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The VQE algorithm has turned out to be quite expensive to run given the waywe currently access quantum processors (i.e. over the cloud). In order toalleviate this issue, we introduce Quantum Sampling Regression (QSR), analternative hybrid quantum-classical algorithm, and analyze some of its usecases based on time complexity in the low qubit number regime. In exchange forsome extra classical resources, this novel strategy is proved to be optimal interms of the number of samples it requires from the quantum processor. Wedevelop a simple analytical model to evaluate when this algorithm is moreefficient than VQE, and, from the same theoretical considerations, establish athreshold above which quantum advantage can occur. Finally, we demonstrate theefficacy of our algorithm for a benchmark problem.

An optimal quantum sampling regression algorithm for variational eigensolving in the low qubit number regime