Digitizing gauge invariant interactions, I: Conserving Abelian constraints with shears

Universal quantum simulations of gauge field theories are exposed to the risk of gauge symmetry violations when it is not known how to compile the desired operations exactly using the available gate set. In this talk, we discuss how interactions can be compiled -- if only approximately -- without compromising Abelian constraints by graphically motivating a block-diagonalization procedure. When gauge invariant interactions can be associated with a "spatial network" in the space of discrete quantum numbers, it is seen that cyclically shearing the spatial network converts simultaneous updates to many quantum numbers into conditional updates of a single quantum number: ultimately, this eliminates any need to pass through (and acquire overlap onto) symmetry-violating intermediate configurations. As examples, we look at how shears can be applied to gauge-matter interactions in (1+1)-dimensional U(1) and SU(2) gauge theories.