Penalty methods for variational quantum eigensolver

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, but systematic analysis on 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 work, we investigate two major types of penalty terms for the VQE that were proposed in the previous studies. We show a penalty term in 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 the faster convergence of the VQE. Meanwhile, we prove that the other type of penalty terms 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.

10 pages, 2 figures, 5 tables; published version

References in corpus (2)

• Quantum algorithm for solving linear systems of equations
• Quantum Computation of Electronic Transitions using a Variational Quantum Eigensolver

Cited by in corpus (29)

• The Variational Quantum Eigensolver: a review of methods and best practices
• Exploiting symmetry in variational quantum machine learning
• Emerging quantum computing algorithms for quantum chemistry
• Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice
• Quantum Machine-Learning for Eigenstate Filtration in Two-Dimensional Materials
• Symmetry enhanced variational quantum spin eigensolver
• Orbital transformations to reduce the 1-norm of the electronic structure Hamiltonian for quantum computing applications
• Noisy intermediate-scale quantum algorithm for semidefinite programming
• Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers
• The Basics of Quantum Computing for Chemists
• Bootstrap Embedding on a Quantum Computer
• Simulating strongly interacting Hubbard chains with the Variational Hamiltonian Ansatz on a quantum computer
• Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer
• Extending the Variational Quantum Eigensolver to Finite Temperatures
• Quantum circuits for the preparation of spin eigenfunctions on quantum computers
• Multiobjective variational quantum optimization for constrained problems: an application to Cash Management
• Molecular Structure Optimization based on Electrons-Nuclei Quantum Dynamics Computation
• A backend-agnostic, quantum-classical framework for simulations of chemistry in C++
• Calculation of core-excited and core-ionized states using variational quantum deflation method and applications to photocatalyst modelling
• Optimization of the Variational Quantum Eigensolver for Quantum Chemistry Applications
• Variational quantum simulation of long-range interacting systems
• Mitigating Quantum Gate Errors for Variational Eigensolvers Using Hardware-Inspired Zero-Noise Extrapolation
• Sequence of penalties method to study excited states using VQE
• Quantum computation of a quasiparticle band structure with the quantum-selected configuration interaction
• Inequality constraints in variational quantum circuits with qudits
• The Levy-Lieb embedding of density functional theory and its Quantum Kernel: Illustration for the Hubbard Dimer using near-term quantum algorithms
• Jerky chiral active particles
• Molecular resonance identification in complex absorbing potentials via integrated quantum computing and high-throughput computing
• Accelerated spin-adapted ground state preparation with non-variational quantum algorithms