Abstract
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.
Highlights
Noisy intermediate-scale quantum (NISQ) devices have a moderate number [O(10) − O(100)] of qubits that we can control very precisely they are not fault tolerant [1,2]
We find that the values of “Effective: ED” and those of “Deep Variational quantum eigensolver (VQE)” are almost identical, which implies that the error of the “Deep VQE” results from the exact ones solely comes from the truncation of the Hilbert space in Step 2 of Deep VQE
We have proposed the improved way for performing Deep VQE, in which we can properly obtain low-energy eigenstates with arbitrary preferable variational quantum circuits on a smaller number of qubits
Summary
Noisy intermediate-scale quantum (NISQ) devices have a moderate number [O(10) − O(100)] of qubits that we can control very precisely they are not fault tolerant [1,2]. By performing the coarse graining of an original large problem based on the solutions of the VQE in smaller subsystems, Deep VQE allows us to obtain the approximate ground state of the original problem with the smaller number of qubits compared to the usual VQE From both physical and technical aspects, it has been still nontrivial and essential whether Deep VQE can deal with low-energy properties, including excited states, which are of great interest both in physics and chemistry. The use of quantum computers (possibly NISQ devices) [26,27,28,34] may circumvent the situation, but it still requires the large number of qubits to perform the calculation In this regards, the coarse graining techniques such as Deep VQE are of great importance for periodic materials.
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