Recent Developments in VQE: Survey and Benchmarking

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.

💡 Research Summary

The paper provides a comprehensive survey of recent developments in the Variational Quantum Eigensolver (VQE) with a focus on three major categories: circuit‑complexity reduction, chemistry‑inspired ansätze, and extensions to excited‑state calculations. It begins by contextualizing VQE within the broader landscape of quantum algorithms, contrasting it with fully quantum approaches such as Quantum Phase Estimation (QPE) that require fault‑tolerant hardware. The authors argue that VQE’s hybrid quantum‑classical nature makes it the most viable method for near‑term NISQ devices, where qubit counts are limited and decoherence is significant.

In the circuit‑complexity reduction section, several strategies are examined. Adaptive Derivative‑Assembled Pseudo‑Trotter VQE (ADAPT‑VQE) dynamically builds the ansatz by selecting operators from a predefined pool based on the magnitude of the measured energy gradient, thereby minimizing the number of gates and parameters needed to capture most of the correlation energy. While ADAPT‑VQE achieves low circuit depth, it incurs a high measurement overhead because each iteration requires gradient evaluation for every candidate operator. Hardware‑efficient ansätze (HEA) are presented as an alternative: they employ fixed, shallow circuit layers with parameter sharing to reduce depth, at the cost of discarding some chemical symmetries and potentially lowering accuracy. Other techniques such as symmetry tapering, operator compression, and parameter sharing are also discussed, highlighting the trade‑offs between quantum resource savings and increased classical computation or measurement requirements.

The chemistry‑inspired ansatz segment reviews the standard unitary coupled‑cluster with singles and doubles (UCCSD) and its limitations due to the large number of excitation operators. To address scalability, the authors introduce variants such as k‑UCCGSD, which includes higher‑order excitations while keeping the number of variational parameters proportional to the active‑space size, and truncated or qubit‑adapted versions of UCCSD that prune negligible excitations. These methods retain physical symmetries and often achieve higher chemical accuracy than hardware‑efficient designs, but they still demand careful circuit optimization to be feasible on NISQ hardware.

For excited‑state extensions, the paper surveys VQE‑E, Subspace‑VQE, and Quantum Subspace Expansion (QSE). These approaches reuse the ground‑state ansatz or construct a low‑dimensional subspace from a set of measured observables to extract excited‑state energies. Benchmarks show that such techniques can reduce excited‑state energy errors by 10–20 % relative to plain VQE, making them attractive for spectroscopy and reaction‑path studies.

A substantial part of the work is devoted to benchmarking. The authors evaluate a suite of molecules (H₂, LiH, BeH₂, H₂O) across different active spaces, comparing energy errors (in μHartree), gate depths, number of measurement shots, and optimizer iterations for each method. The results indicate that a hybrid strategy combining ADAPT‑VQE’s adaptive operator selection with a chemistry‑aware ansatz such as k‑UCCGSD delivers the best balance of accuracy and resource efficiency. In particular, this combination achieves comparable chemical accuracy to full UCCSD while using roughly one‑third the number of two‑qubit gates and requiring fewer measurement shots than fixed‑depth HEA circuits.

The final section surveys the current ecosystem of quantum simulators and software packages. It compares Qiskit, Cirq, Pennylane, and OpenFermion‑QChem in terms of supported Hamiltonian mappings (Jordan‑Wigner, Bravyi‑Kitaev, parity), measurement‑optimization tools (operator grouping, commuting‑set reduction), noise‑model integration, and cloud‑based access. The authors emphasize that realistic benchmarking must account for hardware‑specific noise characteristics, and they provide guidelines for selecting appropriate simulators when evaluating VQE performance.

Overall, the paper concludes that VQE remains the most promising pathway toward quantum advantage in chemistry on NISQ devices, provided that algorithmic advances continue to reduce circuit depth, exploit chemical structure, and efficiently target excited states. The authors outline a roadmap that includes (i) further development of adaptive, symmetry‑aware ansätze, (ii) integration of advanced error‑mitigation techniques, and (iii) systematic benchmarking on both simulators and emerging quantum hardware. This roadmap aims to bridge the gap between current NISQ capabilities and the fault‑tolerant regime required for truly scalable quantum chemistry simulations.