Recent Developments in VQE: Survey and Benchmarking

TL;DR

The paper surveys recent developments in Variational Quantum Eigensolver (VQE) methodologies tailored for NISQ devices, focusing on circuit-complexity reduction, chemistry-inspired approaches, and excited-state extensions. It critically compares flavors such as ADAPT-VQE, qADAPT-VQE, nu-VQE, USCC, and fragmentation-based methods (ClusterVQE, FMO-VQE) through benchmarking against classical reference methods and across basis sets, highlighting the trade-offs between quantum resource demands and accuracy. The survey also covers advanced excited-state techniques (VQD, Folded Spectrum VQE, qEOM variants) and a spectrum of quantum simulators and qubit-mapping tools, underscoring practical considerations for implementation, error mitigation, and scalability. Collectively, the work identifies ADAPT-VQE as a particularly promising pathway for achieving chemical accuracy with limited quantum resources, while outlining future directions such as integrating dynamical and non-dynamical correlation and pursuing MR-CC-like quantum approaches for larger, more complex systems.

Abstract

The Variational Quantum Eigensolver (VQE) algorithm has been developed to target near term Noisy Intermediate Scale Quantum (NISQ) computers as a method to find the eigenvalues of Hamiltonians. Unlike fully quantum algorithms such as Quantum Phase Estimation (QPE), VQE based methods are hybrid algorithms that utilize both quantum and classical hardware to combat issues with the near term quantum hardware such as small numbers of available qubits and the decoherence of qubits. Different adaptations (flavors) of VQE have been implemented to combat these scalability issues on NISQ devices compared to standard VQE. These different flavors are modifications of the underlying VQE ansatz to reduce the computational workload on the quantum hardware. In this review we focus on 3 main areas related to VQE. The first focus is on flavors of VQE that fall under the categories of circuit complexity reduction, chemistry inspired ansatz, and extensions of VQE to excited states. The remaining portion of the review focuses on benchmarking the accuracy of VQE methods and an overview of the current state of quantum simulators.

Figures (7)

• Figure 1: General schematic of the VQE algorithm, showing the fermionic to qubit Hamiltonian mapping and highlighting the interplay between classical and quantum computers. The first step is to map the fermionic Hamiltonian calculated through a Hartree-Fock calculation to a qubit representation of the Hamiltonian. The ansatz is initialized with starting parameters $\overrightarrow{\theta_0}$. The ansatz is then prepared on the quantum computer as a set of gates. This begins the iterative cycle between the quantum and classical computers. Adapted from referenceFedorov2022 Copyright 2022 by Springer Nature under the https://creativecommons.org/licenses/ license.
• Figure 2: The ADAPT-VQE algorithm begins on the quantum computer by generating a pool of the possible qubit operators. The computed VQE ansatz is grown by one operator per iteration. The operator with the largest contribution to the correlation energy (determined by magnitude of the measured gradient) is added each iteration. This pattern is continued, and the ansatz grown, until the convergence tolerance is satisfied. This algorithm neglects terms with near zero contributions to the energy, reducing ansatz size without sacrificing accuracy. Adapted from referenceGrimsley2019. Copyright 2019 by Nature Communication under the https://creativecommons.org/licenses/ license.
• Figure 3: Schematic diagram of the ADAPT-VQE-SCF algorithm, highlighting the interplay between the solution of the wavefunction via VQE and the orbital optimization of CASSCF. Reproduced with permission from referenceFitzpatrick_2024. Copyright 2024 by American Chemical Society.
• Figure 4: RHF, CCSD, CCSD(T), UCCSD(T), UCCSD-VQE and ADAPT-VQE (threshold $\epsilon^{-4}$) energy differences from FCI plotted on a log scale for increasing interatomic distances with the STO-3G minimal basis set. The molecules tested are A. non-symmetric H$_2$O B. non-symmetric BeH$_2$ and C. non-symmetric H$_6$.
• Figure 5: ADAPT-VQE, USCC, and UCCSD-VQE energy differences from FCI plotted on a log scale for increasing interatomic distances with thresholds values of $\epsilon^{-1}$, $\epsilon^{-2}$, $\epsilon^{-3}$, and $\epsilon^{-4}$. The molecules tested are A. H$_2$, B. LiH, C. non-symmetric H$_6$, D. non-symmetric H$_2$O, and E. non-symmetric BeH$_2$. The minimal STO-3G basis set was used for each calculation