State-Averaged Orbital-Optimized VQE

• The paper extends the standard VQE framework by integrating a state-averaged cost function with orbital rotations, enabling simultaneous optimization of multiple quantum states.
• It leverages symmetry-adapted circuit design and classical orbital optimization to reduce quantum resource requirements while enhancing noise resilience.
• The approach provides direct access to analytical gradients, geometry derivatives, and nonadiabatic couplings, facilitating accurate simulation of conical intersections and photochemical processes.

The State-Averaged Orbital-Optimized Variational Quantum Eigensolver (SA-OO-VQE) is a hybrid quantum-classical electronic structure algorithm developed to treat multiple, possibly degenerate, quantum states on an equal footing within the variational quantum eigensolver framework. By combining a state-averaged cost function with explicit optimization over both quantum circuit parameters and molecular orbital rotations, SA-OO-VQE enables a compact and noise-resilient route to accurate potential energy surfaces, geometry derivatives, and nonadiabatic couplings—even in systems featuring strong correlation, conical intersections, or multistate character. It serves as a quantum analog of classical state-averaged multiconfigurational self-consistent field (SA-MCSCF) methods, but leverages quantum processing units (QPUs) for variational wavefunction preparation and measurement, while using classical resources for orbital optimization and optimization routines.

1. Algorithmic Structure and Theoretical Foundations

SA-OO-VQE extends standard VQE—originally designed for ground-state calculations—to the simultaneous optimization of several low-lying eigenstates. The method minimizes a state-averaged energy functional incorporating both quantum circuit (ansatz) parameters $\vec{\theta}$ and orbital rotation parameters $\vec{\kappa}$:

$$E_\text{avg}(\vec{\theta}, \vec{\kappa}) = \sum_{k} w_k \langle \Psi_k(\vec{\theta}, \vec{\kappa}) | \hat{H} |\Psi_k(\vec{\theta}, \vec{\kappa}) \rangle$$

with $$|\Psi_k(\vec{\theta}, \vec{\kappa})\rangle = \hat{U}_\text{VQE}(\vec{\theta})\hat{U}_\text{OO}(\vec{\kappa})|\Phi_k\rangle$$, $w_k$ being the ensemble weights, and $\hat{H}$ the molecular Hamiltonian projected onto an active space.

The orbital-rotation unitary is typically parameterized as:

$$\hat{U}_\text{OO}(\vec{\kappa}) = \exp\left(-\sum_{p>q} \kappa_{pq} (\hat{E}_{pq} - \hat{E}_{qp}) \right)$$

where $\hat{E}_{pq}$ are orbital excitation operators.

State-averaged ansatz construction ensures that the same quantum circuit evolves multiple orthonormal input states $|\Phi_k\rangle$ (typically chosen as Hartree–Fock reference or CI singles) into correlated, mutually orthogonal quantum states, allowing for the automatic preservation of state orthogonality and smooth treatment of degeneracies (Yalouz et al., 2020, Omiya et al., 2021, Yalouz et al., 2021). The orbital optimization subproblem is solved via classical routines (Newton–Raphson, gradient projection, etc.) using one- and two-body reduced density matrices measured on the quantum circuit.

2. Symmetry, Resource Efficiency, and Circuit Design

SA-OO-VQE leverages physically motivated ansätze and symmetry-adapted circuits to restrict optimization to the physically relevant sector of Hilbert space, improving both efficiency and resilience to barren plateaus:

• Particle-number and spin projection symmetry are preserved by construction using cascaded "A-gates" (exchange-type two-qubit gates) and, for higher symmetry, "E-gate" circuits based on hyperspherical angles, ensuring the ansatz exactly spans the physically allowed subspace (Gard et al., 2019).
• The minimal number of variational parameters is determined by subspace dimension: a state in $\mathcal{H}_{n,m}$ requires $2\cdot\text{dim}(\mathcal{H}_{n,m})-2$ real parameters (halved if coefficients are real and time-reversal symmetry is imposed).
• Explicit circuit decompositions minimize two-qubit gate count and allow nearest-neighbor layouts, e.g., the E$_4$ circuit for $(n=4, m=2)$ is reduced from 155 CNOTs (symbolic) to $\sim$28 CNOTs with numerical transpilation (Gard et al., 2019).

Gate-efficient implementations—especially those using pair-correlated (seniority-zero) or bosonic encodings—can halve the qubit count and measurement overhead relative to general-purpose UCCSD, making SA-OO-VQE viable on NISQ-era hardware (Zhao et al., 2022, Kim et al., 2023). Circuit depth is further reduced by offloading singles excitations (orbital rotations) to classical optimization (Mizukami et al., 2019). Adaptive strategies (e.g., ADAPT-VQE-SCF, OE-VQE) iteratively grow the ansatz and active space, constructed from operator pools and gradient information, yielding shallow circuits while preserving chemical accuracy (Fitzpatrick et al., 2022, Wu et al., 2022).

