Variational quantum simulation of ground states and thermal states for lattice gauge theory with multi-objective optimization

Variational quantum algorithms provide feasible approaches for simulating quantum systems and are widely applied. For lattice gauge theory, however, variational quantum simulation faces a challenge as local gauge invariance enforces a constraint on the physical Hilbert space. In this paper, we incorporate multi-objective optimization for variational quantum simulation of lattice gauge theory at zero and finite temperatures. By setting energy or free energy of the system and penalty for enforcing the local gauge invariance as two objectives, the multi-objective optimization can self-adjust the proper weighting for two objectives and thus faithfully simulate the gauge theory in the physical Hilbert space. Specifically, we propose variational quantum eigensolver and variational quantum thermalizer for preparing the ground states and thermal states of lattice gauge theory, respectively. We demonstrate the quantum algorithms for a $Z_2$ lattice gauge theory with spinless fermion in one dimension. With numeral simulations, the multi-objective optimization shows that minimizing energy (free energy) and enforcing the local gauge invariance can be achieved simultaneously at zero temperature (finite temperature). The multi-objective optimization suggests a feasible ingredient for quantum simulation of complicated physical systems on near-term quantum devices.