Optimal Fermionic swap networks for Hubbard models

Abstract

We propose a Fermionic swap network scheme for efficient quantum computing of $n$-dimensional Hubbard-model Hamiltonians, assuming linear qubit connectivity. We establish new lower bounds on swap depth for such networks. These rely on isoperimetric inequalities from the combinatorics literature and are closely connected to graph bandwidth. We show that the scheme is swap-depth optimal for both spin and spinless two-dimensional Hubbard model Hamiltonians. In the first case it is also optimal in the number of Hamiltonian interaction layers, and is one from optimal in the second case.