Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials

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, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, 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.