Variational Quantum Eigensolver

• Variational Quantum Eigensolver is a hybrid algorithm that uses parameterized quantum circuits to variationally minimize energy expectations for estimating ground states.
• It leverages tailored ansätze like UCCSD and hardware-efficient designs, combining quantum state preparation on NISQ devices with classical optimization.
• Applications include quantum chemistry and condensed matter physics, with advancements in error mitigation and measurement grouping enhancing simulation accuracy.

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm designed to estimate the ground-state energy and eigenstates of quantum many-body Hamiltonians, particularly in the context of quantum chemistry and condensed matter physics. By variationally minimizing an energy expectation value over a parameterized class of quantum circuits (ansätze), VQE leverages both near-term quantum devices for state preparation and measurement, and classical optimization routines for parameter updates. VQE’s adaptability to different Hamiltonian structures, its compatibility with noisy intermediate-scale quantum (NISQ) hardware, and ongoing algorithmic improvements have established it as the leading paradigm for quantum simulation in the NISQ era.

1. Fundamental Principles and Workflow

The VQE framework is built on the variational principle, which guarantees that the expectation value of a Hamiltonian $H$ evaluated in any trial quantum state $|\psi(\theta)\rangle$ yields an upper bound to the true ground-state energy $E_0$: $E_0 \leq \langle\psi(\theta)|H|\psi(\theta)\rangle$ The algorithm operates by choosing a parameterized quantum circuit $U(\theta)$ that acts on a simple reference state (such as $|0\rangle^{\otimes n}$ or a product Hartree–Fock state), producing a variational state $$|\psi(\theta)\rangle = U(\theta)|0\rangle^{\otimes n}$$. The energy $E(\theta)$ is estimated by decomposing $H$ into a sum of observables, typically Pauli strings in quantum chemistry applications: $$H = \sum_a w_a P_a\qquad (P_a \in \{I,X,Y,Z\}^{\otimes n})$$ Measured expectation values $\langle P_a \rangle_\theta$ are accumulated, and the result is fed into a classical optimizer, which proposes updates to the variational parameters $\theta$ to decrease $E(\theta)$ iteratively (Fonseca et al., 7 May 2025, Ion et al., 27 Dec 2025).

2. Ansätze and Circuit Design

Effective ansatz construction is critical for VQE’s expressiveness and resource efficiency, as circuit depth and parameter count must balance hardware limitations with the need to capture complex many-body correlations.

• Unitary Coupled Cluster (UCC) and variants: For quantum chemistry, the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz is widely used:

$$|\psi_{\mathrm{UCC}}(\mathbf{t})\rangle = \exp\left(\hat{T} - \hat{T}^\dagger\right)|\Psi_{\mathrm{HF}}\rangle\;$$

where $\hat{T}$ includes particle-hole excitations. The factorized/Trotterized form is common for hardware compatibility (Fonseca et al., 7 May 2025, Xu et al., 2023). Efficient application of high-rank UCC terms is enabled using linear-combination-of-unitaries (LCU) techniques that exploit hidden SU(2) algebras, allowing polynomial (cubic) CNOT scaling for rank-$n$ excitations, which is crucial for strongly correlated systems (Xu et al., 2023).

• Hardware-Efficient Ansatz (HEA): Layers of parameterized single-qubit rotations and entangling gates (such as CNOT or CZ) are composed to form shallow, device-friendly circuits. Block-structured ansätze tailored to hardware connectivity or evolved under genetic algorithms can further reduce CNOT counts and circuit depth, approaching chemical accuracy with an order-of-magnitude fewer two-qubit gates than fixed hardware-efficient circuits (Chivilikhin et al., 2020, Rattew et al., 2019).
• Symmetry-Adapted and Problem-Inspired Design: For lattice models and molecular systems with symmetry, ansätze invariant under $U(1)$, $SU(2)$, spatial, or permutation group actions exploit physical constraints to decrease parameter count and mitigate barren plateaus (Liu et al., 2019, Mondal et al., 2023).
• Tensor Network and Qubit-efficient Constructions: Quantum tensor network representations (e.g., MPS, PEPS) can be constructed via register reuse, enabling simulation of systems larger than available qubit counts, with resource scaling determined by the virtual bond dimension (Liu et al., 2019).

