Variational Quantum Eigensolvers

• Variational quantum eigensolvers are hybrid quantum-classical algorithms that optimize circuit parameters to estimate ground and excited state energies of many-body Hamiltonians.
• They leverage hardware-efficient and problem-inspired ansätze to execute shallow circuits, enabling efficient simulation on NISQ platforms despite noise and errors.
• VQE integrates quantum state preparation, Pauli measurement, and classical optimizers to iteratively refine circuit parameters, enhancing the scalability of quantum simulations.

A variational quantum eigensolver (VQE) is a quantum-classical hybrid algorithm that identifies ground and excited states of many-body Hamiltonians by variationally optimizing a parameterized quantum circuit. The cost function is the energy expectation value of the Hamiltonian with respect to a trial quantum state, and a classical optimizer iteratively updates circuit parameters to minimize this energy. VQE protocols are at the forefront of quantum simulation on noisy intermediate-scale quantum (NISQ) hardware due to their relatively shallow circuit requirements and flexible ansatz design. The method has been extensively extended to general Hamiltonian classes, symmetry-enforced subspaces, excited states, and challenging correlated quantum systems, with numerous advances in ansatz design, encoding strategies, and hybrid workflow optimization.

1. Formulation of the VQE Framework

A VQE protocol is centered around the Rayleigh–Ritz variational principle: $$E(\boldsymbol\theta) = \langle\psi(\boldsymbol\theta)|H|\psi(\boldsymbol\theta)\rangle \geq E_0,$$ where $$|\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)|\psi_0\rangle$$ is a parameterized n-qubit circuit ansatz and $H$ is the target Hamiltonian. $U(\boldsymbol\theta)$ may be hardware-efficient, problem-inspired (e.g., Unitary Coupled Cluster), or constructed to enforce relevant physical or point-group symmetries.

$H$ is typically decomposed after encoding (Jordan–Wigner, Bravyi–Kitaev, or fermion–to–qubit mappings) as a sum of $v$ Pauli strings: $$H = \sum_{r=1}^v a_r P_r, \quad P_r \in \{I, X, Y, Z\}^{\otimes n}.$$

The main computational loop alternates quantum steps—state preparation and observable measurement—with classical optimization:

1. Prepare $|\psi(\boldsymbol\theta)\rangle$.
2. Measure all $\langle P_r \rangle$ to estimate $E(\boldsymbol\theta)$.
3. Update $\boldsymbol\theta$ using classical optimization (e.g., (quasi-)Newton methods, gradient descent, or quantum natural gradients).

All energy and gradient estimates are subject to statistical (shot) noise and, on hardware, to decoherence and gate error effects (Nicoli et al., 29 Jan 2025).

2. Ansatz Construction and Circuit Design

The expressive power and trainability of VQE largely depend on the circuit ansatz. Representative families include:

• Hardware-efficient ansätze: Alternating layers of parametrized single-qubit rotations and nearest-neighbor or fully connected entanglers (e.g., CNOT, iSWAP). Parameter count and circuit depth scale with the number of qubits and layers (Hu et al., 2022, Velury et al., 20 Nov 2025).
• Problem-inspired ansätze: Unitary coupled cluster (UCC), Hamiltonian variational (HVA), symmetry-projected constructions, translationally invariant or momentum-space circuits (Consiglio et al., 2021, Velury et al., 20 Nov 2025).
• Symmetry-adapted circuits: Circuits explicitly preserving global symmetries (e.g., total spin, particle number, SU(N)) are constructed using entangling blocks that commute with the desired symmetry generators, and suitable references are prepared in the correct symmetry sector (Lyu et al., 2022, Consiglio et al., 2021).

Example: For SU($N$) fermions on a ring, the mapping $n = i + sL$ and generalization of the Jordan–Wigner transform enable a compact qubit Hamiltonian. The number- and color-preserving circuit repeats three sublayers (hopping entanglers, color-interaction CR$_z$ gates, and on-site $R_z$ rotations), achieving a per-layer CNOT count of $5NL-3N-2L$ and supporting up to 3 layers to reach ground-state energy convergence in strongly correlated regimes (Consiglio et al., 2021).

