The Variational Quantum Eigensolver: a review of methods and best practices

Abstract

The variational quantum eigensolver (or VQE) uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits. The VQE may be used to model complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims to provide an overview of the progress that has been made on the different parts of the algorithm. All the different components of the algorithm are reviewed in detail including representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process, the post-processing mitigation of errors, and best practices are suggested. We identify four main areas of future research:(1) optimal measurement schemes for reduction of circuit repetitions; (2) large scale parallelization across many quantum computers;(3) ways to overcome the potential appearance of vanishing gradients in the optimization process, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.

156 pages, 19 figures

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• Spontaneous symmetry breaking in a $SO(3)$ non-Abelian lattice gauge theory in $2+1$D with quantum algorithms
• Spectral Gap Optimization for Enhanced Adiabatic State Preparation
• Noise tailoring for Robust Amplitude Estimation
• Inequality constraints in variational quantum circuits with qudits
• A quantum computing approach to fixed-node Monte Carlo using classical shadows
• Quantum Machine Learning of Molecular Energies with Hybrid Quantum-Neural Wavefunction
• Certified Random Number Generation using Quantum Computers
• Minimal evolution times for fast, pulse-based state preparation in silicon spin qubits
• Density matrix representation of hybrid tensor networks for noisy quantum devices
• Benchmarking Variational Quantum Algorithms for Combinatorial Optimization in Practice
• A recipe for local simulation of strongly-correlated fermionic matter on quantum computers: the 2D Fermi-Hubbard model
• Quantum Software Ecosystem Design
• Multi-set variational quantum dynamics algorithm for simulating nonadiabatic dynamics on quantum computers
• Photonic variational quantum eigensolver for NISQ-compatible quantum technology
• QRCC: Evaluating Large Quantum Circuits on Small Quantum Computers through Integrated Qubit Reuse and Circuit Cutting
• Simulating quantum field theories on gate-based quantum computers
• Effective temperature in approximate quantum many-body states
• Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations
• Toward scalable quantum computations of atomic nuclei
• Tangent Space Excitation Ansatz for Quantum Circuits
• Grover's search meets Ising models: a quantum algorithm for finding low-energy states
• Practical limitations of quantum data propagation on noisy quantum processors
• The Levy-Lieb embedding of density functional theory and its Quantum Kernel: Illustration for the Hubbard Dimer using near-term quantum algorithms
• Quantum Computing Beyond Ground State Electronic Structure: A Review of Progress Toward Quantum Chemistry Out of the Ground State
• Almost optimal measurement scheduling of molecular Hamiltonian via finite projective plane
• Simulating methylamine using symmetry adapted qubit-excitation-based variational quantum eigensolver
• Limitations of Quantum Hardware for Molecular Energy Estimation Using VQE
• Coupled-cluster pairing models for radicals with strong correlations
• Extracting the spin excitation spectrum of a fermionic system using a quantum processor
• Multi-target quantum compilation algorithm
• Full-counting statistics of particle distribution on a digital quantum computer
• Entanglement-informed Construction of Variational Quantum Circuits
• Measuring Correlation and Entanglement between Molecular Orbitals on a Trapped-Ion Quantum Computer
• Does the full configuration interaction method based on quantum phase estimation with Trotter decomposition satisfy the size consistency condition?
• Quantum implicit representation of vortex filaments in turbulence
• State Specific Measurement Protocols for the Variational Quantum Eigensolver
• Demonstration of a variational quantum eigensolver with a solid-state spin system under ambient conditions
• QUBO Formulations for Variation of Domination Problem
• Variational simulation of higher-spin systems on qubit-based quantum simulators
• Best-practice aspects of quantum-computer calculations: A case study of hydrogen molecule
• Effects of Topological Boundary Conditions on Bell Nonlocality
• Batched Line Search Strategy for Navigating through Barren Plateaus in Quantum Circuit Training
• Cheaper and more noise-resilient quantum state preparation using eigenvector continuation
• A quantum approach for optimal control
• Investigating Techniques to Optimise the Layout of Turbines in a Windfarm using a Quantum Computer
• Warm Start of Variational Quantum Algorithms for Quadratic Unconstrained Binary Optimization Problems
• Hamiltonian Lattice Formulation of Compact Maxwell-Chern-Simons Theory
• Finding eigenvectors with a quantum variational algorithm
• OnionVQE Optimization Strategy for Ground State Preparation on NISQ Devices
• Adaptive Quantum Generative Training using an Unbounded Loss Function
• Empirical Power of Quantum Encoding Methods for Binary Classification
• On The Study Of Partial Qubit Hamiltonian For Efficient Molecular Simulation Using Variational Quantum Eigensolvers
• Accelerating Quantum Eigensolver Algorithms With Machine Learning
• Symmetry-enhanced Counterdiabatic Quantum Algorithm for Qudits
• Feasibility of performing quantum chemistry calculations on quantum computers
• Sparse Non-Markovian Noise Modeling of Transmon-Based Multi-Qubit Operations
• High-fidelity initialization a logical qubit with multiple injections
• Notes on Quantum Computing for Thermal Science
• Parallelized Givens Ansatz for Molecular ground-states: Bridging Accuracy and Efficiency on NISQ Platforms
• An Error Mitigated Non-Orthogonal Quantum Eigensolver via Shadow Tomography
• Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization
