Bonsai Algorithm: Grow Your Own Fermion-to-Qubit Mappings

Abstract

Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations. In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees. We intuitively discuss the connection between the structure of the generating trees and certain properties of the resulting mapping, such as the Pauli weight and the delocalization of mode occupation. Moreover, we introduce a recipe that guarantees that Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing. Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the swap overhead compared to other paradigmatic mappings. We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers. The resulting mappings have a favorable Pauli-weight scaling O(sqrt[N]) on this connectivity while ensuring that no swap gates are necessary for single-excitation operations.