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arXiv:2608.27861

Vectorized Symmetric and Fermionic Tensor Network Implementations for GPU-Accelerated Variational Monte Carlo

Si-Jing Du, Johnnie Gray, Ao Chen, Garnet Kin-Lic Chan

physics.comp-phcond-mat.str-elquant-phSubmitted Aug 28, 2026• Announced Aug 31, 2026• 0 citations
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Abstract

Variational Monte Carlo (VMC) calculations based on tensor networks (TN) have recently achieved competitive accuracy in ground-state calculations of strongly correlated spin and fermionic systems. However, existing tensor network VMC (TN-VMC) algorithms have not been formulated in a manner that can fully utilize GPU acceleration. We tackle the key missing ingredient, namely, the vectorized evaluation of tensor network amplitudes and tensor network operations. This ensures high GPU utilization by batching over computations with identical structure. In particular, we show how to achieve vectorization for the practically relevant case of abelian symmetric tensor networks (which includes fermionic tensor networks) by developing a ``flat'' tensor network formalism for block-sparse tensor representation and contraction. Using this, we construct a GPU-adapted symmetric TN-VMC workflow with batched tensor network computation. In the two-dimensional Fermi--Hubbard model, we demonstrate a GPU speedup of up to $300 \times$ over single core CPU implementations, for fermionic TN variational wavefunctions.

Comments
28 pages, 9 figures

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Variational Monte Carlo (VMC) calculations based on tensor networks (TN) have recently achieved competitive accuracy in ground-state calculations of strongly correlated spin and fermionic systems. However, existing tensor network VMC (TN-VMC) algorithms have not been formulated in a manner that can fully utilize GPU acceleration. We tackle the key missing ingredient, namely, the vectorized evaluation of tensor network amplitudes and tensor network operations. This ensures high GPU utilization by batching over computations with identical structure. In particular, we show how to achieve vectorization for the practically relevant case of abelian symmetric tensor networks (which includes fermionic tensor networks) by developing a ``flat'' tensor network formalism for block-sparse tensor representation and contraction. Using this, we construct a GPU-adapted symmetric TN-VMC workflow with batched tensor network computation. In the two-dimensional Fermi--Hubbard model, we demonstrate a GPU speedup of up to 300×300 \times300× over single core CPU implementations, for fermionic TN variational wavefunctions.
The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real-space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum, and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to a hole-doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak ddd-wave superconducting order in the ground state for the doping concentration of less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.
Abstract In recent years, variational Monte Carlo (VMC) calculations of projected entangled pair states (PEPS) have emerged as a competitive method for computing the ground states of many-body quantum systems. This approach is particularly important for fermionic systems, where sign problems are prevalent. We derive and explain the algorithms for VMC calculations of fermionic PEPS within the swap-gates formulation. As a separate key result, we prove the detailed balance for sequential sampling of tensor networks.
Efficient sampling of two-dimensional statistical physics systems remains a central challenge in computational statistical physics. Traditional Markov chain Monte Carlo (MCMC) methods, including cluster algorithms, provide only partial solutions, as their efficiency collapses for large systems in the presence of frustration and quenched disorder. The recently proposed Tensor Network Monte Carlo (TNMC) method offers a promising alternative, yet its original implementation suffers from inefficiencies due to the lack of scalable parallel sampling. In this work, we introduce BatchTNMC, a GPU-optimized and parallelized implementation of TNMC tailored for large-scale simulations of two-dimensional spin glasses. By leveraging batch processing and parallel sampling across multiple disorder realizations, our implementation achieves speedups of up to five orders of magnitude compared with the original serial scheme. Benchmarking on two-dimensional spin glasses demonstrates dramatic gains in efficiency: for instance, on a single GPU, BatchTNMC concurrently produces 1000 uncorrelated and unbiased samples across 1000 disorder realizations on 1024×10241024\times 10241024×1024 lattices in just 3.3 hours, with an acceptance probability of 37%. These results establish BatchTNMC as a scalable and powerful computational framework for the study of two-dimensional disordered spin glass systems.
Simulating strongly correlated fermionic systems remains a fundamental challenge in quantum physics, largely due to the sign problem in quantum Monte Carlo (QMC) methods. We present a neural network-based variational Monte Carlo (NN-VMC) approach, leveraging a flexible neural network ansatz to represent the many-body wavefunction. Focusing on quantum dots with up to 30 electrons, we demonstrate that NN-VMC significantly reduces variational bias and achieves ground-state energies surpassing those of fixed-node diffusion Monte Carlo (DMC). A key feature is that the neural network adaptively learns and optimizes nodal structures during energy minimization. We provide qualitative insights into the nodal structure of fermionic wavefunctions by comparing the nodal structures generated by NN-VMC with those obtained from traditional trial functions. Additionally, we reveal spin-resolved radial distributions and electron density profiles, highlighting the versatility and accuracy of NN-VMC. This work underscores the potential of machine learning to advance quantum simulations and deepen our understanding of strongly correlated systems.
We present novel algorithmic solutions together with implementation details utilizing non-Abelian symmetries in order to boost the current limits of tensor network state algorithms on high performance computing infrastructure. In our in-house developed hybrid CPU-multiGPU solution scheduling is decentralized, threads are autonomous and inter-thread communications are solely limited to interactions with globally visible lock-free constructs. Our custom tailored virtual memory management ensures data is produced with high spatial locality, which together with the use of specific sequences of strided batched matrix operations translates to significantly higher overall throughput. In order to lower IO overhead, an adaptive buffering technique is used to dynamically match the level of data abstraction, at which cache repositories are built and reused, to system resources. The non-Abelian symmetry related tensor algebra based on Wigner-Eckhart theorem is fully detached from the conventional tensor network layer, thus massively parallel matrix and tensor operations can be performed without additional overheads. Altogether, we have achieved an order of magnitude increase in performance with respect to results reported in arXiv:2305.05581 in terms of computational complexity and at the same time a factor of three to six in the actual performance measured in TFLOPS. Benchmark results are presented on Hilbert space dimensions up to 2.88×10362.88\times10^{36}2.88×1036 obtained via large-scale SU(2) spin adapted density matrix renormalization group simulations on selected strongly correlated molecular systems. These demonstrate the utilization of NVIDIA's highly specialized tensor cores, leading to performance around 110 TFLOPS on a single node supplied with eight NVIDIA A100 devices. In comparison to U(1) implementations with matching accuracy, our solution has an estimated effective performance of 250-500 TFLOPS.
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.