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

In this paper, we present a set of local fermion-to-qudit mappings for simulating fermionic lattice systems. We focus on the use of multilevel qudits, specifically ququarts. Traditional mappings, such as the Jordan-Wigner transformation (JWT), while useful, often result in nonlocal operators that scale unfavorably with system size. To address these challenges, we introduce mappings that efficiently localize fermionic operators on qudits, reducing the nonlocality 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. Notably, for superconducting devices, the control complexity of qudits compared to qubits does not increase prohibitively for four-level systems, making them practical for near-term demonstrations, though control challenges grow with increasing dimensionality. Therefore, we provide solutions to the operator decompositions of the Trotterized quantum dynamics into one- and two-qudit gates for all mappings. 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. Finally, we show the connection between the plaquette constraint of our mapping and the Z2 toric code. This connection can be exploited to prove that the ground state of the localized qudit mapping can be efficiently prepared using quantum circuits or measurement-based feedback control. Our results highlight the potential of qudit-based quantum simulations in achieving scalability and efficiency for fermionic systems on near-term quantum devices.