Optimal fermion-qubit mappings

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. All fermion-qubit mappings must use a numbering scheme for the fermionic modes in order to make translation to qubit operations possible. We present a new way to design fermion-qubit mappings by making use of the extra degree of freedom – the choice of enumeration for the fermionic modes. This allows us to minimise the average number of Pauli matrices in each term of a mapping’s qubit Hamiltonian – its average Pauli weight. Finding the best enumeration scheme allows one to increase the locality of the target qubit Hamiltonian without expending any additional resources.

We demonstrate how Mitchison-Durbin’s enumeration pattern is optimal for the square fermionic lattice. This leads to a qubit Hamiltonian for simulating the square fermionic lattice consisting of terms with an average Pauli weight 13.9% shorter than previously known. Furthermore, by adding only two ancilla qubits we can reduce the average Pauli weight of Hamiltonian terms by 37.9% on square lattices compared to previous methods. For other natural n-mode fermionic systems we find enumeration patterns which result in n^1/4 improvement in average Pauli weight over naïve enumeration schemes,