State-Averaged Orbital-Optimized VQE

• The SA-OO-VQE method significantly improves multistate balance by optimizing a common orbital basis for both ground and excited states.
• It utilizes a hybrid quantum-classical approach with iterative orbital rotations and VQE energy minimizations to reliably address near-degenerate and crossing states.
• The framework enables accurate treatment of conical intersections and nonadiabatic dynamics while accommodating the limitations of NISQ hardware.

Searching arXiv for the cited SA-OO-VQE literature and closely related OO-VQE work. Searching "State-Averaged Orbital-Optimized Variational Quantum Eigensolver" State-Averaged Orbital-Optimized Variational Quantum Eigensolver (SA-OO-VQE) is a hybrid quantum-classical electronic-structure method that combines three ingredients: the variational framework of VQE, orbital optimization in the spirit of multiconfigurational self-consistent-field theory, and state averaging over several low-lying electronic states. Its defining purpose is to describe ground and excited states on an equal footing in a common orbital basis, particularly in regimes where states are degenerate or quasi-degenerate and where avoided crossings or conical intersections render state-specific approaches unstable or numerically problematic. In the literature, SA-OO-VQE is explicitly positioned as a quantum analogue of state-averaged CASSCF or, more broadly, SA-MCSCF, but adapted to NISQ constraints by delegating state preparation and measurement to a quantum processor while retaining orbital updates and optimization logic on a classical computer (Beseda et al., 2024, Yalouz et al., 2020).

1. Conceptual setting and motivation

The motivation for SA-OO-VQE arises from two simultaneous constraints. The first is chemical: many problems of practical interest require not only a ground-state energy but also excited states, potential energy surfaces, gradients, and non-adiabatic couplings, especially for photochemical processes that traverse avoided crossings and conical intersections. The second is architectural: contemporary quantum hardware is noisy and qubit-limited, so near-term algorithms must be hybrid, measurement-efficient, and compatible with active-space truncations rather than full-Hamiltonian treatments (Beseda et al., 2024, Omiya et al., 2021).

In classical quantum chemistry, state-averaged CASSCF can treat several states simultaneously, but the active spaces remain small because of the classical exponential bottleneck, and dynamical correlation is typically absent unless added perturbatively afterward. SA-OO-VQE is introduced against that background as a method that preserves the multistate, orbital-relaxed logic of SA-CASSCF while replacing the active-space eigensolver by a quantum variational procedure. The central claim is not merely that a quantum circuit may improve active-space correlation, but that the active-space Hamiltonian itself must be optimized in a state-averaged way if one wants a faithful description of near-degenerate manifolds (Yalouz et al., 2020).

This emphasis on multistate balance is crucial. If orbitals are optimized for only one state, the ground state can improve at the expense of excited states; if different states are optimized separately, the resulting orbital sets are not directly comparable, orthogonality can be compromised, and root flipping can occur near degeneracies. SA-OO-VQE addresses these failure modes by using one orbital basis and one state-averaged objective for the targeted manifold. A plausible implication is that the method is best understood not as a generic excited-state VQE add-on, but as a specifically multistate orbital-relaxed framework for nonadiabatic electronic structure (Yalouz et al., 2020, Bezděk, 15 Sep 2025).

2. Variational formulation and ensemble structure

At the circuit level, SA-OO-VQE inherits the standard VQE ansatz structure

$$|\Psi(\boldsymbol{\theta})\rangle = \hat U(\boldsymbol{\theta})|\Phi_0\rangle,$$

but the state-averaged variant applies the same unitary to multiple orthonormal references. In the two-state case used repeatedly in the literature,

$$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$

with $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$. Because the same circuit acts on orthonormal references, orthogonality is preserved throughout the variational optimization (Yalouz et al., 2020, Illésová et al., 9 Oct 2025).

The objective is a weighted state-averaged energy,

$$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$

with

$$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$

Here $\boldsymbol{\theta}$ denotes circuit parameters and $\boldsymbol{\kappa}$ orbital-rotation parameters. In the noisy-optimizer benchmark for H$_2$, the method is explicitly described in terms of the ensemble variational principle of Theophilou–Gross–Oliveira–Kohn, and the two references are chosen as the Hartree–Fock state and the first excited singlet configuration state function, with a generalized unitary coupled-cluster ansatz with singles and doubles and 3 trainable parameters in that specific setup (Illésová et al., 9 Oct 2025).