3. Measurement and Classical Optimization

The energy functional $E(\theta)$ requires the estimation of expectation values for each term in the Hamiltonian decomposition, often numbering $O(N^4)$ in molecular systems. Key strategies include:

• Measurement grouping: Exploit commutativity by partitioning the set of Pauli terms into groups that can be measured in a single basis, reducing the number of circuit evaluations (Fonseca et al., 7 May 2025, Ion et al., 27 Dec 2025).
• Hamiltonian sampling: Variance-minimizing approaches (VVQE) and stochastic gradient descent via Hamiltonian term sampling can dramatically reduce measurement overhead by sub-sampling Hamiltonian terms to estimate both the energy and its variance, while retaining unbiasedness (Zhang et al., 2020).
• Gradient estimation: Use of parameter-shift rules and analytic gradients enables efficient derivative computation even on hardware, essential for gradient-based optimizers. Gradient-free optimizers such as SPSA, COBYLA, and classical evolutionary strategies are routinely employed, with trade-offs in convergence reliability and shot efficiency (Ion et al., 27 Dec 2025, Chivilikhin et al., 2020).

4. Algorithmic Variants, Extensions, and Error Mitigation

VQE admits a broad family of algorithmic extensions, addressing both performance and hardware limitations:

• Variance Minimization and Excited States: Optimizing a weighted sum of energy and variance (VVQE) allows direct targeting of arbitrary eigenstates (including low-lying excited states) without explicit orthogonality constraints; energy variance provides a certificate for eigenstate accuracy (Zhang et al., 2020).
• Contextual Subspace Approaches: Hamiltonians are decomposed into a large “noncontextual” component (optimized classically) and a smaller “contextual” remainder handled by VQE with reduced qubit count, trading classical cost for quantum resource efficiency (Kirby et al., 2020).
• Measurement-Based VQE: Alternatives to circuit-based state preparation leverage measurement-based quantum computation (MBQC), where entangled graph states and single-qubit measurements replace gate sequences, potentially lowering coherence time requirements and offering new ansatz families (Ferguson et al., 2020).
• Photonic and Multi-platform Implementations: VQE is hardware-agnostic and can be adapted to photonic, trapped-ion, or superconducting platforms. Photonic VQE leverages path, polarization, or orbital angular momentum modes for qubit and qudit encodings, exploiting low decoherence and mature optical infrastructure for NISQ applications (Hu et al., 22 Dec 2025).
• Noise Mitigation and Denoising: Techniques such as variational denoising with quantum autoencoders, zero-noise extrapolation, and Pauli noise inversion are used to recover ground-state energy accuracy and fidelity from noisy VQE outputs, leveraging either additional quantum neural networks or classical post-processing (Tran et al., 2023, Ion et al., 27 Dec 2025).
• Quasi-dynamical and Annealing-inspired Heuristics: Sequential layer-wise VQE, where each new variational cycle is initialized to the previously optimized state with new parameters set to zero, can systematically lower the energy, reduce the impact of barren plateaus, and aid in escaping local minima (Jattana et al., 2022).

5. Applications and Practical Performance

VQE has been extensively deployed on problems in quantum chemistry, strongly correlated electron systems (Hubbard, Heisenberg, XXZ, SU(N) rings), and combinatorial optimization. Notable benchmarks include:

• Quantum chemistry: For diatomics and small polyatomics (H₂, LiH, BeH₂, H₂O), VQE achieves chemical accuracy with $\sim$4–12 qubits, using UCC or evolved ansätze, matching full configuration-interaction energies within milli-Hartree errors on simulators and hardware (Fonseca et al., 7 May 2025, Ion et al., 27 Dec 2025, Chivilikhin et al., 2020, Mondal et al., 2023, Huggins et al., 2019).
• Strongly correlated lattice models: Simulations of SU(N) Hubbard rings, XXZ spin chains, and frustrated systems demonstrate the ability of VQE pipelines (ansatz mapping, measurement grouping, classical optimization) to resolve quantum phases (superfluid, Mott, beat), excitation spectra, and correlation functions within a few percent of exact results, even on NISQ devices (Consiglio et al., 2021, Velury et al., 20 Nov 2025, Uvarov et al., 2020).
• Qubit resource and circuit depth efficiency: Advanced evolutionary and genetic ansatz optimization protocols have yielded ground-state fidelities $>90-99\%$ with up to $18\times$ shallower circuits and $12\times$ fewer CNOTs than traditional UCCSD formulations. State-specific VQE methods directly target excited-state manifolds with the same hardware cost as ground-state VQE (Rattew et al., 2019, Chivilikhin et al., 2020, Mondal et al., 2023).
• Hardware demonstration: VQE routines have been realized on IBM Q and other quantum hardware, with optimization over ansatz depth, error mitigation, and warm-start heuristics required to robustly track ground-state energy curves and chemical reaction pathways (Ion et al., 27 Dec 2025).

