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.
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.
Abstract We propose an extension of the variational quantum eigensolver (VQE) that leads to more accurate energy estimations and can be used to study excited states. The method is based on the introduction of a sequence of increasing penalties in the cost function. This approach does not require circuit modifications and thus can be applied with no additional depth cost. Through numerical simulations, we show that we are able to produce variational states with desired physical properties, such as total spin and charge. We assess its performance both on classical simulators and on currently available quantum devices, calculating the potential energy curves of small molecular systems in different physical configurations. Finally, we compare our method to the original VQE and to another extension, obtaining a better agreement with exact simulations for both energy and targeted physical quantities.
This work studies the variational quantum eigensolver (VQE) algorithm, which is designed to determine the ground state of a quantum mechanical system by combining classical and quantum hardware. Two methods of reducing the number of required qubit manipulations, prone to induce errors, for the variational quantum eigensolver are studied. First, we formally justify the multiple ℤ2 symmetry qubit reduction scheme first sketched by Bravyi et al. [arXiv:1701.08213 (2017)]. Second, we show that even in small, but non-trivial systems such as H2, LiH, and H2O, the choice of entangling methods (gate based or native) gives rise to varying rates of convergence to the ground state of the system. Through both the removal of qubits and the choice of entangler, the demands on the quantum hardware can be reduced. We find that in general, analyzing the VQE problem is complex, where the number of qubits, the method of entangling, and the depth of the search space all interact. In specific cases however, concrete results can be shown, and an entangling method can be recommended over others as it outperforms in terms of difference from the ground state energy.
Variational Quantum Eigensolvers (VQEs) have become a standard tool for use in preparing eigenstates in quantum simulations of many-body systems. In the case of the nuclear shell model, various adaptations of the VQE have been shown to be able to produce ground and excited states of nuclei. The VQE algorithm works by seeking minima in a cost function, which is often the expectation value of the energy. For a well-formulated problem, symmetries that are respected by the Hamiltonian should appear in the final converged solution provided by the VQE, even if those symmetries are broken by the wave function ansatz. We demonstrate explicit appearance of the particle–hole symmetry, and explicit recovery of angular momentum conservation due to rotational symmetry, during VQE runs in the case of the nuclear shell model. We conduct analyses of how the symmetry convergence appears in different ansatz formulations, and suggest that information obtained could be used in future VQE design.
Variational quantum algorithms have been one of the most intensively studied applications for near-term quantum computing applications. The noisy intermediate-scale quantum (NISQ) regime, where small enough algorithms can be run successfully on noisy quantum computers expected during the next 5 years, is driving both a large amount of research work and a significant amount of private sector funding. Therefore, it is important to understand whether variational algorithms are effective at successfully converging to the correct answer in presence of noise. We perform a comprehensive study of the variational quantum eigensolver (VQE) and its individual quantum subroutines. Building on asymptotic bounds, we show through explicit simulation that the VQE algorithm effectively collapses already when single errors occur during a quantum processing call. We discuss the significant implications of this result in the context of being able to run any variational type algorithm without resource expensive error correction protocols.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.
The variational quantum eigensolver (VQE) is an algorithm to find eigenenergies and eigenstates of systems in quantum chemistry and quantum many-body physics.The VQE is one of the most promising applications of near-term quantum devices to investigate such systems.Here we propose an extension of the VQE to calculate the nonadiabatic couplings of molecules in quantum chemical systems and Berry's phase in quantum many-body systems.Both quantities play an important role in understanding the properties of a system beyond the naive adiabatic picture, e.g., nonadiabatic dynamics and topological phase of matter.We provide quantum circuits and classical postprocessings to calculate the nonadiabatic couplings and Berry's phase.Specifically, we show that the evaluation of the nonadiabatic couplings reduces to that of expectation values of observables while that of Berry's phase also requires one additional Hadamard test.Furthermore, we simulate the photodissociation dynamics of a lithium fluoride molecule using the nonadiabatic couplings evaluated on a real quantum device.Our proposal widens the applicability of the VQE and the possibility of near-term quantum devices to study molecules and quantum many-body systems.