Efficient Quantum Estimation of Hamiltonian Spectra via Shallow Circuits.

Quantum computers promise to provide groundbreaking speed in solving complex problems. However, in the present-day noisy intermediate-scale quantum era, algorithms that require fewer resources are highly desired. In this work, we develop a new method called the variational rodeo eigensolver (VRE) for efficiently searching eigenstates and estimating eigenvalues with shallow circuits. We experimentally demonstrate this method on a programmable photonic chip with a single-qubit exciton transfer Hamiltonian whose eigenstates are searched with fidelities of more than 99% and their eigenvalues are estimated, reaching chemical accuracy. Furthermore, we experimentally estimate the ground energies of the simplified Hamiltonian of the hydrogen molecule with different atomic separations. To verify the scalability of VRE, we numerically search eigenstates of a tapered two-qubit Hamiltonian of hydrogen-helium ion. Our work provides a systematic and promising approach for the efficient estimation of Hamiltonian spectra.