  Utilizing quantum computer to investigate quantum chemistry is an important
research field nowadays. In addition to the ground-state problems that have
been widely studied, the determination of excited-states plays a crucial role
in the prediction and modeling of chemical reactions and other physical
processes. Here, we propose a non-variational full circuit-based quantum
algorithm for obtaining the excited-state spectrum of a quantum chemistry
Hamiltonian. Compared with previous classical-quantum hybrid variational
algorithms, our method eliminates the classical optimization process, reduces
the resource cost caused by the interaction between different systems, and
achieves faster convergence rate and stronger robustness against noise without
barren plateau. The parameter updating for determining the next energy-level is
naturally dependent on the energy measurement outputs of the previous
energy-level and can be realized by only modifying the state preparation
process of ancillary system, introducing little additional resource overhead.
Numerical simulations of the algorithm with hydrogen, LiH, H2O and NH3
molecules are presented. Furthermore, we offer an experimental demonstration of
the algorithm on a superconducting quantum computing platform, and the results
show a good agreement with theoretical expectations. The algorithm can be
widely applied to various Hamiltonian spectrum determination problems on the
fault-tolerant quantum computers.