The equal-weight case has a special status. In the original SA-OO-VQE analysis and its later quasi-diabatic reinterpretation, the equi-ensemble objective is invariant under rotations within the targeted low-energy subspace. Consequently, the optimization identifies the correct subspace before it identifies a unique adiabatic basis within that subspace. This is why the method is often described as treating states “democratically”: it does not privilege one root or impose an artificial ordering across a degeneracy. A common misconception is that the raw equi-ensemble output must already be the adiabatic eigenbasis. The later literature makes clear that this is not generally true; state resolution is a separate, post-variational step (Yalouz et al., 2021, Illésová et al., 25 Feb 2025).

3. Orbital optimization and hybrid algorithmic structure

The “OO” component is the defining extension beyond ordinary SA-VQE. Molecular orbitals are rotated by an anti-Hermitian generator,

$$\hat U_{\rm OO}(\boldsymbol{\kappa}) = e^{-\hat\kappa},\qquad \hat\kappa = \sum_{p>q}\kappa_{pq}(\hat E_{pq}-\hat E_{qp}),$$

so that the electronic Hamiltonian becomes explicitly basis dependent,

$$\hat H(\boldsymbol{\kappa})=\hat U_{\rm OO}^\dagger(\boldsymbol{\kappa})\hat H\,\hat U_{\rm OO}(\boldsymbol{\kappa}).$$

The active-space Hamiltonian is therefore co-optimized with the quantum state rather than being fixed once at the Hartree–Fock level (Beseda et al., 2024, Yalouz et al., 2020).

In practice, SA-OO-VQE is organized as a macro-iterative loop. For fixed orbitals, the quantum part optimizes $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$0 by minimizing the state-averaged energy and measuring the energies and reduced density matrices of the targeted states. For fixed $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$1, the classical part uses state-averaged gradients and Hessians to update $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$2, rebuilds the molecular integrals in the rotated basis, and returns the new Hamiltonian to the quantum step. In the original derivation, the state-averaged orbital update is written in terms of averaged gradient and Hessian tensors,

$$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$3

followed by a Newton–Raphson step. This is the direct quantum analogue of state-averaged orbital optimization in multiconfigurational theory (Yalouz et al., 2020, Bezděk, 15 Sep 2025).

The division of labor between hardware and software is consistent across descriptions. The quantum device prepares trial states and measures energies and 1- and 2-RDMs; the classical computer handles the optimization of $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$4 and $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$5, evaluation of the state-averaged objective, construction of orbital derivatives, and Hamiltonian updates. This architecture is explicitly justified as NISQ-friendly because orbital optimization is a classical post-processing layer and therefore does not increase circuit depth, while only selected subroutines are offloaded to the QPU (Beseda et al., 2024, Bezděk, 15 Sep 2025).

The software note associated with the 2024 package presents a production-oriented implementation written in Python 3, interfaced with Qiskit, supporting Qiskit optimizers and including a custom Particle Swarm Optimization implementation. The same paper contrasts this package with an earlier pedagogical version based on matrix-vector multiplication rather than actual quantum measurements, and frames the newer code as intended for both quantum hardware and classical high-performance computing resources (Beseda et al., 2024).

4. State resolution, analytical derivatives, and quasi-diabatic interpretation

Because the equi-ensemble objective is invariant under rotations in the optimized state subspace, the converged SA-OO-VQE states span the target manifold but are not necessarily the final adiabatic eigenstates. The 2021 extension introduces an efficient state-resolution step that is deliberately postponed to the end of the full optimization. After convergence, the two optimized states are rotated by an angle $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$6,

$$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$7

and $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$8 is optimized so that one rotated state has minimal energy in the final Hamiltonian. This recovers approximate adiabatic states without repeated intermediate diagonalizations during the outer loop (Yalouz et al., 2021).

The same line of work extends SA-OO-VQE to analytical nuclear gradients and nonadiabatic couplings, which are required for geometry optimization, minimum-energy paths, conical-intersection searches, and nonadiabatic dynamics. The derivations use Lagrangian and coupled-perturbed formulations rather than naive Hellmann–Feynman differentiation, since the optimized circuit and orbital parameters depend implicitly on nuclear geometry. In the corresponding measurement model, the quantum processor supplies 1-RDMs, 2-RDMs, transition RDMs, and parameter-shift derivatives, while the classical side solves linear response equations and contracts them with integral derivatives (Omiya et al., 2021, Yalouz et al., 2021).

