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arXiv:1008.3647

Preconditioning the non-relativistic many-fermion problem

Timour Ten, Joaquín E. Drut, Timo A. Lähde

Published Aug 21, 2010• 1 citation
arXivHTMLPDF

Abstract

Preconditioning is at the core of modern many-fermion Monte Carlo algorithms, such as Hybrid Monte Carlo, where the repeated solution of a linear problem involving an ill-conditioned matrix is needed. We report on a performance comparison of three preconditioning strategies, namely Chebyshev polynomials, strong-coupling approximation and weak-coupling expansion. We use conjugate gradient (CG) on the normal equations as well as stabilized biconjugate gradient (BiCGStab) as solvers and focus on the fermion matrix of the unitary Fermi gas. Our results indicate that BiCGStab is by far the most efficient strategy, both in terms of the number of iterations and matrix-vector operations.

DOI
10.48550/arxiv.1008.3647

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Preconditioning is at the core of modern many-fermion Monte Carlo algorithms, such as Hybrid Monte Carlo, where the repeated solution of a linear problem involving an ill-conditioned matrix is needed. We report on a performance comparison of three preconditioning strategies, namely Chebyshev polynomials, strong-coupling approximation and weak-coupling expansion. We use conjugate gradient (CG) on the normal equations as well as stabilized biconjugate gradient (BiCGStab) as solvers and focus on the fermion matrix of the unitary Fermi gas. Our results indicate that BiCGStab is by far the most efficient strategy, both in terms of the number of iterations and matrix-vector operations.
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An improved algorithm is proposed for Monte Carlo methods to study fermion systems interacting with adiabatical fields. To obtain a weight for each Monte Carlo sample with a fixed configuration of adiabatical fields, a series expansion using Chebyshev polynomials is applied. By introducing truncations of matrix operations in a systematic and controlled way, it is shown that the cpu time is reduced from O(N^3) to O(N) where N is the system size. Benchmark results show that the implementation of the algorithm makes it possible to perform systematic investigations of critical phenomena using system-size scalings even for an electronic model in three dimensions, within a realistic cpu timescale.
In this study, we propose the Quantum Gradient Flow Algorithm (QGFA), a novel quantum algorithm for solving symmetric positive definite (SPD) linear systems based on the variational formulation and time-evolution dynamics. Conventional quantum linear solvers, such as the quantum matrix inverse algorithm (QMIA), focus on approximating the matrix inverse through quantum signal processing (QSP). However, QMIA suffers from a crucial drawback: its computational efficiency deteriorates as the condition number increases. In contrast, classical SPD linear solvers, such as the steepest descent and conjugate gradient methods, are known for their fast convergence, which stems from the variational optimization principle of SPD systems. Inspired by this, we develop QGFA, which obtains the solution vector through the gradient-flow process of the corresponding quadratic energy functional. To validate the proposed method, we apply QGFA to the displacement-based finite element method (FEM) for two-dimensional linear elastic problems under plane stress conditions. The algorithm demonstrates accurate convergence toward classical FEM solutions even with a moderate number of QSP phase factors. Compared with QMIA, QGFA achieves lower relative errors and faster convergence when initialized with suitable initial states, demonstrating its potential as an efficient preconditioned quantum linear solver. The proposed framework provides a physically interpretable connection between classical iterative solvers and quantum computational paradigms. These findings suggest that QGFA can serve as a foundation for future developments in Quantum Computer-Aided Engineering (Quantum CAE), including nonlinear and multiphysics simulations.