The Bravyi-Kitaev transformation for quantum computation of electronic structure

TL;DR

This paper introduces the Bravyi-Kitaev transformation as an efficient method for simulating fermionic systems in quantum computing, reducing the computational cost compared to the traditional Jordan-Wigner approach, demonstrated through quantum chemistry examples.

Contribution

The paper develops and applies the Bravyi-Kitaev transformation to quantum simulation of electronic structure, showing improved efficiency over Jordan-Wigner for molecular hydrogen.

Findings

Bravyi-Kitaev reduces qubit operation count to O(log n)

Fewer gates needed for simulating H2 molecule

Demonstrates superior asymptotic scaling over Jordan-Wigner

Abstract

Quantum simulation is an important application of future quantum computers with applications in quantum chemistry, condensed matter, and beyond. Quantum simulation of fermionic systems presents a specific challenge. The Jordan-Wigner transformation allows for representation of a fermionic operator by O(n) qubit operations. Here we develop an alternative method of simulating fermions with qubits, first proposed by Bravyi and Kitaev [S. B. Bravyi, A.Yu. Kitaev, Annals of Physics 298, 210-226 (2002)], that reduces the simulation cost to O(log n) qubit operations for one fermionic operation. We apply this new Bravyi-Kitaev transformation to the task of simulating quantum chemical Hamiltonians, and give a detailed example for the simplest possible case of molecular hydrogen in a minimal basis. We show that the quantum circuit for simulating a single Trotter time-step of the Bravyi-Kitaev derived Hamiltonian for H2 requires fewer gate applications than the equivalent circuit derived from the Jordan-Wigner transformation. Since the scaling of the Bravyi-Kitaev method is asymptotically better than the Jordan-Wigner method, this result for molecular hydrogen in a minimal basis demonstrates the superior efficiency of the Bravyi-Kitaev method for all quantum computations of electronic structure.