Classical and Quantum Algorithmic Developments in Lattice Field Theory

Abstract

Lattice field theory provides a powerful framework for studying non-perturbative aspects of quantum chromodynamics (QCD), enabling first-principles numerical calculations of hadron properties and high-precision theoretical predictions of the Standard Model. Despite its successes, several longstanding computational challenges hinder progress in frontier regimes. Among these are the sign problem, which makes simulations exponentially costly in certain settings; the signal-to-noise problem, which limits precision for key observables; and the infinite variance problem, which can render estimators ill-defined. Addressing these issues is essential for advancing Monte Carlo simulations. This thesis explores new strategies for overcoming these obstacles using classical and quantum computational methods. On the classical side, it develops and analyzes algorithms for variance reduction in Monte Carlo simulations. Approaches include reweighting schemes to eliminate the infinite variance problem; contour deformations and complex normalizing flows to mitigate the sign problem; and control variates---constructed to respect symmetry and using machine learning---to reduce the signal-to-noise problem. These methods are benchmarked on representative lattice field theory models. On the quantum side, the thesis investigates algorithms for efficient eigenstate preparation, a central requirement for simulating lattice field theory on quantum devices. Methods based on Hamiltonian paths are studied, including the rodeo projection algorithm, projection-based schemes inspired by the quantum Zeno effect, and adiabatic evolution. Their computational costs are analyzed in terms of path length and volume of the system, with proposed improvements that reduce fluctuations and demonstrate a potential quantum advantage in simulating quantum field theory on quantum computers. Together, these developments contribute to tackling the numerical barriers that arise in simulating strongly interacting quantum many-body systems. The results presented here highlight crucial advancements in both classical variance reduction and quantum computational methodologies, laying the groundwork for more accurate simulations of non-perturbative QCD in the future.