  We introduce novel algorithms for the quantum simulation of molecular systems
which are asymptotically more efficient than those based on the Trotter-Suzuki
decomposition. We present the first application of a recently developed
technique for simulating Hamiltonian evolution using a truncated Taylor series
to obtain logarithmic scaling with the inverse of the desired precision, an
exponential improvement over all prior methods. The two algorithms developed in
this work rely on a second quantized encoding of the wavefunction in which the
state of an $N$ spin-orbital system is encoded in ${\cal O}(N)$ qubits. Our
first algorithm requires at most $\widetilde{\cal O}(N^8 t)$ gates. Our second
algorithm involves on-the-fly computation of molecular integrals, in a way that
is exponentially more precise than classical sampling methods, by using the
truncated Taylor series simulation technique. Our second algorithm has the
lowest gate count of any approach to second quantized quantum chemistry
simulation in the literature, scaling as $\widetilde{\cal O}(N^{5} t)$. The
approaches presented here are readily applicable to a wide class of fermionic
models, many of which are defined by simplified versions of the chemistry
Hamiltonian.
