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arXiv:2511.21651

Pauli propagation enables fast classical simulation of strongly correlated quantum systems

Chinmay Shrikhande, Arnab Bachhar, Aaron Rodriguez Jimenez, Edward F. Valeev, Nicholas J. Mayhall

quant-phphysics.chem-phphysics.comp-phSubmitted Nov 26, 2025• Announced Nov 27, 2025• 0 citations• Revised Sep 22, 2026 (v3)
v1 · Nov 26, 2025v2 · May 31, 2026v3 · Sep 22, 2026
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Abstract

Ground state energy estimation for strongly correlated quantum systems remains a central challenge in computational physics and chemistry. While tensor network methods like DMRG provide efficient solutions for one-dimensional systems, higher-dimensional problems remain difficult. Here we present a variational double bracket flow (vDBF) algorithm that leverages Pauli Propagation, a technique originally developed for classical simulation of quantum circuits, to efficiently approximate ground state energies. By combining greedy operator selection with coefficient-based fluctuation truncation and energy-variance extrapolation, we obtain sub-1% relative errors for Heisenberg and Hubbard lattices of up to 128 qubits. For a 10$\times$10 Heisenberg lattice (100 qubits), vDBF obtains accurate results in approximately 2 minutes on a single CPU thread, compared to over 50 hours on 64 threads for DMRG. Accurate results are also obtained for systems that are difficult for quantum Monte Carlo: on the maximally frustrated $8\times8$ $J_1$-$J_2$ lattice and the hole-doped $8\times8$ Hubbard model, the tightest thresholds extrapolate to within a few tenths of a percent of the best published variational estimates. We further test vDBF on the 84-qubit $π$-valence active space of hexabenzocoronene, where the most reliable extrapolated correlation energies agree with DMRG to within 1%. These results demonstrate that classical simulation techniques developed in the context of quantum advantage benchmarking can provide practical tools for many-body physics.

Comments
This revision adds molecular example and clarifies fluctuation truncation

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Ground state energy estimation for strongly correlated quantum systems remains a central challenge in computational physics and chemistry. While tensor network methods like DMRG provide efficient solutions for one-dimensional systems, higher-dimensional problems remain difficult. Here we present a variational double bracket flow (vDBF) algorithm that leverages Pauli Propagation, a technique originally developed for classical simulation of quantum circuits, to efficiently approximate ground state energies. By combining greedy operator selection with coefficient-based fluctuation truncation and energy-variance extrapolation, we obtain sub-1% relative errors for Heisenberg and Hubbard lattices of up to 128 qubits. For a 10×\times×10 Heisenberg lattice (100 qubits), vDBF obtains accurate results in approximately 2 minutes on a single CPU thread, compared to over 50 hours on 64 threads for DMRG. Accurate results are also obtained for systems that are difficult for quantum Monte Carlo: on the maximally frustrated 8×88\times88×8 J1J_1J1​-J2J_2J2​ lattice and the hole-doped 8×88\times88×8 Hubbard model, the tightest thresholds extrapolate to within a few tenths of a percent of the best published variational estimates. We further test vDBF on the 84-qubit πππ-valence active space of hexabenzocoronene, where the most reliable extrapolated correlation energies agree with DMRG to within 1%. These results demonstrate that classical simulation techniques developed in the context of quantum advantage benchmarking can provide practical tools for many-body physics.
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Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32\ifmmode\times\else\texttimes\fi{}32$ sites, as well as a highly frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.
Simulating the Hubbard model is of great interest to a wide range of applications within condensed matter physics, however its solution on classical computers remains challenging in dimensions larger than one. The relative simplicity of this model, embodied by the sparseness of the Hamiltonian matrix, allows for its efficient implementation on quantum computers, and for its approximate solution using variational algorithms such as the variational quantum eigensolver. While these algorithms have been shown to reproduce the qualitative features of the Hubbard model, their quantitative accuracy in terms of producing true ground state energies and other properties, and the dependence of this accuracy on the system size and interaction strength, the choice of variational ansatz, and the degree of spatial inhomogeneity in the model, remains unknown. Here we present a rigorous classical benchmarking study, demonstrating the potential impact of these factors on the accuracy of the variational solution of the Hubbard model on quantum hardware, for systems with up to 32 qubits. We find that even when using the most accurate wavefunction ansätze for the Hubbard model, the error in its ground state energy and wavefunction plateaus for larger lattices, while stronger electronic correlations magnify this issue. Concurrently, spatially inhomogeneous parameters and the presence of off-site Coulomb interactions only have a small effect on the accuracy of the computed ground state energies. Our study highlights the capabilities and limitations of current approaches for solving the Hubbard model on quantum hardware, and we discuss potential future avenues of research.
Solving the ground state of quantum many-body systems remains a fundamental challenge in physics and chemistry. Recent advancements in quantum hardware have opened new avenues for addressing this challenge. Inspired by the quantum-enhanced Markov chain Monte Carlo (QeMCMC) algorithm, which was originally designed for sampling the Boltzmann distribution of classical spin models using quantum computers, we introduce a quantum-assisted variational Monte Carlo (QA-VMC) algorithm for solving the ground state of quantum many-body systems by adapting QeMCMC to sample the distribution of a (neural-network) wave function in VMC. The central question is whether such a quantum-assisted proposal can potentially offer a computational advantage over classical methods. Through numerical investigations for the Fermi-Hubbard model and molecular systems, we demonstrate that the quantum-assisted proposal exhibits larger absolute spectral gaps and reduced autocorrelation times compared to conventional classical proposals, leading to more efficient sampling and faster convergence to the ground state in VMC as well as a more accurate and precise estimation of physical observables. This advantage is especially pronounced for specific parameter ranges, where the ground-state configurations are more concentrated in some configurations separated by large Hamming distances. Our results underscore the potential of quantum-assisted algorithms to enhance classical variational methods for solving the ground state of quantum many-body systems.
The variational quantum eigensolver (VQE) is a promising algorithm for demonstrating quantum advantage in the noisy intermediate-scale quantum (NISQ) era. However, optimizing VQE from random initial starting parameters is challenging due to a variety of issues including barren plateaus, optimization in the presence of noise, and slow convergence. While simulating quantum circuits classically is generically difficult, classical computing methods have been developed extensively, and powerful tools now exist to approximately simulate quantum circuits. This opens up various strategies that limit the amount of optimization that needs to be performed on quantum hardware. Here we present and benchmark an approach where we find good starting parameters for parameterized quantum circuits by classically simulating VQE by approximating the parameterized quantum circuit (PQC) as a matrix product state (MPS) with a limited bond dimension. Calling this approach the variational tensor network eigensolver (VTNE), we apply it to the 1D and 2D Fermi-Hubbard model with system sizes that use up to 32 qubits. We find that in 1D, VTNE can find parameters for PQC whose energy error is within 0.5% relative to the ground state. In 2D, the parameters that VTNE finds have significantly lower energy than their starting configurations, and we show that starting VQE from these parameters requires non-trivially fewer operations to come down to a given energy. The higher the bond dimension we use in VTNE, the less work needs to be done in VQE. By generating classically optimized parameters as the initialization for the quantum circuit one can alleviate many of the challenges that plague VQE on quantum computers.