  A programmable quantum device that has a large number of qubits without
fault-tolerance has emerged recently. Variational Quantum Eigensolver (VQE) is
one of the most promising ways to utilize the computational power of such
devices to solve problems in condensed matter physics and quantum chemistry. As
the size of the current quantum devices is still not large for rivaling
classical computers at solving practical problems, Fujii et al. proposed a
method called "Deep VQE" which can provide the ground state of a given quantum
system with the smaller number of qubits by combining the VQE and the technique
of coarse-graining [K. Fujii, et al, arXiv:2007.10917]. In this paper, we
extend the original proposal of Deep VQE to obtain the excited states and apply
it to quantum chemistry calculation of a periodic material, which is one of the
most impactful applications of the VQE. We first propose a modified scheme to
construct quantum states for coarse-graining in Deep VQE to obtain the excited
states. We also present a method to avoid a problem of meaningless eigenvalues
in the original Deep VQE without restricting variational quantum states.
Finally, we classically simulate our modified Deep VQE for quantum chemistry
calculation of a periodic hydrogen chain as a typical periodic material. Our
method reproduces the ground-state energy and the first-excited-state energy
with the errors up to O(1)% despite the decrease in the number of qubits
required for the calculation by two or four compared with the naive VQE. Our
result will serve as a beacon for tackling quantum chemistry problems with
classically-intractable sizes by smaller quantum devices in the near future.
