# A comparison of the Bravyi-Kitaev and Jordan-Wigner transformations for the quantum simulation of quantum chemistry

## TL;DR

This paper compares the Bravyi-Kitaev and Jordan-Wigner transformations for quantum chemistry simulations, showing that Bravyi-Kitaev often reduces quantum gate counts and is similarly efficient for various molecular systems.

## Contribution

It provides a comprehensive resource comparison of the two transformations across 86 molecules, highlighting the efficiency of Bravyi-Kitaev in quantum simulations.

## Key findings

- Bravyi-Kitaev typically reduces gate counts compared to Jordan-Wigner.
- Both transformations are similarly efficient for small systems.
- Bravyi-Kitaev offers substantial gate reduction with circuit optimizations.

## Abstract

The ability to perform classically intractable electronic structure calculations is often cited as one of the principal applications of quantum computing. A great deal of theoretical algorithmic development has been performed in support of this goal. Most techniques require a scheme for mapping electronic states and operations to states of and operations upon qubits. The two most commonly used techniques for this are the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. However, comparisons of these schemes have previously been limited to individual small molecules. In this paper we discuss resource implications for the use of the Bravyi-Kitaev mapping scheme, specifically with regard to the number of quantum gates required for simulation. We consider both small systems which may be simulatable on near-future quantum devices, and systems sufficiently large for classical simulation to be intractable. We use 86 molecular systems to demonstrate that the use of the Bravyi-Kitaev transformation is typically at least approximately as efficient as the canonical Jordan-Wigner transformation, and results in substantially reduced gate count estimates when performing limited circuit optimisations.