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The efficient calculation of Hamiltonian spectra, a problem often intractableon classical machines, can find application in many fields, from physics tochemistry. Here, we introduce the concept of an "eigenstate witness" andthrough it provide a new quantum approach which combines variational methodsand phase estimation to approximate eigenvalues for both ground and excitedstates. This protocol is experimentally verified on a programmable siliconquantum photonic chip, a mass-manufacturable platform, which embeds entangledstate generation, arbitrary controlled-unitary operations, and projectivemeasurements. Both ground and excited states are experimentally found withfidelities >99%, and their eigenvalues are estimated with 32-bits of precision.We also investigate and discuss the scalability of the approach and study itsperformance through numerical simulations of more complex Hamiltonians. Thisresult shows promising progress towards quantum chemistry on quantum computers.

Witnessing eigenstates for quantum simulation of Hamiltonian spectra