Block Encoding Non-Abelian Lattice Gauge Theory

Abstract

Gauge theories like lattice QCD present a complex problem for quantum simulation. In a basis where the electric part of the Hamiltonian is simple, the magnetic part, generally expressed as a sum over the plaquette operators of the lattice, is quite complicated, producing correlated transitions between several link and site degrees of freedom. We provide an efficient block encoding of the plaquette operator in the irrep basis, a refinement of the electric basis where the internal gauge-variant degrees of freedom are integrated out. The construction removes the plaquette matrix element scaling wall which has been a significant barrier for other approaches in this basis. The algorithm leverages a convenient factorization property of the matrix elements, cheap classical precomputation, and quantum oracles built from lookup tables and programmed rotations.

Comments: 35 pages. v2: fixed a bug in the corner stage, minor impact on resource cost; added a section on explicit circuit construction; circuits available as qasm files from https://github.com/pidraper/magnetic-oracle , including sparse simulation script validating the block encoding against the target matrix elements