The Bonsai algorithm: grow your own fermion-to-qubit mapping

TL;DR

The paper introduces the Bonsai algorithm, a novel method for designing fermion-to-qubit mappings tailored to specific qubit connectivities, reducing SWAP overhead and optimizing quantum simulations.

Contribution

It presents a formalism based on ternary trees for creating flexible fermion-to-qubit mappings and introduces the Bonsai algorithm for topology-aware mapping optimization.

Findings

Mappings for IBM's heavy-hexagon topology with $\\mathcal{O}(\sqrt{N})$ Pauli weight scaling.

Mappings ensure no SWAP gates are needed for single excitation operations.

The formalism guarantees Fock basis states map to computational basis states.

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 discuss in an intuitive manner the connection between the generating trees' structure and certain properties of the resulting mapping, such as Pauli weight and the delocalisation of mode occupation. Moreover, we introduce a recipe that guarantees 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 with respect 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 favourable Pauli weight scaling $\mathcal{O}(\sqrt{N})$ on this connectivity, while ensuring that no SWAP gates are necessary for single excitation operations.