  It is crucial to reduce the resources required to run quantum algorithms and
simulate physical systems on quantum computers due to coherence time
limitations. With regards to Hamiltonian simulation, a significant effort has
focused on building efficient algorithms using various factorizations and
truncations, typically derived from the Hamiltonian alone. We introduce a new
paradigm for improving Hamiltonian simulation and reducing the cost of ground
state problems based on ideas recently developed for classical chemistry
simulations. The key idea is that one can find efficient ways to reduce
resources needed by quantum algorithms by making use of two key pieces of
information: the Hamiltonian operator and an approximate ground state
wavefunction. We refer to our algorithm as the $\textit{Self-consistent Quantum
Iteratively Sparsified Hamiltonian}$ (SQuISH). By performing our scheme
iteratively, one can drive SQuISH to create an accurate wavefunction using a
truncated, resource-efficient Hamiltonian. Utilizing this truncated Hamiltonian
provides an approach to reduce the gate complexity of ground state calculations
on quantum hardware. As proof of principle, we implement SQuISH using
configuration interaction for small molecules and coupled cluster for larger
systems. Through our combination of approaches, we demonstrate how SQuISH
performs on a range of systems, the largest of which would require more than
200 qubits to run on quantum hardware. Though our demonstrations are on a
series of electronic structure problems, our approach is relatively generic and
hence likely to benefit additional applications where the size of the problem
Hamiltonian creates a computational bottleneck.
