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

The Bravyi-Kitaev transformation for quantum computation of electronic structure