Efficient quantum implementation of 2+1 U(1) lattice gauge theories with Gauss law constraints

The study of real-time evolution of lattice quantum field theories using classical computers is known to scale exponentially with the number of lattice sites. Due to a fundamentally different computational strategy, quantum computers hold the promise of allowing for detailed studies of these dynamics from first principles. However, much like with classical computations, it is important that quantum algorithms do not have a cost that scales exponentially with the volume. Recently, it was shown how to break the exponential scaling of a naive implementation of a U(1) gauge theory in two spatial dimensions through an operator redefinition. In this work, we describe modifications to how operators must be sampled in the new operator basis to keep digitization errors small. We compare the precision of the energies and plaquette expectation value between the two operator bases and find they are comparable. Additionally, we provide an explicit circuit construction for the Suzuki-Trotter implementation of the theory using the Walsh function formalism. The gate count scaling is studied as a function of the lattice volume, for both exact circuits and approximate circuits where rotation gates with small arguments have been dropped. We study the errors from finite Suzuki-Trotter time-step, circuit approximation, and quantum noise in a calculation of an explicit observable using IBMQ superconducting qubit hardware. We find the gate count scaling for the approximate circuits can be further reduced by up to a power of the volume without introducing larger errors.