A further conceptual development reinterprets SA-OO-VQE as a least-transformed block-diagonalization procedure and, in that sense, as a generator of a quasi-diabatic representation “for free.” The key statement is that equal-weight SA-OO-VQE does not fundamentally diagonalize the Hamiltonian inside the selected low-energy space; it block-diagonalizes it, and the intermediate basis obtained in that block-diagonalization is the quasi-diabatic one. In the formaldimine analysis, intrinsic diabaticity is reported as approximately $$|\Psi_{\rm A}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm A}\rangle,\qquad |\Psi_{\rm B}(\boldsymbol{\theta})\rangle=\hat U(\boldsymbol{\theta})|\Phi_{\rm B}\rangle,$$9 to $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$0, residual diabaticity between $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$1 and about $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$2, with the residual exactly zero on symmetry lines such as $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$3 and $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$4. In the same study, adiabatic CI-NACs become numerically large near the conical intersection, on the order of $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$5–$\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$6, whereas quasi-diabatic CI-NACs remain around a few $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$7, and CSF-NACs are also around $\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$8 (Illésová et al., 25 Feb 2025).

This interpretation clarifies a recurrent source of confusion. SA-OO-VQE is not solely an excited-state energy algorithm; in its equal-weight form it is also a multistate orbital-relaxed subspace optimization that naturally supports quasi-diabatic analysis. This suggests that its relevance extends beyond static spectroscopy to vibronic Hamiltonians and nonadiabatic dynamics, provided that the target subspace and orbital basis are chosen appropriately (Illésová et al., 25 Feb 2025).

5. Benchmark systems and photochemical applications

The canonical benchmark is formaldimine, CH$\langle \Phi_{\rm A}|\Phi_{\rm B}\rangle=0$9NH, treated as a minimal Schiff-base model relevant to photoisomerization in rhodopsin-related chemistry. In the original study, the calculation used a cc-pVDZ basis and a CAS(4,3) active space consisting of the nitrogen nonbonding orbital together with the $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$0 and $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$1 orbitals, corresponding to a six-qubit mapping. In the canonical Hartree–Fock basis, CASCI(4,3) fails to capture the conical intersection, and SA-VQE alone reproduces the CASCI(4,3) energies while still missing that intersection. When state-averaged orbital optimization is added, SA-OO-VQE recovers the conical intersection around $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$2 at $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$3, with individual state energies reported to be within chemical accuracy on the one-dimensional cut studied. The convergence is described as rapid: about 5 SA-OO-VQE cycles at a representative geometry, and at most 10 cycles across the scan. The converged states have an average fidelity of about 99.85% against SA-CASSCF wavefunctions, with a lower bound of about 99.75% (Yalouz et al., 2020).

The later extension on formaldimine sharpens this picture. At $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$4 and $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$5, three full SA-OO-VQE cycles are sufficient for convergence of the state-averaged energy, and after the final state-resolution rotation the energies match SA-CASSCF to about $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$6 Ha. Analytical gradients and NACs are reported to agree very closely with SA-CASSCF, with deviations around $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$7 mHa/degree for gradients and $$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$8 degree$$E_{\mathrm{SA}}=\sum_k w_k E_k,\qquad \sum_k w_k = 1,$$9 for NAC amplitudes. On the symmetry plane $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$0, the conical intersection is located around $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$1, and a steepest-descent optimization gives $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$2, $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$3; a full minimal-energy conical-intersection optimization converges in 78 iterations to approximately $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$4 (Yalouz et al., 2021).

A second photochemical demonstration treats the cis–trans isomerization of 1,3,3,3-tetrafluoropropene. There the method is formulated as SA-OO-VQD and used for the three lowest singlet states $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$5, $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$6, and $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$7 with a 2-electron-in-2-orbital active space, a 6-31G basis, equal state-averaging weights of $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$8, and a real-valued symmetry-preserving circuit of depth $$E_k(\boldsymbol{\theta},\boldsymbol{\kappa})= \langle \Psi_k(\boldsymbol{\theta})|\hat H(\boldsymbol{\kappa})|\Psi_k(\boldsymbol{\theta})\rangle.$$9 with 30 parameters. The study reports a nearly degenerate $\boldsymbol{\theta}$0 crossing at the optimized conical-intersection minimum, with a gap of $\boldsymbol{\theta}$1 kcal/mol in the quantum simulation and $\boldsymbol{\theta}$2 kcal/mol in the SA-CASSCF reference, and a smooth minimum-energy path consistent with the classical benchmark (Omiya et al., 2021).

