U(1) Lattice Gauge Theory Simulations with Truncated Hilbert Spaces Achieve Accuracy Using a Functional Basis with Minimal Basis States

Understanding the fundamental forces governing matter requires increasingly sophisticated theoretical calculations, and Lattice Gauge Theory provides a powerful framework for these investigations. Timo Jacobs, Marco Garofalo, and Tobias Hartung, alongside Karl Jansen, Paul Ludwig, and Johann Ostmeyer, have rigorously tested different methods for simplifying these complex calculations, specifically focusing on U(1) Lattice Gauge Theory. Their work addresses a key challenge in applying new computational techniques, such as tensor networks, by comparing various ways to reduce the enormous computational demands of simulating these theories. The team demonstrates that a novel approach, utilising a functional basis derived from single plaquette Hamiltonians, significantly outperforms existing methods in both two and three dimensions, delivering accurate results with a minimal number of computational resources and paving the way for more efficient and precise calculations of fundamental particle interactions.

This research employs Green Function Monte Carlo (GFMC), a powerful computational technique, to independently confirm results obtained from digitised Hamiltonian methods. The team focused on U(1) gauge theory as a starting point for developing and testing their simulation techniques. GFMC solves the complex equations governing quantum systems by simulating a product of the wave function and a trial wave function, reducing computational demands. The simulation evolves the system using a probabilistic process involving a ‘drift step’ and a ‘branching step’ that creates copies of configurations, favouring those with lower energy.

This process generates an ensemble of gauge field configurations, allowing scientists to calculate physical quantities by averaging over these configurations. The team carefully optimised a trial wave function and employed a population control scheme to maintain a stable ensemble size. Results obtained with a finite time step were carefully extrapolated to ensure accuracy, and rigorous statistical methods were used to estimate errors and confirm reliability. The research team recognised that practical simulations require projecting this space onto finite resources, necessitating a digitisation scheme to approximate the Hamiltonian’s eigenstates. They systematically compared several truncation schemes, ultimately demonstrating the superior performance of a functional basis derived from single plaquette Hamiltonians, termed the ‘plaquette state basis’. This innovative approach involves constructing a basis from the eigenstates of the single plaquette Hamiltonian, effectively interpolating between regimes dominated by kinetic and potential energy.

The team rigorously benchmarked this basis against other established methods using tensor network states across a vast range of coupling strengths and system sizes in both two and three spatial dimensions. To validate the results, scientists employed exact diagonalisation and Green’s function Monte Carlo techniques, providing a comprehensive assessment of accuracy and reliability. The study’s methodology extended to both U(1) and SU(2) gauge theories, demonstrating the general applicability of the plaquette state basis. Crucially, simulations utilising this basis achieved significantly larger system sizes than previously feasible, enabling investigations into previously intractable problems. The research team developed a novel functional basis, termed the “plaquette state basis,” derived from single plaquette Hamiltonians, and demonstrated its superior performance compared to existing truncation schemes. Results show this basis accurately delivers ground state energy and mass gap measurements across a wide range of coupling strengths with a minimal number of basis states in two spatial dimensions. Experiments reveal the plaquette state basis outperforms alternative methods in accurately representing the system’s lowest energy state and the energy required to create particle-antiparticle pairs, known as the mass gap.

The team benchmarked different bases for various system sizes and coupling strengths, verifying results with exact diagonalisation and Green’s function Monte Carlo methods. Measurements confirm the new basis allows for significantly larger system sizes in tensor network simulations than previously achievable, even extending to three spatial dimensions. Data shows the plaquette state basis maintains accuracy even when simulating strongly interacting systems, a challenging regime for many computational methods. Specifically, the team computed the mass gap for a wide range of coupling values, providing a rigorous test of the basis’s effectiveness. This approach allows for efficient simulations using tensor network states and potentially quantum computers by truncating the infinite-dimensional Hilbert space to a manageable size. The team demonstrates that the plaquette-state basis outperforms other existing truncation schemes in two spatial dimensions, delivering accurate results for ground state energy and mass gap across a wide range of coupling strengths while using a minimal number of basis states. The study confirms the effectiveness of this basis in three spatial dimensions and highlights its versatility in accessing a broader range of observable quantities.

Importantly, the results align with highly accurate Green’s function Monte Carlo calculations, validating the approach across different coupling regimes. The researchers are actively pursuing a GPU implementation to enable simulations of larger three-dimensional systems and are investigating the extension of this construction to non-Abelian theories. This work represents a significant step towards more efficient and versatile simulations of lattice gauge theories, with potential applications in both fundamental physics research and quantum computing.