Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in $d$ spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the $(2+1)$-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner's lattice models $M_{(d,n)}$ that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary. Our procedure can be generalized to the simulation of Kitaev's Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model $M_{(d,n)}$ is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.