The 2024 package note includes a smaller but implementation-oriented formaldimine illustration using 3 active orbitals, 4 electrons, and the STO-3G basis, comparing potential energy curves, ground-state gradients, and total non-adiabatic couplings against Molcas CASSCF. The stated outcome is that the implementation reproduces the expected qualitative behavior and can be used for PESs, gradients, and NACs in a small but nontrivial molecular system (Beseda et al., 2024).

6. Implementations, numerical optimization, related extensions, and limitations

Subsequent work has focused heavily on the classical optimizer inside the SA-OO-VQE loop. A 2025 benchmark on H$\boldsymbol{\theta}$3 at equilibrium, using a neutral singlet geometry at $\boldsymbol{\theta}$4, a $\boldsymbol{\theta}$5 active space, and the cc-pVDZ basis, compares BFGS, SLSQP, Nelder–Mead, Powell, COBYLA, and iSOMA under ideal, finite-shot, and decoherence-noisy conditions. In the noiseless case all local methods converge to the same optimal state-averaged energy $\boldsymbol{\theta}$6, matching the Psi4 CASSCF reference, with mean evaluation counts of 37.1 for BFGS, 52.0 for SLSQP, 295.2 for Nelder–Mead, 396.0 for Powell, 433.6 for COBYLA, and 1135.8 for iSOMA. Across 21 noise categories, the reported mean distances to the Psi4 reference are 0.209 for Powell, 0.211 for Nelder–Mead, 0.216 for BFGS, 0.328 for iSOMA, 0.329 for COBYLA, and 0.523 for SLSQP, while the Friedman test gives $\boldsymbol{\theta}$7, $\boldsymbol{\theta}$8, and $\boldsymbol{\theta}$9. The practical recommendation of that study is to use BFGS by default, to regard COBYLA as a low-cost alternative, and to avoid SLSQP in noisy SA-OO-VQE settings because of consistent instability (Illésová et al., 9 Oct 2025).

A separate thesis-level study examines Gradient Descent, BFGS, COBYLA, SLSQP, and several Differential Evolution variants on H$\boldsymbol{\kappa}$0, H$\boldsymbol{\kappa}$1, LiH, and formaldimine. In that comparison, BFGS and SLSQP are reported as the clear winners in accuracy, speed, and stability across H$\boldsymbol{\kappa}$2, H$\boldsymbol{\kappa}$3, and LiH, with specific best state-averaged energies of $\boldsymbol{\kappa}$4 Ha for H$\boldsymbol{\kappa}$5, $\boldsymbol{\kappa}$6 Ha for H$\boldsymbol{\kappa}$7, and $\boldsymbol{\kappa}$8 Ha for LiH. The same work also reiterates, using formaldimine at fixed $\boldsymbol{\kappa}$9, that fixed-orbital SA-VQE yields an avoided crossing where SA-OO-VQE yields a near-degenerate crossing around $_2$0, and concludes that orbital optimization is essential for the correct topology near a conical intersection (Bezděk, 15 Sep 2025).

Two adjacent research directions delimit the method’s scope. First, orbital-optimized VQE combined with adiabatic-connection corrections recovers dynamical correlation outside the active space and adds no extra quantum requirements beyond 1- and 2-RDMs, but this framework is explicitly state-specific rather than a formal SA-OO-VQE functional over multiple states. Its relevance lies in showing how OO-VQE references may be complemented by classical postprocessing, not in replacing state averaging (Matoušek et al., 2023). Second, more general orbital-rotation studies in VQE and neural quantum states show that joint basis optimization can improve expressivity and the optimization landscape and can act as a classical layer that does not increase circuit depth in VQE, which provides a broader formal context for why the orbital-relaxed component of SA-OO-VQE is effective (Moreno et al., 2023).

The principal limitations stated across the literature are consistent. Performance depends on ansatz quality and optimizer behavior; quantum noise and limited qubit counts still constrain scale; repeated measurements and careful numerical optimization remain necessary; active-space truncation leaves missing dynamical correlation outside the active space; and orbital optimization itself carries classical overhead, with repeated two-electron integral transformations scaling roughly as $_2$1 without approximation in the original analysis. A further practical point is that the best-developed demonstrations remain modest in active-space size, so the method’s long-term significance depends on whether larger active spaces and lower-noise quantum hardware can be brought to bear on the multistate orbital-relaxed framework already established (Beseda et al., 2024, Yalouz et al., 2020).