Local fermion-to-qudit mappings: a practical recipe for four-level systems

TL;DR

This paper introduces local fermion-to-qudit mappings using multi-level qudits, improving the efficiency of simulating fermionic systems by reducing non-locality and gate complexity, validated on key models.

Contribution

The paper presents novel local fermion-to-qudit mappings that localize fermionic operators, enhancing quantum simulation efficiency for fermionic lattice models.

Findings

Reduced non-locality compared to Jordan-Wigner transformation

Lower circuit depth and gate complexity in simulations

Successful validation on spinless t-V and Fermi-Hubbard models

Abstract

In this paper, we present a new set of local fermion-to-qudit mappings for simulating fermionic lattice systems. We focus on the use of multi-level qudits, specifically ququarts. Traditional mappings, such as the Jordan-Wigner transformation (JWT), while useful, often result in non-local operators that scale unfavorably with system size. To address these challenges, we introduce mappings that efficiently localize fermionic operators on qudits, reducing the non-locality and operator weights associated with JWT. We propose one mapping for spinless fermions and two mappings for spinful fermions, comparing their performance in terms of qudit-weight, circuit depth, and gate complexity. By leveraging the extended local Hilbert space of qudits, we show that these mappings enable more efficient quantum simulations in terms of two-qudit gates, reducing hardware requirements without increasing computational complexity. We validate our approach by simulating prototypical models such as the spinless t-V model and the Fermi-Hubbard model in two dimensions, using Trotterized time evolution. Our results highlight the potential of qudit-based quantum simulations in achieving scalability and efficiency for fermionic systems on near-term quantum devices.