Abstract
The variational quantum eigensolver (VQE) is a promising algorithm to compute eigenstates and eigenenergies of a given quantum system that can be performed on a near-term quantum computer. Obtaining eigenstates and eigenenergies in a specific symmetry sector of the system is often necessary for practical applications of the VQE in various fields ranging from high-energy physics to quantum chemistry. It is common to add a penalty term in the cost function of the VQE to calculate such a symmetry-resolving energy spectrum; however, systematic analysis of the effect of the penalty term has been lacking, and the use of the penalty term in the VQE has not been justified rigorously. In this paper, we investigate two major types of penalty terms for the VQE that were proposed in previous studies. We show that a penalty term of one of the two types works properly in that eigenstates obtained by the VQE with the penalty term reside in the desired symmetry sector. We further give a convenient formula to determine the magnitude of the penalty term, which may lead to faster convergence of the VQE. Meanwhile, we prove that the other type of penalty term does not work for obtaining the target state with the desired symmetry in a rigorous sense and even gives completely wrong results in some cases. We finally provide numerical simulations to validate our analysis. Our results apply to general quantum systems and lay the theoretical foundation for the use of the VQE with the penalty terms to obtain the symmetry-resolving energy spectrum of the system, which fuels the application of a near-term quantum computer.Received 29 October 2020Revised 13 January 2021Accepted 21 January 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.013197Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum simulationTechniquesSymmetries in condensed matterQuantum InformationCondensed Matter, Materials & Applied Physics
Highlights
Quantum computers have been attracting attention because they are expected to solve some classes of computation problems remarkably faster than classical computers [1,2,3,4]
We study two cost functions F (1)(θ) [Eq (5)] and F (2)(θ) [Eq (6)] of the constrained variational quantum eigensolver (VQE) and variational quantum deflation (VQD), which can be exploited to compute eigenstates of the Hamiltonian of a given system that reside in the desired symmetry sector
Our theoretical analysis revealed that minimization of the cost function F (1)(θ) can yield the desired state and energy when the penalty coefficient μC is larger than a certain threshold [Eq (11)], and we derived a simple and practical formula to estimate it [Eq (13)]
Summary
Quantum computers have been attracting attention because they are expected to solve some classes of computation problems remarkably faster than classical computers [1,2,3,4]. Quantum computers that are expected to be realized in the near future are called noisy intermediate-scale quantum (NISQ) devices, which contain a few hundred qubits and do not perform error correction on their qubits [5]. The variational quantum eigensolver (VQE) [6], which was already performed on a real NISQ device [6,7,8], computes the ground-state energy and the ground state of a given quantum system, for example, molecules [9,10]. In the VQE, we prepare a parametrized state, an ansatz state, on a quantum computer and minimize the expectation value of energy for this state by updating the parameters.
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