  There has been an increasing research focus on quantum algorithms for
condensed matter systems recently, particularly on calculating energy band
structures. Here, we propose a quantum algorithm, the powered full quantum
eigensolver(P-FQE), by using the exponentiation of operators of the full
quantum eigensolver(FQE). 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 exponentiation of operators
exhibit a log polynomial dependence on the number of orbitals. Furthermore, we
conduct numerical calculations for 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 (NISQ) devices.
