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arXiv:1810.00864

A unified theory of variational and quantum Monte Carlo methods and beyond

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

Published Oct 1, 2018• 2 citations
arXivHTMLPDF

Abstract

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 $τ\sim O(1)$ 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.

DOI
10.48550/arxiv.1810.00864

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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.
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A new computational method for finite-temperature properties of strongly correlated electrons is proposed by extending the variational Monte Carlo method originally developed for the ground state. The method is based on the path integral in the imaginary-time formulation, starting from the infinite-temperature state that is well approximated by a small number of certain random initial states. Lower temperatures are progressively reached by the imaginary-time evolution. The algorithm follows the framework of the quantum transfer matrix and finite-temperature Lanczos methods, but we extends them to treat much larger system sizes without the negative sign problem by optimizing the truncated Hilbert space on the basis of the time-dependent variational principle (TDVP). This optimization algorithm is equivalent to the stochastic reconfiguration (SR) method that has been frequently used for the ground state to optimally truncate the Hilbert space. The obtained finite-temperature states allow an interpretation based on the thermal pure quantum (TPQ) state instead of the conventional canonical-ensemble average. Our method is tested for the one- and two-dimensional Hubbard models and its accuracy and efficiency are demonstrated.
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.
By precisely writing down the matrix element of the local Boltzmann operator ([Formula: see text], where [Formula: see text] is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With the current formula, the Hubbard-Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principles calculations. The current formula also improves the accuracy of the Suzuki-Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be applied to various studies in future.
Abstract By precisely writing down the matrix element of the local Boltzmann operator (e-th, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With current formula, the Hubbard–Stratonovich transformation is not necessary, and is not based on the determinant approach, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principle calculation. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm can be used in various studies.