Careful matching between ansatz locality, circuit depth, and the target Hamiltonian is required: more localized encodings (e.g. Bravyi–Kitaev) mitigate global cost function flatness ("barren plateaus"), while matchgate or checkerboard-based ansätze preserve particle number and are especially effective for frustrated fermionic models (Uvarov et al., 2020).

3. Quantum Measurement Protocols and Error Mitigation

Post-circuit measurement yields expectation values for each Pauli string in $H$. For Pauli-decomposed Hamiltonians, commuting term grouping optimizes measurement parallelism and minimizes shot cost. For models with non-local or sparse Hamiltonian structure (as in sparse-mode representations or occupation-number bases), Hamiltonian decomposition into self-inverse, one-sparse, Hermitian terms enables efficient measurement via Hadamard test circuits, at a cost scaling linearly with the number of monomials and logarithmically with the inverse of the desired precision (Kirby et al., 2020).

Error mitigation protocols directly integrated within VQE include:

• Measurement grouping across Pauli bases for maximal parallel extraction of $\langle P_r \rangle$.
• Readout error correction, e.g., twisted readout extinction (TREX) or Pauli noise inversion (Nicoli et al., 29 Jan 2025, Hu et al., 22 Dec 2025).
• Zero-noise extrapolation (ZNE) by systematically amplifying gate noise and extrapolating observables to the zero-noise limit (Hu et al., 22 Dec 2025).
• Post-selection (e.g., on preserved Hamming weight or symmetry outcomes) and adaptive mid-circuit measurement for resource reduction (Consiglio et al., 2021, Chan et al., 2023).

Shot noise scales as $\mathcal{O}(1/\sqrt{n})$ per term, and total measurement cost depends on Hamiltonian support, number of non-commuting Pauli strings, and error-mitigation overhead.

4. Classical Optimization Strategies

VQE performance is intimately linked to the optimizer:

• Quasi-Newton (L-BFGS, BFGS) and gradient-based approaches minimize $E(\boldsymbol\theta)$ using parameter-shift or finite-difference gradient rules (Hu et al., 2022, Uvarov et al., 2020).
• Gradient-free methods: Sequential minimal optimization ("NFT"/Rotosolve), COBYLA, and Bayesian/GP-based algorithms operate robustly in high noise, finite-shot, and shallow-circuit scenarios (Consiglio et al., 2021, Nicoli et al., 29 Jan 2025).
• Quantum natural gradient (QNG) and Hamiltonian-aware QNG (H-QNG): QNG rescales parameter updates with the Fubini–Study metric tensor (pullback from quantum state space), improving convergence at the cost of extra metric estimation per step (Shi et al., 18 Nov 2025). H-QNG leverages the Riemannian pullback metric induced by the Hamiltonian subspace, inheriting QNG's geometric benefits with greatly reduced quantum measurement cost (matching that of vanilla gradient descent) and demonstrating order-of-magnitude convergence speedup in molecular benchmarks.

Recent directions include integrating gradient-enhanced optimizers with advanced noise models, shot-adaptive step sizes, and error-mitigation-aware batch schedules.

5. Symmetry, Excited States, and Advanced Formulations

Incorporating physical symmetries can dramatically shrink the search space and accelerate convergence. Approaches include:

• Hardware symmetry-preserving ansätze: All symmetry operations are hardwired into circuit structure (e.g., total $S_z$, $S_{\mathrm{tot}}^2$, SU(N)), reducing eligible states and measurement overhead (Lyu et al., 2022, Consiglio et al., 2021).
• Penalty-based and hybrid symmetry enforcement: Classical cost penalties maintain correct symmetry sector, especially when hardware implementation is infeasible. Hybrid approaches employ hardware for local symmetries and penalties for global constraints (Lyu et al., 2022).
• Excited-state targeting: Subspace-search VQE (SSVQE) and penalty-augmented cost functions generalize VQE to excited states. Symmetry adaptation is critical for suppressing arbitrary ground–excited state mixing and minimizing measurement cost for each targeted sector.

Extensions to multi-parameter VQE, variance minimization (VVQE), Gaussian-filtered ground-state projection, and variational quantum state diagonalization (VQSE) have demonstrated efficient excited-state extraction, self-verification, or principal component identification within the VQE paradigm (Cerezo et al., 2020, Zhang et al., 2020, Liu et al., 2024).