6. Scalability, Barren Plateaus, and Resource Estimates

Scalability remains a central challenge for VQE:

• Resource scaling: Number of qubits is set by the number of spin-orbitals in the basis for quantum chemistry ($n=N_f$), while circuit depth per evaluation is determined by ansatz form—$O(N^5)$ for UCC, $O(kN^3)$ for $k$-UpCCGSD, $O(NL)$ for hardware-efficient circuits. Measurement circuits scale naively as $O(N^4)$ but can be compressed by grouping and sub-sampling (Fonseca et al., 7 May 2025, Ion et al., 27 Dec 2025).
• Barren plateaus: Gradient vanishing with increasing system size or sign problem is acute for deep, unstructured ansätze, global cost functions, and certain fermion-to-qubit mappings (notably Jordan–Wigner in non-local models). Approaches such as symmetry constraints, adaptive ansatz growth, contextually-motivated subspace restriction, and quasi-dynamical evolution all serve to mitigate these effects (Jattana et al., 2022, Uvarov et al., 2020, Liu et al., 2019).
• Classical optimization cost: Number of parameter updates is governed by convergence rate (gradient-based vs gradient-free), landscape ruggedness, and noise. Hybrid loop costs are dominated by quantum circuit evaluations per iteration.

A summary of scaling is provided in the following table, as presented in (Fonseca et al., 7 May 2025):

Ansatz/Approach	Spatial complexity (qubits)	Circuit depth per energy	Energy measurements
UCC(UCCSD)/UCCG	$n=N_f$	$O(N^5)$	$O(N^4)$
k-UpCCGSD	$n=N_f$	$O(k N^3)$	$O(N^4)$
Hardware-efficient	$n$	$O(nL)$	$O(M)$, $M\sim N_f^4$

7. Variants and Advanced Algorithms

Recent advances have generalized or extended VQE in several directions:

• Non-orthogonal Variational Quantum Eigensolver (NOVQE): By solving a generalized eigenvalue problem in a non-orthogonal ansatz subspace, NOVQE systematically improves accuracy without circuit deepening, using efficient off-diagonal measurement protocols and adaptive shot allocation to manage uncertainties (Huggins et al., 2019).
• State-Specific VQE and Excitation Targeting: Construction of symmetry-adapted, multi-determinant references and totally (spatial) symmetric, spin-scalar unitaries allows for direct targeting of ground or excited electronic states with arbitrary symmetry, eliminating the need for additional ancillas or orthogonalization constraints (Mondal et al., 2023).
• Sparse Hamiltonian VQE: Application to general sparse-matrix problems (fermionic, bosonic, quantum field theory) using one-sparse, self-inverse Hermitian decompositions, enlarging the applicable problem domains beyond Pauli-based models (Kirby et al., 2020).

• "Variational Quantum Eigensolver for SU($N$) Fermions" (Consiglio et al., 2021)
• "Contextual Subspace Variational Quantum Eigensolver" (Kirby et al., 2020)
• "Variational quantum eigensolvers by variance minimization" (Zhang et al., 2020)
• "Variational Denoising for Variational Quantum Eigensolver" (Tran et al., 2023)
• "Efficient application of the factorized form of the unitary coupled-cluster ansatz for the variational quantum eigensolver algorithm by using linear combination of unitaries" (Xu et al., 2023)
• "Improved variational quantum eigensolver via quasi-dynamical evolution" (Jattana et al., 2022)
• "Tangent Vector Variational Quantum Eigensolver: A Robust Variational Quantum Eigensolver against the inaccuracy of derivative" (Wakaura et al., 2021)
• "Variational quantum eigensolvers for sparse Hamiltonians" (Kirby et al., 2020)
• "A measurement-based variational quantum eigensolver" (Ferguson et al., 2020)
• "A Non-Orthogonal Variational Quantum Eigensolver" (Huggins et al., 2019)
• "Photonic variational quantum eigensolver for NISQ-compatible quantum technology" (Hu et al., 22 Dec 2025)
• "Variational quantum eigensolver for chemical molecules" (Ion et al., 27 Dec 2025)
• "Quasiparticle Variational Quantum Eigensolver" (Velury et al., 20 Nov 2025)
• "MoG-VQE: Multiobjective genetic variational quantum eigensolver" (Chivilikhin et al., 2020)
• "An Introduction to Variational Quantum Eigensolver Applied to Chemistry" (Fonseca et al., 7 May 2025)
• "Ground or Excited State: a State-Specific Variational Quantum Eigensolver for Them All" (Mondal et al., 2023)
• "Variational Quantum Eigensolver for Frustrated Quantum Systems" (Uvarov et al., 2020)
• "Variational Quantum Eigensolver with Fewer Qubits" (Liu et al., 2019)
• "A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver" (Rattew et al., 2019)

For a comprehensive treatment of VQE algorithmic advances, especially as applied to specific model Hamiltonians, quantum platforms, error mitigation, and extensions for excited-state calculations or hardware-optimized ansatz design, see the cited works.