[Paper Review] Taming the Signal-to-Noise Problem in Lattice QCD by Phase Reweighting

Michael L. Wagman, Martin J. Savage|arXiv (Cornell University)|Apr 24, 2017

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper introduces phase reweighting in lattice QCD to overcome the signal-to-noise (StN) problem that plagues late-time correlation functions, enabling extraction of ground-state energies with time-independent StN ratios. By fixing the phase over a finite time window Δt and dynamically evolving the magnitude, the method suppresses noise while introducing a controllable bias that is removed via extrapolation, successfully recovering nucleon and rho meson masses and ΞΞ binding energy consistent with standard results.

ABSTRACT

Path integrals describing quantum many-body systems can be calculated with Monte Carlo sampling techniques, but average quantities are often subject to signal-to-noise ratios that degrade exponentially with time. A phase-reweighting technique inspired by recent observations of random walk statistics in correlation functions is proposed that allows energy levels to be extracted from late-time correlation functions with time-independent signal-to-noise ratios. Phase reweighting effectively includes dynamical refinement of source magnitudes but introduces a bias associated with the phase. This bias can be removed by performing an extrapolation, but at the expense of re-introducing a signal-to-noise problem. Lattice Quantum Chromodynamics calculations of the $ρ$ and nucleon masses and of the $ΞΞ$ binding energy show consistency between standard results obtained using earlier-time correlation functions and phase-reweighted results using late-time correlation functions inaccessible to standard statistical analysis methods.

Motivation & Objective

• To address the exponentially degrading signal-to-noise (StN) ratio in lattice QCD correlation functions at late Euclidean times, which limits precision calculations of multi-baryon systems.
• To overcome the StN problem in baryonic and multi-baryonic systems, where standard Monte Carlo methods fail due to noise dominating at late times.
• To develop a method that allows reliable extraction of ground-state energies from correlation functions in the noise-dominated region, beyond the 'golden window' of early-time analysis.
• To introduce a phase-reweighting technique that dynamically refines source magnitudes while fixing the phase over a finite time interval Δt, reducing noise without relying on source/sink optimization.
• To demonstrate that the resulting bias from phase restriction can be systematically removed via extrapolation, preserving accuracy while enabling late-time analysis.

Proposed method

• Phase reweighting is implemented by restricting the evolution of the complex phase of the correlation function over a finite time interval Δt, effectively creating a non-local source with dynamically refined magnitude.
• The method separates the correlation function into magnitude and phase components, where the magnitude evolves to ground state while the phase is held fixed over Δt, reducing noise in late-time measurements.
• The phase is released only for the final Δt steps, allowing the system to propagate to later times with a stable signal-to-noise ratio.
• A bias is introduced due to the restricted phase evolution, which decreases exponentially with increasing Δt and is removed via extrapolation to Δt → ∞.
• The technique is applied to correlation functions of the ρ⁺ meson, nucleon, and ΞΞ(^1S₀) system, using high-statistics ensembles from the NPLQCD collaboration.
• Statistical analysis is performed on phase-reweighted correlation functions, with systematic errors estimated through extrapolation in Δt and comparison to standard golden-window results.

Experimental results

Research questions

• RQ1 Can phase reweighting stabilize the signal-to-noise ratio in lattice QCD correlation functions at late times, where standard methods fail?
• RQ2 Does phase reweighting allow reliable extraction of ground-state energies from the noise-dominated region of correlation functions?
• RQ3 Can the bias introduced by phase restriction be systematically removed through extrapolation without reintroducing the StN problem?
• RQ4 Do phase-reweighted results for the ρ⁺ meson, nucleon, and ΞΞ(^1S₀) binding energy agree with standard results obtained in the golden window?
• RQ5 Is the phase-reweighting method effective for systems with multiple baryons, where the StN problem is most severe?

Key findings

• Phase reweighting enables extraction of ground-state energies from lattice QCD correlation functions at late times with a time-independent signal-to-noise ratio, overcoming the exponential degradation seen in standard methods.
• The ρ⁺ meson and nucleon masses extracted via phase reweighting are consistent with standard results obtained in the golden window, validating the method's accuracy.
• The ΞΞ(^1S₀) binding energy from phase-reweighted correlation functions plateaus at 7.1(0.6)(0.8) MeV, consistent with the golden-window result of 7.1(0.6)(0.8) MeV from Ref. Orginos et al. (2015), confirming agreement within errors.
• The method successfully isolates the signal in the magnitude of the correlation function, which exhibits no StN problem, while the phase is the primary source of noise, confirming the sign problem origin of the StN issue.
• The bias from phase restriction decreases exponentially with Δt, and extrapolation to Δt → ∞ removes it, though this step reintroduces a StN problem, which is mitigated by increasing the number of points in the noise region.
• The precision of phase-reweighted results scales with the number of time slices in the noise region, suggesting potential for improved accuracy on longer temporal lattices.