6. Scalability, NISQ Feasibility, and Resource Estimation

Scalability arguments must address qubit count, circuit depth, parameter count, measurement overhead, and optimizer cost:

• Qubit requirements scale with physical Hilbert-space size (e.g. $N_q = N_e$ electrons × $N_{\rm orbitals}$, or $N\,L$ for SU($N$) fermion rings) (Consiglio et al., 2021, Langkabel et al., 2024).
• Depth per circuit layer, gate count, and entanglement capacity depend strongly on ansatz design (e.g., per-layer CNOT count $O(NL)$ for SU(N) fermions; $O(N^2)$ for fully connected hardware-efficient or UCC types) (Consiglio et al., 2021, Hu et al., 2022).
• State-of-the-art results typically require total depths $<200$ for realistic Hamiltonians on $\sim$10–15 qubits (Consiglio et al., 2021), matching constraints for current superconducting qubit platforms, with resource-efficient ansätze or MRA orbital tailoring enabling further resource reductions (Langkabel et al., 2024).
• Measurement cost is $O(v/\epsilon^2)$ for energy convergence at precision $\epsilon$, with error mitigation protocols adding minor overheads but yielding substantial gains in effective accuracy (Nicoli et al., 29 Jan 2025, Shi et al., 18 Nov 2025).
• Hybrid classical–quantum strategies such as collective VQE, tensor-network pre-optimization, or machine-learning-in-the-loop can further compress the quantum workload through parameter sharing and improved initializations (Zhang et al., 2019, Khan et al., 2023).

Sampling costs are particularly sensitive to persistent-current or phase-swept observables; efficient protocols reuse parameter points to minimize circuit executions per $\phi$ in response profiles (Consiglio et al., 2021).

7. Representative Applications and Recent Advances

VQE protocols have been validated across molecular and condensed-matter ground states, frustrated fermion ladders, SU($N$) symmetry-protected phases, and ab initio electronic structure:

• SU($N$) Fermi–Hubbard ring: Direct simulation of persistent-current profiles captures superfluid-to-Mott transitions and fractional flux quantization. A 3-layer symmetry- and number-preserving circuit with $N L$ qubits recapitulates exact energies and bipartite entanglement (Consiglio et al., 2021).
• Frustrated spinless-fermion chains: Hardware-efficient checkerboard ansätze and BK vs JW encoding reveal the critical impact of mapping on barren-plateau severity and correlation function recovery (Uvarov et al., 2020).
• Quantum chemistry: Extensive benchmarking indicates that UCCSD0 and k-UpCCGSD are optimal for moderate system sizes (10–20 qubits), while adaptive and hardware-efficient circuits remain necessary for NISQ applications, though none achieve chemical accuracy for stretched geometries (Hu et al., 2022).
• Momentum-space VQE for quasiparticle dispersion: Translationally symmetric ansatz with FFFT and HVA blocks reconstructs the full quasiparticle dispersion of the XXZ chain, in close agreement with Bethe ansatz results (Velury et al., 20 Nov 2025).
• Excited-state, state-diagonalization, and quantum PCA: Variance minimization, Gaussian filters, symmetry mixing, and adaptive cost landscapes permit systematic excited-state extraction, density-matrix principal component identification, and robust error mitigation (Liu et al., 2024, Zhang et al., 2020, Cerezo et al., 2020).

These advances, together with error mitigation, neural-network post-processing, measurement-based and photonic implementations, and classical pre-optimization, have dramatically extended the reach and physical scope of VQE protocols on progressively more powerful quantum platforms.

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References:

(Consiglio et al., 2021, Uvarov et al., 2020, Lyu et al., 2022, Liu et al., 2024, Zhang et al., 2019, Nicoli et al., 29 Jan 2025, Cerezo et al., 2020, Velury et al., 20 Nov 2025, Hu et al., 2022, Shi et al., 18 Nov 2025, Langkabel et al., 2024, Zhang et al., 2020, Zhang et al., 2021, Hu et al., 22 Dec 2025, Khan et al., 2023, Chan et al., 2023, Kirby et al., 2020, Tamiya et al., 2020, Jattana et al., 2022, Ferguson et al., 2020)