Efficient Simulation of Sparse, Non-Local Fermion Models

Abstract: Efficient simulation of interacting fermionic systems is a key application of near-term quantum computers, but is hindered by the overhead required to encode fermionic operators on qubit hardware. Here, we consider models with $N$ fermionic modes in which each participates in at most a constant number $d$ of interactions and study the circuit depth required to implement the Trotterized time evolution on qubit hardware with all-to-all connectivity. We introduce an encoding that augments each physical fermionic mode with a small number of auxiliary fermions, enabling the removal of Jordan--Wigner strings. Although the preparation of the auxiliary fermion state incurs an initial overhead, this state remains invariant under time evolution. As a result, long-time evolution can be implemented with asymptotically optimal circuit depth, reducing a previously multiplicative $O(\log N)$ overhead to an additive overhead. Our results thus establish that the simulation of sparse fermionic models on qubit hardware matches the performance achievable on ideal fermionic hardware up to constant factors and $O(dN)$ ancillary qubits.