Compiling Quantum Gauge Theories for Quantum Computation

Abstract

This dissertation gives a survey of formal issues of Hamiltonian lattice gauge theories in the context of simulation by quantum computers. The basic properties of gauge field theories and their lattice regularizations are first reviewed, especially as they pertain to the local constraint that arises in canonical quantization: Gauss’s law, the satisfaction of which is synonymous with gauge invariance and charge conservation. Digital quantum algorithms are developed for the basic task of checking Gauss’s law in U(1) and Z(N) Abelian gauge theories, as they are conventionally formulated. We then analyze U(1) gauge theories by reconstructing them in terms of dual variables that make Gauss’s law manifest. The task of quantum simulation is then studied for the non-Abelian gauge group SU(2). The first quantum simulation of SU(2) gauge bosons using existing IBM quantum hardware is presented, made possible by partially solving the Gauss law constraints in a small system. The quest for the “right” variables to use for quantum algorithms begun with U(1) is then taken up for SU(2). Building on the prepotential formulation of lattice gauge theories, a complete ‘loop-string-hadron’ (LSH) framework is developed for one fermion flavor interacting with SU(2) gauge bosons, in terms of strictly SU(2)-invariant variables. The LSH Hamiltonian is unpacked at a low level, making it transparent what would have to be implemented on a quantum computer. This LSH framework is then applied to provide the first quantum circuits for validating wave functions in SU(2) gauge theories and the associated resource requirements are discussed.