  The ability to simulate a fermionic system on a quantum computer is expected
to revolutionize chemical engineering, materials design, nuclear physics, to
name a few. Thus, optimizing the simulation circuits is of significance in
harnessing the power of quantum computers. Here, we address this problem in two
aspects. In the fault-tolerant regime, we optimize the $\rzgate$ and $\tgate$
gate counts along with the ancilla qubit counts required, assuming the use of a
product-formula algorithm for implementation. We obtain a savings ratio of two
in the gate counts and a savings ratio of eleven in the number of ancilla
qubits required over the state of the art. In the pre-fault tolerant regime, we
optimize the two-qubit gate counts, assuming the use of the variational quantum
eigensolver (VQE) approach. Specific to the latter, we present a framework that
enables bootstrapping the VQE progression towards the convergence of the
ground-state energy of the fermionic system. This framework, based on
perturbation theory, is capable of improving the energy estimate at each cycle
of the VQE progression, by about a factor of three closer to the known
ground-state energy compared to the standard VQE approach in the test-bed,
classically-accessible system of the water molecule. The improved energy
estimate in turn results in a commensurate level of savings of quantum
resources, such as the number of qubits and quantum gates, required to be
within a pre-specified tolerance from the known ground-state energy. We also
explore a suite of generalized transformations of fermion to qubit operators
and show that resource-requirement savings of up to more than $20\%$, in small
instances, is possible.
