Investigating a (3+1)D Topological $θ$-Term in the Hamiltonian Formulation of Lattice Gauge Theories for Quantum and Classical Simulations

Abstract

Quantum technologies offer the prospect to efficiently simulate sign-problem afflicted regimes in lattice field theory, such as the presence of topological terms, chemical potentials, and out-ofequilibrium dynamics. In this work, we derive the 3+1D topological θ-term for Abelian and non-Abelian lattice gauge theories in the Hamiltonian formulation, paving the way towards Hamiltonianbased simulations of such terms on quantum and classical computers. We further study numerically the zero-temperature phase structure of a 3+1D U(1) lattice gauge theory with the θ-term via exact diagonalization for a single periodic cube. In the strong coupling regime, our results suggest the occurrence of a phase transition at constant values of θ, as indicated by an avoided level-crossing and abrupt changes in the plaquette expectation value, the electric energy density, and the topological charge density. These results could in principle be cross-checked by the recently developed 3+1D tensor network methods and quantum simulations, once sufficient resources become available.

Lattice Quantum Chromodynamics and Electrodynamics on a Universal Quantum Computer

“…In this paper, we explored quantum simulations of U(1), SU (2), and SU (3) LGTs with mass, kinetic, electric, and magnetic terms. Possible avenues of extensions of our work include considering more terms, such as the Higgs term [35] or topological θ-term [36,37]. The Higgs term allows the study of the Higgs mechanism and the confinement of Higgs matter.…”

Standard model physics and the digital quantum revolution: thoughts about the interface

“…Formalisms for simulating QED in higher dimensions have been put in place for a number of systems, e.g., Refs. [356,362,364,475,[482][483][484][485][486][487][488][489][490][491][492][493][494][495], but remain to be executed on a quantum device.…”

Noisy Bayesian optimization for variational quantum eigensolvers

“…Their choice has an impact on the geometry of the posterior GP of Eq. (7). The possibility of using different 𝜇(𝜃) and 𝑘 (𝜃 𝛼 , 𝜃 𝛼 ) can then be an advantage as it can be used to impose certain properties that are motivated from the physics of the considered problem.…”

“…( 5) is in 𝐶 ∞ when using commonly chosen quantum circuits [15], and selecting 𝜇(𝜃 𝛼 ), 𝑘 (𝜃 𝛼 , 𝜃 𝛼 ) ∈ 𝐶 ∞ will impose this property on the posterior mean of Eq. (7). A common choice is to set 𝜇(𝜃 𝛼 ) to a constant:…”

“…However, many interesting problems of LGTs can already be studied with NISQ devices [6]. In particular, if LGTs are studied in their Hamiltonian formulation, quantum algorithms do not generally suffer from the sign problem [7,8]. An important ready-to-use algorithm is the variational quantum eigensolver (VQE) [9], which is a hybrid quantum-classical algorithm for finding the ground (and excited) state of a given Hamiltonian H using the variational principle.…”