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arXiv:2408.10311

Pfaffian quantum Monte Carlo: solution to Majorana sign ambiguity and applications

Ze-Yao Han, Zhou‐Quan Wan, Hong Yao

Published Aug 19, 2024• 0 citations
arXivHTMLPDF

Abstract

Determinant quantum Monte Carlo (DQMC), formulated in complex-fermion representation, has played a key role in studying strongly-correlated fermion systems. However, its applicability is limited due to the requirement of particle-number conservation after Hubbard-Stratonovich transformation. In going beyond the conventional DQMC, one encouraging development occurred when Majorana fermions were introduced for QMC [1,2]. But in previous Majorana-based QMC, Boltzmann weight is determined often with a sign ambiguity. Here we successfully resolved this ambiguity by deriving a closed-form Pfaffian formula for the weight, enabling efficient calculation of the weight with its sign in polynomial time. We call it ''Pfaffian quantum Monte Carlo'' (PfQMC), which can be applied to generic interacting fermion models. We have successfully employed PfQMC to explore how robust Majorana edge modes in Kitaev chain are against strong interactions. By offering greater flexibility, PfQMC can potentially enhance existing sign-mitigating and approximation methods and help address challenging issues such as the ground-state properties of the doped Hubbard model.

DOI
10.48550/arxiv.2408.10311

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Determinant quantum Monte Carlo (DQMC), formulated in complex-fermion representation, has played a key role in studying strongly-correlated fermion systems. However, its applicability is limited due to the requirement of particle-number conservation after Hubbard-Stratonovich transformation. In going beyond the conventional DQMC, one encouraging development occurred when Majorana fermions were introduced for QMC [1,2]. But in previous Majorana-based QMC, Boltzmann weight is determined often with a sign ambiguity. Here we successfully resolved this ambiguity by deriving a closed-form Pfaffian formula for the weight, enabling efficient calculation of the weight with its sign in polynomial time. We call it ''Pfaffian quantum Monte Carlo'' (PfQMC), which can be applied to generic interacting fermion models. We have successfully employed PfQMC to explore how robust Majorana edge modes in Kitaev chain are against strong interactions. By offering greater flexibility, PfQMC can potentially enhance existing sign-mitigating and approximation methods and help address challenging issues such as the ground-state properties of the doped Hubbard model.
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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The sign problem is the fundamental limitation to quantum Monte Carlo simulations of the statistical mechanics of interacting fermions. Determinant quantum Monte Carlo (DQMC) is one of the leading methods to study lattice fermions, such as the Hubbard Hamiltonian, which describe strongly correlated phenomena including magnetism, metal-insulator transitions, and possibly exotic superconductivity. Here, we provide a comprehensive dataset on the geometry dependence of the DQMC sign problem for different densities, interaction strengths, temperatures, and spatial lattice sizes. We supplement these data with several observations concerning general trends in the data, including the dependence on spatial volume and how this can be probed by examining decoupled clusters, the scaling of the sign in the vicinity of a particle-hole symmetric point, and the correlation between the total sign and the signs for the individual spin species.
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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.
We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps τ∼O(1)τ\sim O(1)τ∼O(1) and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.