The Bravyi–Kitaev Transformation: Properties and Applications

Andrew Tranter
Sarah Sofia
Jake Seeley
Michael Kaicher
Jarrod McClean
Peter Coveney
Florian Mintert
Frank Wilhelm
Peter Love
International Journal of Quantum Chemistry, 115(2015), pp. 1431-1441


Quantum chemistry is an important area of application for quantum computation. In particular, quantum algorithms applied to the electronic structure problem promise exact, efficient methods for determination of the electronic energy of atoms and molecules. The Bravyi-Kitaev transformation is a method of mapping the occupation state of a fermionic system onto qubits. This transformation maps the Hamiltonian of n interacting fermions to an O(log n)-local Hamiltonian of n qubits. This is an improvement in locality over the Jordan-Wigner transformation, which results in an O(n)-local qubit Hamiltonian. We present the Bravyi-Kitaev transformation in detail, introducing the sets of qubits which must be acted onto change occupancy and parity of states in the occupation number basis. We give recursive definitions of these sets and of the transformation and inverse transformation matrices,which relate the occupation number basis and the Bravyi–Kitaev basis. We then compare the use of the Jordan-Wigner and Bravyi-Kitaev Hamiltonians for the quantum simulation of methane using the STO-6G basis.

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