Padé - Z$_2$ Estimator of Determinants

TL;DR

The paper introduces the Padé--$Z_2$ estimator to efficiently compute determinants and determinant ratios of large sparse fermion matrices in lattice QCD by coupling a high-order Padé approximation of $\log z$ with complex $Z_2$ noise trace estimation. A variance-reducing unbiased subtraction scheme based on hopping-parameter expansion is implemented alongside the multi-shift MR solver ($M$^3$R) to handle multiple inverses simultaneously. On an $8^3\times12$ Wilson lattice, high-order Padé accuracy combined with subtraction yields dramatic reductions in stochastic error, making determinant-based updates (and determinant ratios) feasible with modest noise counts; the method also extends to density-of-states calculations for Hermitian Hamiltonians and is applicable to non-Hermitian cases. The approach provides a practical, scalable alternative to purely pseudofermion-based methods for dynamical fermion simulations and related large-scale linear-algebra problems in physics.

Abstract

We introduce the Padé--Z$_2$ (PZ) stochastic estimator for calculating determinants and determinant ratios. The estimator is applied to the calculation of fermion determinants from the two ends of the Hybrid Monte Carlo trajectories with pseudofermions. Our results on the $8^3 \times 12$ lattice with Wilson action show that the statistical errors from the stochastic estimator can be reduced by more than an order of magnitude by employing an unbiased variational subtraction scheme which utilizes the off-diagonal matrices from the hopping expansion. Having been able to reduce the error of the determinant ratios to about 20 % with a relatively small number of noise vectors, this may become a feasible algorithm for simulating dynamical fermions in full QCD. We also discuss the application to the density of states in Hamiltonian systems.

Figures (7)

• Figure 1: $\epsilon(z)$ for the Padé approximation of $\log z$ on the positive real axis for different orders of the Padé expansion at $z_0 = 1.0$.
• Figure 2: The same as Fig. 1 with a larger domain of $z$.
• Figure 3: The real part of Padé approximated $\log z$ along a circle of radius 1 as a function of arg(z) for several different orders of the Padś expansion. The expansion point is at $z_0 = 1$.
• Figure 5: The same as in Fig. 3 for the circle of radius 10.
• Figure 7: The Green function approximation to $\log$ with 11 terms and several $\epsilon$ are compared to the $\log$ function itself.