# Analysis of Superfast Encoding Performance for Electronic Structure Simulations

## TL;DR

This paper compares the efficiency of the Bravyi-Kitaev Superfast encoding with the Jordan-Wigner transform in quantum chemistry simulations, highlighting how basis set locality influences resource requirements and performance.

## Contribution

It provides a comparative analysis of OSE and JW mappings, emphasizing the impact of basis set locality on their resource efficiency in quantum simulations.

## Key findings

- OSE outperforms JW with localized orbitals like in Hydrogen lattices.
- JW is more efficient with highly overlapping orbitals.
- Choosing the right basis set and mapping improves quantum simulation efficiency.

## Abstract

In our recent work, we have examined various fermion to qubit mappings in the context of quantum simulation including the original Bravyi-Kitaev Superfast encoding (OSE) as well as a generalized version (GSE). We return to OSE and compare it against the Jordan-Wigner (JW) transform for quantum chemistry considering the number of qubits required, the Pauli weight of terms in the transformed Hamiltonians, and the $L_1$ norm of the Hamiltonian. We considered a test set of molecular systems known as the Atomization Energy 6 (AE6) as well as Hydrogen lattices. Our results showed that the resource efficiency of OSE is strongly affected by the spatial locality of the underlying single-particle basis. We find that OSE is outperformed by JW when the orbitals in the underlying single-particle basis are highly overlapping, which limits its applicability to near-term quantum chemistry simulations utilizing standard basis sets. In contrast, when orbitals are overlapping with only few others, as is the case of Hydrogen lattices with very tight orbitals, OSE fares comparatively better. Our results illustrate the importance of choosing the right combination of basis sets and fermion to qubit mapping to get the most out of a quantum device when simulating physical systems.