3. Analytical Gradients, Geometry Optimization, and Nonadiabatic Couplings

A major advantage of the fully variational formulation is the possibility for direct evaluation of analytical gradients and nonadiabatic couplings:

• The total derivative of any state-averaged electronic energy with respect to an external parameter $x$ (e.g., a nuclear coordinate) is governed by a Lagrangian formalism which yields:

$$\frac{d E_A}{d x} = \langle \psi | \hat{U}^\dagger(\vec{\theta}_A^*) \frac{\partial \hat{H}(\vec{x}, \vec{\kappa}^*)}{\partial x} \hat{U}(\vec{\theta}_A^*) | \psi \rangle + \text{response terms}$$

with additional terms capturing the orbital and ansatz parameter response through Lagrange multipliers obtained by solving coupled-perturbed equations (Omiya et al., 2021, Yalouz et al., 2021). Standard parameter-shift or commutator techniques enable gradient measurement on quantum hardware.

• Nonadiabatic couplings are similarly evaluated by extended Lagrangian techniques, accounting for the lack of full stationarity with respect to all variational parameters, and yield equations of the form:

$$D_{IJ} = \frac{1}{E_J - E_I} \left\{ \sum_{pq} \left( \frac{\partial h_{pq}}{\partial x} \gamma_{pq}^{IJ, \text{eff}} \right) + \ldots \right\} + \ldots$$

where the effective density matrices include orbital response corrections (Yalouz et al., 2021).

• This framework enables robust localization of conical intersections and branching plane characterization. Minimization of the energy gap driven by gradient and NAC evaluation can directly target minimum-energy conical intersections (MECIs) (Omiya et al., 2021, Yalouz et al., 2021), essential for photochemistry and ultrafast nonadiabatic dynamics.

4. Robustness to Noise and Quantum Resource Constraints

SA-OO-VQE is explicitly designed for NISQ-era quantum hardware, with strategies to mitigate device noise and resource bottlenecks:

• The quantum workloads (expectation value measurement) are limited to the state-averaged evaluation; circuit and orbital optimization, gradient evaluation, and orbital transformation are all handled with classical compute.
• By using symmetry-preserving circuits and restricting the parameter space to the physically relevant subspace, the method reduces convergence failures associated with symmetry breaking and “getting lost” in irrelevant Hilbert space sectors (Gard et al., 2019, Beseda et al., 22 Jan 2024).
• Benchmarking under noisy models (dephasing, depolarizing, thermal relaxation) on SA-OO-VQE shows that local gradient-based optimizers (notably BFGS) are the most robust and efficient in the presence of quantum noise, with a clear preference over SLSQP and global methods (e.g., iSOMA) for low-dimensional parameter spaces (Illésová et al., 9 Oct 2025). COBYLA is efficient for rapid, low-cost approximations; global methods are not beneficial at low dimensionality.
• Natural orbital updates (i.e., representing the state in the NO basis at each iteration) further suppress circuit depth requirements, offering robustness against noise-induced circuit expressivity limitations (Besserve et al., 20 Jun 2024).

5. Quasi-Diabatization and Multistate Dynamics

A key conceptual result from recent work is that the block-diagonalization inherent in state-averaged variational optimization naturally yields a quasi-diabatic representation of the electronic Hamiltonian (Illésová et al., 25 Feb 2025). In this setting:

• The block-diagonalized subspace generated by the converged state-averaged variational optimization constitutes a “least-transformed” basis relative to an initial chemically motivated “model” subspace.
• Two descriptors—intrinsic and residual diabaticity—quantify the proximity to an exact diabatic representation: d(q) (intrinsic, based on SVD overlap with the model subspace) and r(q) (residual, based on additional rotations needed for full diagonalization).
• The quasi-diabatic representation minimizes the nonadiabatic couplings (NACs) by construction, sidestepping ad hoc adiabatic-to-diabatic transformations and phase tracking. Numerical results in the formaldimine molecule show three-orders-of-magnitude NAC suppression in the quasi-diabatic basis compared to the adiabatic representation (Illésová et al., 25 Feb 2025).
• This realization is highly significant for quantum molecular dynamics, as it enables robust diabatization and trajectory surface hopping on quantum computers without the numerical pathologies associated with phase/sign changes or CI singularities.

6. Applications, Benchmarks, and Future Perspective

SA-OO-VQE has been benchmarked for complex electronic structure applications including:

• Photochemical reactions and conical intersection dynamics: Minimal active space calculations (e.g., on formaldimine and tetrafluoropropene) correctly reproduce degeneracies and potential energy surface topology essential for nonadiabatic processes (Yalouz et al., 2020, Omiya et al., 2021, Yalouz et al., 2021).
• Small molecule dissociation and spectral gap predictions: State-averaged orbital optimization systematically improves energy accuracy and allows the use of smaller qubit counts or active spaces while capturing correlation effects relevant for ground and excited states (Matoušek et al., 2023, Zhao et al., 2022, Bierman et al., 2022).
• Full variational orbital optimization using adaptive multi-wavelet orbitals enables basis-set-limit accuracy and further resource reduction, facilitating simulation of larger systems on NISQ hardware (Langkabel et al., 24 Oct 2024).
• Statistical benchmarking under various quantum noise models substantiates the practicality and numerical efficiency of the algorithm with proper optimizer selection (Illésová et al., 9 Oct 2025).

The ongoing development of orbital-optimized, state-averaged techniques—including natural-orbital updating, adaptive circuit design, and resource-efficient mapping—positions SA-OO-VQE as a central algorithm for NISQ quantum chemistry, especially for cases where multistate descriptions, nonadiabatic phenomena, or nontrivial electronic degeneracies are essential. The framework is expected to generalize to larger active spaces, open-shell/multireference systems, and more sophisticated quantum dynamics protocols as quantum hardware and hybrid numerical strategies further evolve.