  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.
