Discovering optimal fermion-qubit mappings through algorithmic enumeration

TL;DR

This paper introduces a novel approach to optimize fermion-qubit mappings by leveraging enumeration schemes for fermionic modes, significantly reducing the complexity of qubit Hamiltonians in quantum simulations.

Contribution

It demonstrates a new method for designing optimal fermion-qubit mappings using enumeration schemes, achieving substantial reductions in Pauli weight without additional resources.

Findings

Minimizes average Pauli weight in Jordan-Wigner transformations by 13.9%.

Reduces Hamiltonian complexity by 37.9% with two ancilla qubits.

Achieves an $n^{1/4}$ improvement in average Pauli weight for cellular arrangements.

Abstract

Simulating fermionic systems on a quantum computer requires a high-performing mapping of fermionic states to qubits. A characteristic of an efficient mapping is its ability to translate local fermionic interactions into local qubit interactions, leading to easy-to-simulate qubit Hamiltonians. All fermion-qubit mappings must use a numbering scheme for the fermionic modes in order for translation to qubit operations. We make a distinction between the unordered labelling of fermions and the ordered labelling of the qubits. This separation shines light on a new way to design fermion-qubit mappings by making use of the enumeration scheme for the fermionic modes. The purpose of this paper is to demonstrate that this concept permits notions of fermion-qubit mappings that are optimal with regard to any cost function one might choose. Our main example is the minimisation of the average number of Pauli matrices in the Jordan-Wigner transformations of Hamiltonians for fermions interacting in square lattice arrangements. In choosing the best ordering of fermionic modes for the Jordan-Wigner transformation, and unlike other popular modifications, our prescription does not cost additional resources such as ancilla qubits. We demonstrate how Mitchison and Durbin's enumeration pattern minimises the average Pauli weight of Jordan-Wigner transformations of systems interacting in square lattices. This leads to qubit Hamiltonians consisting of terms with average Pauli weights 13.9% shorter than previously known. By adding only two ancilla qubits we introduce a new class of fermion-qubit mappings, and reduce the average Pauli weight of Hamiltonian terms by 37.9% compared to previous methods. For $n$-mode fermionic systems in cellular arrangements, we find enumeration patterns which result in $n^{1/4}$ improvement in average Pauli weight over na\"ive schemes.