  We present a hybrid quantum-classical Green's function Monte Carlo (GFMC)
algorithm for estimating the excited states of the nuclear shell model. The
conventional GFMC method, widely used to find the ground state of a quantum
many-body system, is plagued by the sign problem, which leads to an
exponentially increasing variance with the growth of system size and evolution
time. This issue is typically mitigated by applying classical constraints but
at the cost of introducing bias. Our approach uses quantum subspace
diagonalization (QSD) on a quantum computer to prepare a quantum trial state,
replacing the classical trial state in the GFMC process. We also incorporated a
modified classical shadow technique in the implementation of QSD to optimize
quantum resource utilization. Besides, we extend our hybrid GFMC algorithm to
find the excited states of a given quantum system. Numerical results suggest
our method largely enhances accuracy in determining excited state energies,
offering an improvement over the conventional method.
