Improving the full quantum eigensolver with exponentiated operators

There has been an increasing research focus on quantum algorithms for condensed matter systems recently, with a particular emphasis on calculating energy band structures. Here, we propose a quantum algorithm, the powered full quantum eigensolver (P-FQE), by using the power of operators of the full quantum eigensolver. This leads to an exponential increase in the success probability of measuring the target state in certain circumstances where the number of generating elements involved in the power of operators exhibits a log polynomial dependence on the number of orbitals. Furthermore, we conduct numerical calculations for the energy spectrum of the Fermi-Hubbard model and band-structure determination of the twisted double-layer graphene. We experimentally demonstrate the feasibility and robustness of the P-FQE algorithm using superconducting quantum computers for graphene and Weyl semimetal. One significant advantage of our algorithm is its ability to reduce the requirements of extremely high-performance hardware, making it more suitable for energy spectra determination on noisy intermediate-scale quantum devices.