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It is crucial to reduce the resources required to run quantum algorithms andsimulate physical systems on quantum computers due to coherence timelimitations. With regards to Hamiltonian simulation, a significant effort hasfocused on building efficient algorithms using various factorizations andtruncations, typically derived from the Hamiltonian alone. We introduce a newparadigm for improving Hamiltonian simulation and reducing the cost of groundstate problems based on ideas recently developed for classical chemistrysimulations. The key idea is that one can find efficient ways to reduceresources needed by quantum algorithms by making use of two key pieces ofinformation: the Hamiltonian operator and an approximate ground statewavefunction. We refer to our algorithm as the $\textit{Self-consistent QuantumIteratively Sparsified Hamiltonian}$ (SQuISH). By performing our schemeiteratively, one can drive SQuISH to create an accurate wavefunction using atruncated, resource-efficient Hamiltonian. Utilizing this truncated Hamiltonianprovides an approach to reduce the gate complexity of ground state calculationson quantum hardware. As proof of principle, we implement SQuISH usingconfiguration interaction for small molecules and coupled cluster for largersystems. Through our combination of approaches, we demonstrate how SQuISHperforms on a range of systems, the largest of which would require more than200 qubits to run on quantum hardware. Though our demonstrations are on aseries of electronic structure problems, our approach is relatively generic andhence likely to benefit additional applications where the size of the problemHamiltonian creates a computational bottleneck.

Self-consistent Quantum Iteratively Sparsified Hamiltonian method (SQuISH): A new algorithm for efficient Hamiltonian simulation and compression