# Operator Locality in Quantum Simulation of Fermionic Models

## TL;DR

This paper reviews and analyzes various fermion-to-qubit mappings for quantum simulation, focusing on locality improvements and encoding efficiency, especially for lattice models like the Hubbard model.

## Contribution

It reformulates the Bravyi-Kitaev transform with a classical data structure and introduces locality improvements for fermionic models, comparing different mappings' efficiencies.

## Key findings

- Bravyi-Kitaev transform can be reformulated for better locality.
- Auxiliary Fermion scheme achieves the most compact encoding with ancillas.
- A variant of Bravyi-Kitaev is optimal without ancillas for Hubbard models.

## Abstract

Simulating fermionic lattice models with qubits requires mapping fermionic degrees of freedom to qubits. The simplest method for this task, the Jordan-Wigner transformation, yields strings of Pauli operators acting on an extensive number of qubits. This overhead can be a hindrance to implementation of qubit-based quantum simulators, especially in the analog context. Here we thus review and analyze alternative fermion-to-qubit mappings, including the two approaches by Bravyi and Kitaev and the Auxiliary Fermion transformation. The Bravyi-Kitaev transform is reformulated in terms of a classical data structure and generalized to achieve a further locality improvement for local fermionic models on a rectangular lattice. We conclude that the most compact encoding of the fermionic operators can be done using ancilla qubits with the Auxiliary Fermion scheme. Without introducing ancillas, a variant of the Bravyi-Kitaev transform provides the most compact fermion-to-qubit mapping for Hubbard-like models.