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Register a new userTaming the Signal-to-Noise Problem in Lattice QCD by Phase Reweighting
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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 $rho$ and nucleon masses and of the $XiXi$ 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.
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The theory of quantum chromodynamics (QCD) encodes the strong interactions that bind quarks and gluons into nucleons and that bind nucleons into nuclei. Predictive control of QCD would allow nuclear structure and reactions as well as properties of supernovae and neutron stars to be theoretically studied from first principles. Lattice QCD can represent generic QCD predictions in terms of well-defined path integrals, but the sign and signal-to-noise problems have obstructed lattice QCD calculations of large nuclei and nuclear matter in practice. This thesis presents a statistical study of lattice QCD correlation functions, with a particular focus on characterizing the structure of the noise associated with quantum fluctuations. The signal-to-noise problem in baryon correlation functions is demonstrated to arise from a sign problem associated with Monte Carlo sampling of complex correlation functions. The phases of complex correlation functions are analyzed in the framework of circular statistics, and the time evolution of the phase is shown to resemble a heavy-tailed random walk on the unit circle. Building on these observations, a new technique called phase reweighting is investigated that involves calculations of phase differences over fixed-length time intervals. Phase reweighting allows results for hadronic observables to be extracted from large-time correlation functions with constant signal-to-noise ratios. The signal-to-noise problem re-emerges as the length of the phase-difference interval is increased. Applications of phase reweighting to meson, baryon, and two-baryon systems are discussed.
A systematic analysis of the structure of single-baryon correlation functions calculated with lattice QCD is performed, with a particular focus on characterizing the structure of the noise associated with quantum fluctuations. The signal-to-noise problem in these correlation functions is shown, as long suspected, to result from a sign problem. The log-magnitude and complex phase are found to be approximately described by normal and wrapped normal distributions respectively. Properties of circular statistics are used to understand the emergence of a large time noise region where standard energy measurements are unreliable. Power-law tails in the distribution of baryon correlation functions, associated with stable distributions and Levy flights, are found to play a central role in their time evolution. A new method of analyzing correlation functions is considered for which the signal-to-noise ratio of energy measurements is constant, rather than exponentially degrading, with increasing source-sink separation time. This new method includes an additional systematic uncertainty that can be removed by performing an extrapolation, and the signal-to-noise problem re-emerges in the statistics of this extrapolation. It is demonstrated that this new method allows accurate results for the nucleon mass to be extracted from the large-time noise region inaccessible to standard methods. The observations presented here are expected to apply to quantum Monte Carlo calculations more generally. Similar methods to those introduced here may lead to practical improvements in analysis of noisier systems.
Lattice QCD simulations of multi-baryon correlation functions can predict the structure and reactions of nuclei without encountering the baryon chemical potential sign problem. However, they suffer from a signal-to-noise problem where Monte Carlo estimates of observables have quantum fluctuations that are exponentially larger than their average values. Recent lattice QCD results demonstrate that the complex phase of baryon correlations functions relates the baryon signal-to-noise problem to a sign problem and exhibits unexpected statistical behavior resembling a heavy-tailed random walk on the unit circle. Estimators based on differences of correlation function phases evaluated at different Euclidean times are discussed that avoid the usual signal-to-noise problem, instead facing a signal-to-noise problem as the time interval associated with the phase difference is increased, and allow hadronic observables to be determined from arbitrarily large-time correlation functions.
The reweighting method is widely used in numerical studies of QCD, in particular, for the cases in which the conventional Monte-Carlo method cannot be applied directly, e.g., finite density QCD. However, the application range of the reweighing method is restricted due to several problems. One of the most severe problems here is the overlap problem. To solve it, we examine a multipoint reweighting method in which simulations at several simulation points are combined in the data analyses. We systematically study the applicability and limitation of the multipoint reweighting method in two-flavor QCD at zero density. Measuring histograms of physical quantities at a series of simulation points, we apply the multipoint reweighting method to calculate the meson masses as continuous functions of the gauge coupling $beta$ and the hopping parameters $kappa$. We then determine lines of constant physics and beta functions, which are needed in a calculation of the equation of state at finite temperature.
Gauge theories are of paramount importance in our understanding of fundamental constituents of matter and their interactions. However, the complete characterization of their phase diagrams and the full understanding of non-perturbative effects are still debated, especially at finite charge density, mostly due to the sign-problem affecting Monte Carlo numerical simulations. Here, we report the Tensor Network simulation of a three dimensional lattice gauge theory in the Hamiltonian formulation including dynamical matter: Using this sign-problem-free method, we simulate the ground states of a compact Quantum Electrodynamics at zero and finite charge densities, and address fundamental questions such as the characterization of collective phases of the model, the presence of a confining phase at large gauge coupling, and the study of charge-screening effects.
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