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arXiv:2609.21631

Reformulating Pfaffian Quantum Monte Carlo with the Hybrid Monte Carlo formalism

Thomas Hauschild, Lin Wang, Finn L. Temmen, Thomas Luu

Published Sep 18, 2026• 0 citations
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Abstract

Pfaffian quantum Monte Carlo extends auxiliary field methods to anomalous fermionic Hamiltonians, opening access to pairing systems and Hubbard--Stratonovich decompositions that are unavailable in determinant formulations. We show that this flexibility can be combined with the global updates of Hybrid Monte Carlo (HMC), leading to an algorithm scaling as $\mathcal{O}(N_τL^3)$, similar to its determinant quantum Monte Carlo (DQMC) counterpart without pseudofermions. The central observation is that the molecular dynamics force depends only on the magnitude of the Pfaffian weight, which admits a determinant representation and leads to a simplified force expression. This construction also yields unequal-time Majorana correlators at essentially no additional cost. We demonstrate the method for the interacting Kitaev chain using the imaginary and symmetric Majorana hopping channel. The resulting noncommuting auxiliary-field bond operators are treated through an even-odd decomposition. Their block structure permits an analytic reduction to a sparse effective interaction matrix of bandwidth two and allows explicit force expressions. Results agree with exact diagonalization and resolve low-energy modes localized at opposite boundaries, together with their increasing overlap near the topological transition of the non-interacting system. These findings establish HMC as an efficient sampling framework for Pfaffian auxiliary-field simulations.

DOI
10.48550/arxiv.2609.21631

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Pfaffian quantum Monte Carlo extends auxiliary field methods to anomalous fermionic Hamiltonians, opening access to pairing systems and Hubbard--Stratonovich decompositions that are unavailable in determinant formulations. We show that this flexibility can be combined with the global updates of Hybrid Monte Carlo (HMC), leading to an algorithm scaling as O(NτL3)\mathcal{O}(N_τL^3)O(Nτ​L3), similar to its determinant quantum Monte Carlo (DQMC) counterpart without pseudofermions. The central observation is that the molecular dynamics force depends only on the magnitude of the Pfaffian weight, which admits a determinant representation and leads to a simplified force expression. This construction also yields unequal-time Majorana correlators at essentially no additional cost. We demonstrate the method for the interacting Kitaev chain using the imaginary and symmetric Majorana hopping channel. The resulting noncommuting auxiliary-field bond operators are treated through an even-odd decomposition. Their block structure permits an analytic reduction to a sparse effective interaction matrix of bandwidth two and allows explicit force expressions. Results agree with exact diagonalization and resolve low-energy modes localized at opposite boundaries, together with their increasing overlap near the topological transition of the non-interacting system. These findings establish HMC as an efficient sampling framework for Pfaffian auxiliary-field simulations.
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We introduce determinant quantum-quantum Monte Carlo (DQ2^22MC), a quantum algorithm that lifts the auxiliary-field sampling and averaging at the operational core of determinant quantum Monte Carlo onto a quantum computer. A determinant oracle synthesizes the DQMC amplitudes directly from a block encoding of the single-particle action matrix via quantum singular value transformations, so that the exponentially many Hubbard-Stratonovich weights are never enumerated, precomputed, or stored. Since the fermions are free for fixed auxiliary fields, the construction operates entirely at the single-particle level, requiring O(log⁡Nst)O(\log N_{\mathrm{st}})O(logNst​) system qubits and no Jordan-Wigner or Bravyi-Kitaev encoding, where NstN_{\mathrm{st}}Nst​ is the space-time volume. A full-quantum protocol makes observables interference amplitudes, eliminating the Markov chain and its autocorrelation time altogether; a hybrid quantum-classical protocol retains a constant-size active block of qubits and replaces the Metropolis-Hastings acceptance step with an exact heat-bath draw, so that cluster updates of any size are rejection-free, and passes only classical information between updates, admitting parallel tempering and distributed execution across quantum processors. The circuit-depth scales more favorably with spatial volume than classical DQMC, at the price of a post-selection overhead determined exactly by the largest target probability --- polynomial for smooth distributions, exponential for sharply peaked ones. Finally, the reweighting estimator underlying the fermion sign problem maps exactly onto a quantum weak value, placing the exponential cost of sign-problematic DQMC in precise correspondence with the post-selection overhead of weak-value extraction.
Strongly correlated fermionic systems are of great interest in condensed matter physics and numerical methods are indispensable tools for their study. However, existing approaches such as exact diagonalization (ED) and stochastic quantum Monte Carlo methods each suffer from fundamental limitations: ED is hindered by exponential scaling in system size, while Monte Carlo methods are plagued by sign problems and long autocorrelation times. These limitations restrict the accessible parameter space and developing algorithms that efficiently alleviate them remains a central challenge in computational physics. In this work, we propose a algorithm that combines ED and Hamiltonian Monte Carlo (HMC) to simulate two-dimensional (2D) arrays of coupled quantum wires, modeled as interacting fermionic Hubbard chains. We demonstrate how our hybrid implementation of HMC, which we dub H 2 MC , outperforms either method alone across several key simulation facets. When compared to pure ED, H 2 MC has a much more favorable computational scaling, which allows us to push simulations to much larger 2D arrays. H 2 MC also greatly alleviates the sign problem and reduces autocorrelation times when compared to pure HMC formulations utilizing either real or imaginary auxiliary fields. Our formalism demonstrates how complementary strengths of seemingly disparate methods can be leveraged to enable feasible simulations in an extended parameter space.
We introduce algorithmic improvements to quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC) that reduce the dominant per-step classical scaling from O~(N5.5)\tilde{\mathcal{O}}(N^{5.5})O~(N5.5) to O~(N4.5)\tilde{\mathcal{O}}(N^{4.5})O~(N4.5) as a function of the number of molecular spin-orbitals NNN. Central to this improvement is the application of Aitken's block transformation to handle singular Pfaffians arising in the estimation of overlaps between a quantum trial state and classical Slater-determinant walkers. Together with the use of algorithmic differentiation for the computation of the force bias, this yields a 248×248\times248× estimated runtime improvement for a system of 100 molecular orbitals. Using our workflow, we demonstrate a ground-state energy calculation for H8H_8H8​ from quantum data collected on IQM Emerald and post-processed with a tensor-network-based error-mitigation technique. We further validate the method's scalability through noiseless simulation of hydrogen chains up to H12H_{12}H12​, and on the lithium-air battery related rearrangement pathway of the Li2O4Li_2O_4Li2​O4​ lithium superoxide dimer in a (26e, 20o) active space. We estimate both quantum and classical runtimes for a potential fault-tolerant implementation of QC-AFQMC, showing that the method holds promise for the early fault-tolerant era. These results move QC-AFQMC a step closer to treating chemically relevant systems.
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Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.