[Paper Review] Polynomial Subtraction Method for Disconnected Quark Loops

Quan Liu, Walter Wilcox+1 more|arXiv (Cornell University)|May 7, 2014

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper introduces the polynomial subtraction method, a novel numerical technique in lattice QCD that reduces noise variance in disconnected quark loop calculations by approximating the inverse quark matrix using MinRes polynomial expansion. The method significantly outperforms perturbative subtraction, achieving up to a 30% reduction in computational cost for small quark masses (κ ≤ 0.15), with variance reduction increasing with quark mass.

ABSTRACT

The polynomial subtraction method, a new numerical approach for reducing the noise variance of Lattice QCD disconnected matrix elements calculation, is introduced in this paper. We use the MinRes polynomial expansion of the QCD matrix as the approximation to the matrix inverse and get a significant reduction in the variance calculation. We compare our results with that of the perturbative subtraction and find that the new strategy yields a faster decrease in variance which increases with quark mass.

Motivation & Objective

• To address the high statistical noise in lattice QCD calculations of disconnected quark loop matrix elements.
• To reduce computational cost in stochastic trace estimation for large-scale QCD simulations.
• To develop a robust, low-overhead alternative to existing subtraction methods like perturbative and eigenvalue subtraction.
• To improve the efficiency of nucleon form factor, strangeness, and scattering length calculations in lattice QCD.

Proposed method

• Uses MinRes polynomial expansion to approximate the inverse of the QCD matrix, serving as a noise-reduction proxy.
• Applies a traceless correction matrix derived from polynomial expansion of the quark matrix to subtract dominant noise components.
• Employs Z(4) noise vectors for stochastic trace estimation, ensuring unbiased matrix element evaluation.
• Constructs a polynomial approximation of the matrix inverse up to 10th order in κ to optimize variance reduction.
• Compares performance against perturbative subtraction using variance ratios (V_pert / V_poly) across different κ values and operators.
• Validates the method on both 16⁴ and 24³×32 lattices, including large-scale 2.6 million × 2.6 million matrices.

Experimental results

Research questions

• RQ1 Can polynomial approximation of the quark matrix inverse reduce noise variance more effectively than perturbative subtraction in disconnected quark loop calculations?
• RQ2 How does the performance of the polynomial subtraction method vary with quark mass (κ) and operator type?
• RQ3 What is the optimal polynomial order (up to 10th) for variance reduction in the stochastic trace estimation of disconnected diagrams?
• RQ4 Does the method maintain efficiency and variance reduction on large-scale lattices (e.g., 24³×32)?
• RQ5 Can the polynomial subtraction method be combined with other noise reduction techniques like eigenspectrum subtraction for further improvement?

Key findings

• The polynomial subtraction method reduces variance more effectively than perturbative subtraction, with variance ratios (V_pert / V_poly) reaching 1.40 at κ = 0.15, corresponding to a 29% reduction in computational cost.
• At κ = 0.155, the method achieves a 28% reduction in computational cost (variance ratio ≈ 1.28), and at κ = 0.1571, a 9% reduction (variance ratio ≈ 1.10), showing consistent performance across operators.
• The 7th-order polynomial approximation yields the best variance reduction, outperforming 4th and 10th orders, suggesting numerical saturation at higher orders.
• On a 24³×32 lattice (2.6 million × 2.6 million matrix), the method maintains a 1.39–1.40 variance reduction for κ = 0.15, confirming scalability.
• The method is computationally efficient, requiring minimal extra cost to compute polynomial coefficients while delivering significant variance reduction.
• The results suggest that combining polynomial subtraction with eigenspectrum subtraction could yield further improvements, especially for larger κ values.