Optimal fermion-qubit mappings

Abstract

Simulating fermionic systems on a quantum computer requires a high-performing mapping of fermionic states to qubits. The key characteristic of an efficient mapping is its ability to translate local fermionic interactions into local qubit interactions, leading to easy-to-simulate qubit Hamiltonians. Improvements in the locality of fermion-qubit mappings have traditionally come at higher resource costs elsewhere, such as in the form of a significant number of additional qubits. We present a new way to design fermion-qubit mappings by making use of the extra degree of freedom: the choice of numbering scheme for the fermionic modes, a feature all mappings must have. This allows us to minimse the average Pauli weight of a qubit Hamiltonian -- its average number of Pauli matrices per term. Our approach leads to a rigorous notion of optimality by viewing fermion-qubit mappings as functions of their enumeration schemes. Furthermore, finding the best enumeration scheme allows one to increase the locality of the target qubit Hamiltonian without expending any additional resources. Minimising the average Pauli weight of a mapping is an NP-complete problem in general. We show how one solution, Mitchison and Durbin's enumeration pattern, leads to a qubit Hamiltonian for simulating the square fermionic lattice consisting of terms with an average Pauli weight 13.9% shorter than previously any previously known. Adding just two ancilla qubits, we can reduce the average Pauli weight of Hamiltonian terms by 37.9% on square lattices compared to previous methods. Lastly, we demonstrate the potential of our techniques to polynomially reduce the average Pauli weight by exhibiting $n$-mode fermionic systems where optimisation yields patterns that achieve $n^{\frac{1}{4}}$ improvement in average Pauli weight over na\"ive enumeration schemes.Comment: 29 pages, 30 figure