otherwise intractable quantum many-body systems. We first derive general analytical bounds on the error introduced by reduced precision on Metropolis-Hastings MCMC, and then empirically validate these bounds on the use-case of VMC. We demonstrate that significant portions of the algorithm, in particular, sampling the quantum state, can be executed in half precision without loss of accuracy. More broadly, this work provides a theoretical framework to assess the applicability of mixed-precision arithmetic in machine-learning approaches that rely on MCMC sampling. In the context of VMC, we additionally demonstrate the practical effectiveness of mixed-precision strategies, enabling more scalable and energy-efficient simulations of quantum many-body systems.

Journal reference
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:114287-114310, 2026
Comments
22 pages, 12 figures

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Scientific computing has long relied on double precision (64-bit floating point) arithmetic to guarantee accuracy in simulations of real-world phenomena. However, the growing availability of hardware accelerators such as Graphics Processing Units (GPUs) has made low-precision formats attractive due to their superior performance, reduced memory footprint, and improved energy efficiency. In this work, we investigate the role of mixed-precision arithmetic in neural-network based Variational Monte Carlo (VMC), a widely used method for solving computationally otherwise intractable quantum many-body systems. We first derive general analytical bounds on the error introduced by reduced precision on Metropolis-Hastings MCMC, and then empirically validate these bounds on the use-case of VMC. We demonstrate that significant portions of the algorithm, in particular, sampling the quantum state, can be executed in half precision without loss of accuracy. More broadly, this work provides a theoretical framework to assess the applicability of mixed-precision arithmetic in machine-learning approaches that rely on MCMC sampling. In the context of VMC, we additionally demonstrate the practical effectiveness of mixed-precision strategies, enabling more scalable and energy-efficient simulations of quantum many-body systems.
Artificial neural networks were recently shown to be an efficient representation of highly entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in variational Monte Carlo method, most notably the use of Markov-chain Monte Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized neural-network architecture that supports efficient and exact sampling, completely circumventing the need for Markov-chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcasing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states.
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Quantum many-body computation is a central challenge across multiple domains of scientific research, and Monte Carlo method has emerged as one of the most effective numerical techniques for investigating large-scale many-body systems. In recent years, the integration of Monte Carlo methods with machine learning has led to the development of neural quantum state approach. Quantum neural networks (QNNs) offer superior expressiveness compared to classical neural networks. In this paper, we propose a novel Quantum Neural Network Quantum State (QNNQS) method, based on the variational Monte Carlo framework. This approach leverages a quantum neural network to represent the many-body quantum state, and optimizes it iteratively through gradient descent to determine the ground-state energy. To assess the performance of this method, we conduct simulations of the one-dimensional transverse-field Ising model, investigating the influence of quantum circuit architecture on circuit expressibility, entangling capability, and computational accuracy. The results demonstrate that the QNNQS method achieves comparable simulation accuracy while utilizing significantly fewer parameters and requiring less training resources than classical network quantum state methods.
Neural Network Variational Monte Carlo (NNVMC) has emerged as a promising paradigm for solving quantum many-body problems by combining variational Monte Carlo with expressive neural-network wave-function ansätze. Although NNVMC can achieve competitive accuracy with favorable asymptotic scaling, practical deployment remains limited by high runtime and memory cost on modern graphics processing units (GPUs). Compared with language and vision workloads, NNVMC execution is shaped by physics-specific stages, including Markov-Chain Monte Carlo sampling, wave-function construction, and derivative/Laplacian evaluation, which produce heterogeneous kernel behavior and nontrivial bottlenecks. This paper provides a workload-oriented survey and empirical GPU characterization of four representative ansätze: PauliNet, FermiNet, Psiformer, and Orbformer. Using a unified profiling protocol, we analyze model-level runtime and memory trends and kernel-level behavior through family breakdown, arithmetic intensity, roofline positioning, and hardware utilization counters. The results show that end-to-end performance is often constrained by low-intensity elementwise and data-movement kernels, while the compute/memory balance varies substantially across ansätze and stages. Based on these findings, we discuss algorithm--hardware co-design implications for scalable NNVMC systems, including phase-aware scheduling, memory-centric optimization, and heterogeneous acceleration.
The variational quantum Monte Carlo (VQMC) method has received significant attention because of its ability to overcome the curse of dimensionality inherent in many-body quantum systems, by representing the exponentially complex quantum states variationally with machine learning models. We develop novel training strategies to improve the scalability of VQMC, and build parallelization frameworks for solving large-scale problems. The application of our method is extended to quantum chemistry and financial derivative pricing. For quantum chemistry, we build a pre-processing pipeline serving as an interface connecting molecular information and VQMC, and achieve remarkable performance in comparison with the classical approximate methods. On the other hand, we present a simple generalization of VQMC applicable to arbitrary linear PDEs, showcasing the technique in the Black-Scholes equation for pricing European contingent claims dependent on many underlying assets. We also introduce meta-learning and multi-fidelity active learning as exotic components to VQMC, which, under some reasonable assumptions on the problem formulation, can further improve the convergence and the sampling efficiency of our method.