• Quantum algorithm for solving generalized eigenvalue problems with application to the Schrödinger equation
• Performance-centric roadmap for building a superconducting quantum computer
• Optimizing Quantum Chemistry Simulations with a Hybrid Quantization Scheme
• OrQstrator: An AI-Powered Framework for Advanced Quantum Circuit Optimization
• The Dual Role of Low-Weight Pauli Propagation: A Flawed Simulator but a Powerful Initializer for Variational Quantum Algorithms
• Quantum State Preparation Of Multiconfigurational States For Quantum Chemistry
• Mitigating the measurement overhead of ADAPT-VQE with optimised informationally complete generalised measurements
• Hybrid quantum gap estimation algorithm using a filtered time series
• Simple and efficient end-to-end quantum thermal and ground state preparation
• Orbital Expansion Variational Quantum Eigensolver: Enabling Efficient Simulation of Molecules with Shallow Quantum Circuit
• Determining the ensemble N-representability of Reduced Density Matrices
• A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures
• Quantum-Assisted Space Logistics Mission Planning
• Quantum circuit evolutionary framework applied on set partitioning problem
• Many-body thermal states on a quantum computer: a variational approach
• Resource-Optimized Grouping Shadow for Efficient Energy Estimation
• Inference of maximum parsimony phylogenetic trees with model-based classical and quantum methods
• Imaginary Time Spectral Transforms for Excited State Preparation
• Evaluating Ground State Energies of Chemical Systems with Low-Depth Quantum Circuits and High Accuracy
• Quantum Algorithms for Causal Estimands
• Towards Classical Software Verification using Quantum Computers
• Protocol for Purifying Noisy Preparation and Measurements of Qubits
• Variational Quantum Algorithm for Solving the Liouvillian Gap
• Qubit frugal entanglement determination with the deep multi-scale entanglement renormalization ansatz
• Preparation of the single-spinon wave function on a quantum computer
• Quantum reservoir computing for predicting and characterizing chaotic maps
• Classical Neural Networks on Quantum Devices via Tensor Network Disentanglers: A Case Study in Image Classification
• A Taxonomy of Real Faults in Hybrid Quantum-Classical Architectures
• A penalty-free quantum algorithm to find energy eigenstates
• Perturbation theory, irrep truncations, and state preparation methods for quantum simulations of SU(3) lattice gauge theory
• Slice-Wise Initial State Optimization to Improve Cost and Accuracy of the VQE on Lattice Models
• A quantum eigenvalue solver based on tensor networks
• Circuit Folding: Scalable and Graph-Based Circuit Cutting via Modular Structure Exploitation
• Utility-Scale Quantum State Preparation: Classical Training using Pauli Path Simulation
• Data Efficient Prediction of excited-state properties using Quantum Neural Networks
• Benchmarking Quantum Simulation of Chemical Hamiltonians using the Sorted-List Encoding
• Exploiting biased noise in variational quantum models
• Molecular resonance identification in complex absorbing potentials via integrated quantum computing and high-throughput computing
• Resource-Efficient Quantum Optimization via Higher-Order Encoding
• Exploring fixed points and eigenstates of quantum systems with reinforcement learning
• Quantum imaginary time evolution and UD-MIS problem
• Zero-Noise Extrapolation via Cyclic Permutations of Quantum Circuit Layouts
• Resource Estimation for VQE on Small Molecules: Impact of Fermion Mappings and Hamiltonian Reductions
• Optimizing Unitary Coupled Cluster Wave Functions on Quantum Hardware: Error Bound and Resource-Efficient Optimizer
• Noisy Monitored Quantum Circuits
• Variational matrix product states for combinatorial optimization
• Noise-resilient and resource-efficient hybrid algorithm for robust quantum gap estimation
• Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space
• Verifiable cloud-based variational quantum algorithms
• Topology-Aware Block Coordinate Descent for Qubit Frequency Allocation of Superconducting Quantum Processors
• Spin-$s$ $U(1)$-eigenstate preparation
• Quantum Phase Recognition via Quantum Attention Mechanism
• Resource-efficient Quantum Algorithms for Selected Hamiltonian Subspace Diagonalization
• Effective Bethe Ansatz for Spin-1 Non-integrable Models
• Error Mitigation in Dynamic Circuits for Hamiltonian Simulation
• How Many Shots Are Enough for a Quantum Circuit?
• Velocity Verlet-based optimization for variational quantum eigensolvers
• Efficiently preparing chiral states via fermionic cooling on bosonic quantum hardware
• Double-bracket quantum algorithms for high-fidelity ground state preparation
• Improving the trainability of VQE on NISQ computers for solving portfolio optimization using convex interpolation
• Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver
• Extension of the Jordan-Wigner mapping to nonorthogonal spin orbitals for quantum computing application to valence bond approaches
• Universal Euler-Cartan Circuits for Quantum Field Theories
• Hybrid Quantum-Classical Clustering for Preparing a Prior Distribution of Eigenspectrum
• Quantum Computing Solution of the Bethe-Salpeter Equation for Relativistic Scalar Bound States via Tensor-Network VQE
• Quantum-Classical Computing for Time-Dependent Ion-Atom Collision Dynamics: Applications to Charge Transfer Cross Section Simulations
• Qudit-native measurement protocol for dynamical correlations using Hadamard tests
• Hamiltonian Expressibility for Ansatz Selection in Variational Quantum Algorithms
• Digital quantum simulation of squeezed states via enhanced bosonic encoding in a superconducting quantum processor
• Analog Circuit-QED Simulator of Quantum Spin Dynamics Through the Extended Bose-Hubbard Model
• Shot-Efficient ADAPT-VQE via Reused Pauli Measurements and Variance-Based Shot Allocation
• Qubit encodings for lattices of dipolar planar rotors
• Learning Variational Quantum Circuit Parameters with Classical Artificial Intelligence for Quantum Phase Transition Detection