  Quantum computers promise to efficiently solve important problems that are
intractable on a conventional computer. For quantum systems, where the
dimension of the problem space grows exponentially, finding the eigenvalues of
certain operators is one such intractable problem and remains a fundamental
challenge. The quantum phase estimation algorithm can efficiently find the
eigenvalue of a given eigenvector but requires fully coherent evolution. We
present an alternative approach that greatly reduces the requirements for
coherent evolution and we combine this method with a new approach to state
preparation based on ans\"atze and classical optimization. We have implemented
the algorithm by combining a small-scale photonic quantum processor with a
conventional computer. We experimentally demonstrate the feasibility of this
approach with an example from quantum chemistry: calculating the ground state
molecular energy for He-H+, to within chemical accuracy. The proposed approach,
by drastically reducing the coherence time requirements, enhances the potential
of the quantum resources available today and in the near future.
