Variational Quantum Eigensolvers (VQE)

Updated 2 January 2026

Variational Quantum Eigensolvers (VQE) are hybrid algorithms that approximate quantum system eigenstates using parameterized circuits and classical optimization.
They leverage various ansatz designs, such as UCCSD and hardware-efficient circuits, to balance circuit complexity with experimental feasibility on NISQ devices.
Recent advances include adaptive ansatz strategies, variance minimization approaches, and efficient measurement techniques to improve convergence and reduce error.

The variational quantum eigensolver (VQE) is a hybrid quantum–classical algorithm designed to approximate the ground or low-lying excited eigenstates of a quantum Hamiltonian. VQE leverages parameterized quantum circuits (ansätze) to generate trial wavefunctions, whose energy or related functionals are estimated via quantum measurements and minimized using classical optimization. VQE has become a central paradigm for quantum chemistry, materials simulation, and quantum many-body physics on noisy intermediate-scale quantum (NISQ) devices due to its hardware efficiency and adaptability (Fedorov et al., 2021, Hu et al., 2022). Substantial developments include variance-based generalizations, collective and transfer-learning strategies, neural-augmented protocols, constraint handling, measurement-based approaches, circuit compression, and extensions to nonstandard Hamiltonians and excited states.

1. Core Principles and Standard Workflow

A VQE begins with a structured ansatz circuit $U(\theta)$ acting on an initial product state $|0\rangle^{\otimes n}$ , parametrized by a vector $\theta \in \mathbb{R}^d$ . The goal is to minimize the variational energy functional: $E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle = \langle 0 | U^\dagger(\theta) H U(\theta) | 0 \rangle,$ where $H = \sum_j \alpha_j P_j$ is typically expressed as a sum of Pauli strings via a fermion-to-qubit mapping (e.g., Jordan–Wigner, Bravyi–Kitaev) (Fedorov et al., 2021, Hu et al., 2022).

The hybrid workflow entails:

Quantum state preparation and measurement: given $\theta$ , generate $|\psi(\theta)\rangle$ , measure the expectation values $\langle P_j \rangle_\theta$ , and calculate $E(\theta)$ .
Classical optimization: update $\theta$ using gradient-based (e.g., parameter-shift rule) or gradient-free methods (e.g., COBYLA, SPSA) to descend $E(\theta)$ . The parameter-shift rule applies for gates $e^{-i \theta_k V_k/2}$ with $V_k^2=I$ :

$\frac{\partial E}{\partial \theta_k} = \frac{E(\theta_k+\pi/2) - E(\theta_k-\pi/2)}{2}.$

Iterate until convergence to an approximate ground state $|\psi(\theta^*)\rangle$ (Hutchings et al., 2024).

Optimization may get stuck in local minima or “barren plateaus” (regions of exponentially vanishing gradients), motivating algorithmic innovations as discussed below (Uvarov et al., 2020, Jattana et al., 2022).

2. Ansatz Classes and Circuit Design

VQE performance critically depends on the expressiveness and circuit depth of the chosen ansatz (Fedorov et al., 2021, Hu et al., 2022).

Chemistry-Inspired Ansatzes (e.g., UCCSD/ADAPT):

Unitary coupled-cluster (UCCSD):

$|\psi_\text{UCCSD}(\theta)\rangle = e^{T(\theta) - T^\dagger(\theta)} | \phi_0 \rangle$ , where $T$ encodes single and double fermionic excitations; circuit depth and parameter number grow rapidly with system size.

Adaptive ansätze (ADAPT-VQE, qubit-ADAPT): Iteratively construct the ansatz by selecting the operator from a pool with the largest energy gradient, appending its corresponding gate, and reoptimizing. ADAPT ansätze achieve higher accuracy with smaller parameter sets for small to intermediate system sizes (Hu et al., 2022).

Hardware-Efficient Ansatzes:

Layers of single-qubit rotations and entanglers (e.g., RY–CZ, brickwork), designed for shallow depth and hardware compatibility, but may suffer from barren plateaus or optimization traps for highly expressive circuits (Fedorov et al., 2021, Hu et al., 2022).

Problem-Specific and Reduced-Complexity Ansatzes:

ClusterVQE: Qubit clustering based on quantum mutual information, combining intra-cluster circuit execution with classical “dressing” to account for inter-cluster interactions. Reduces both qubit count and depth, offering efficiency on NISQ devices at the expense of increased classical postprocessing (Zhang et al., 2021).
Quasiparticle VQE: Tailored for symmetry-resolved excitations in many-body systems, constructs a free-particle (e.g., particle-hole) momentum eigenstate, applies symmetrized variational circuits (e.g., Hamiltonian Variational Ansatz, FFFT layers), and explicitly targets the excitation spectrum (Velury et al., 20 Nov 2025).

3. Measurement Strategies and Sampling Complexity

A central cost in VQE arises from the need to estimate many expectation values with statistical error $\epsilon$ , incurring a measurement cost of $O(N_\text{terms}/\epsilon^2)$ (Fedorov et al., 2021, Hutchings et al., 2024). For Hamiltonians with a large number of non-commuting terms (Pauli or more general sparse matrices), recent extensions include:

Sparse Hamiltonian VQE: For general $d$ -sparse matrices, Hamiltonians can be decomposed into $O(d^2)$ one-sparse self-inverse terms via digit decomposition, with expectation values estimated via the Hadamard test and efficient oracle queries (Kirby et al., 2020).
Hamiltonian Sampling/Stochastic Gradient Descent: Instead of evaluating all $N^2$ Pauli covariance terms for the energy variance (see below), sampling-based estimates can reduce early-stage measurement overhead by an order of magnitude while preserving optimization guidance (Zhang et al., 2020).

Grouping commuting Pauli terms and advanced measurement scheduling further alleviates measurement overhead, a critical consideration on NISQ hardware (Hu et al., 22 Dec 2025, Chan et al., 2023).