  Quantum chemistry provides a target for quantum simulation of considerable
scientific interest and industrial importance. The majority of algorithms to
date have been based on a second-quantized representation of the electronic
structure Hamiltonian - necessitating qubit requirements that scale linearly
with the number of orbitals. The scaling of the number of gates for such
methods, while polynomial, presents some serious experimental challenges.
However, because the number of electrons is a good quantum number for the
electronic structure problem it is unnecessary to store the full Fock space of
the orbitals. Representation of the wave function in a basis of Slater
determinants for fixed electron number suffices. However, to date techniques
for the quantum simulation of the Hamiltonian represented in this basis - the
CI-matrix - have been lacking. We show how to apply techniques developed for
the simulation of sparse Hamiltonians to the CI-matrix. We prove a number of
results exploiting the structure of the CI-matrix, arising from the Slater
rules which define it, to improve the application of sparse Hamiltonian
simulation techniques in this case. We show that it is possible to use the
minimal number of qubits to represent the wavefunction, and that these methods
can offer improved scaling in the number of gates required in the limit of
fixed electron number and increasing basis set size relevant for high-accuracy
calculations. We hope these results open the door to further investigation of
sparse Hamiltonian simulation techniques in the context of the quantum
simulation of quantum chemistry.
