VQE with Automatically-Adjusted Constraints

• The paper introduces VQE-AC, a constrained excited-state variational quantum eigensolver that eliminates manual penalty tuning by enforcing hard overlap constraints via COBYLA.
• VQE-AC employs a spin-restricted, chemistry-inspired ansatz that maintains the correct singlet subspace, essential for accurate state tracking at Frank-Condon and conical intersection geometries.
• Experiments on ethylene and phenol blue show that VQE-AC produces smooth potential energy surfaces with energy errors as low as 0.14 to 2 kcal/mol, demonstrating improved stability over VQD.

Searching arXiv for the cited papers to ground the article in current records. VQE with Automatically-Adjusted Constraints (VQE-AC) is a constrained excited-state variational quantum eigensolver strategy in which the excited-state search is performed by direct constrained optimization rather than by adding a manually tuned penalty term. It was introduced together with a chemistry-inspired spin-restricted ansatz for ground and excited state calculations at key geometries such as the Frank-Condon and conical intersection geometries, where states can be close in energy or nearly degenerate. In the reported applications to ethylene and phenol blue at the complete active space self-consistent field level of theory, the method was designed to avoid pre-determination of constraint weights and to describe smooth potential energy surfaces; the published real-device results on ibm_kawasaki reported energy errors at most $2\ \mathrm{kcal\ mol^{-1}}$ (Gocho et al., 2021).

1. Problem setting and motivation

The motivation for VQE-AC is the difficulty of computing excited-state potential energy surfaces on noisy intermediate-scale quantum devices, especially near Frank-Condon and conical intersection geometries. Standard VQE relies on the ground-state variational principle,

$$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$

but excited states are not minima of the energy functional in the full Hilbert space; they are typically saddle points. Excited-state optimization therefore requires constraints that keep the variational search orthogonal to lower states (Gocho et al., 2021).

The immediate antecedent considered in the same work is variational quantum deflation (VQD). For the first excited state, VQD minimizes

$$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$

where $\Psi_0$ is the previously found ground state and $\beta$ is a manually chosen penalty weight. A singlet-targeting form is also given: $$C_2(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2 +\gamma \langle\Psi(\boldsymbol{\theta})|\hat{S}^2|\Psi(\boldsymbol{\theta})\rangle.$$ The reported limitation is that $\beta$ must be chosen in advance and tuned manually. If $\beta$ is too small, the optimization may collapse toward the ground state; if $\beta$ is too large, the objective may be distorted and an incorrect higher state may be obtained. The best $\beta$ was reported to depend on geometry, molecule, and ansatz, which is especially problematic for potential-energy-surface scans.

This dependence is particularly acute near conical intersections, where the first excited singlet is very close to or degenerate with the ground state. The paper explicitly reports that with insufficient penalty, the optimization converged to $$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$0 rather than $$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$1 for ethylene at the conical intersection geometry. VQE-AC was introduced to remove this hyperparameter-tuning step while retaining explicit orthogonality control.

2. Constrained optimization formulation

VQE-AC replaces the manually weighted VQD penalty with a hard overlap constraint enforced by a classical constrained optimizer. In the implementation described in the original work, the constraint is

$$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$2

and the optimization problem becomes

$$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$3

The paper states that the “weight of the constraint” is automatically adjusted inside COBYLA, which is the reason for the designation “automatically-adjusted constraints” (Gocho et al., 2021).

This formulation is derivative-free and uses COBYLA’s internal linearized constrained optimization. The methodological distinction from VQD is therefore not the presence of orthogonality itself, but the way orthogonality is enforced. In VQD, orthogonality is mediated by a user-chosen penalty coefficient; in VQE-AC, orthogonality is imposed directly as a constraint.

For higher excited states, the same paper notes that more orthogonality constraints can be added to all lower states,

$$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$4

with the energy minimized under all such constraints. The authors state that this can be done without an exponential increase in complexity. This suggests a route from first-excited-state targeting to a more general constrained state-specific eigensolver framework.

A recurrent theme in the reported discussion is the relevance of this direct constrained formulation to smooth potential energy surfaces. Because the same overlap constraint can be enforced consistently by COBYLA across geometries, the workflow avoids repeated penalty-weight retuning when the state character changes rapidly.

3. Spin-restricted ansatz and chemistry-inspired structure

The VQE-AC formulation was combined with a spin-restricted ansatz designed to stay entirely inside the target spin-multiplicity subspace, specifically the singlet subspace (Gocho et al., 2021). In the minimal active-space example of two electrons in two orbitals, the parity-mapped basis states support the following singlet and triplet combinations:

• doubly occupied HOMO: $$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$5
• doubly occupied LUMO: $$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$6
• open-shell singlet:

$$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$7

• open-shell triplet:

$$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$8

The ansatz is constructed to generate only the singlet combinations. The reported low-depth circuit prepares a doubly excited configuration via an $$\langle \Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle \ge E_0,$$9 gate, applies an $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$0 rotation and CNOT, and then applies additional $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$1 transformations to build the singlet superposition. The resulting parameterized state is given in Eq. (5) of the paper as a linear combination of singlet configuration state functions with trigonometric coefficients.

The characterization “chemistry-inspired” is justified in the source text by four properties: it is built from configuration state functions rather than arbitrary hardware-efficient rotations; it respects particle number and spin symmetry; it is aligned with the physical singlet subspace relevant for organic photoexcited states; and it uses the natural electronic-structure structure of the active space rather than generic entangling layers. The same discussion emphasizes two practical consequences: a minimal number of parameters to span the singlet subspace and shorter circuits than generic heuristic ansätze, which reduces noise sensitivity and barren plateau risk.

The paper also notes a limitation: spin-restricted ansätze can be extended to larger active spaces, but circuit depth grows with active-space size. On current hardware, this may become demanding. Within the reported benchmarks, however, the short-depth, symmetry-preserving structure is part of the method rather than an auxiliary implementation detail.

4. Molecular targets, software stack, and computational workflow

The main molecular targets were ethylene and phenol blue, and the calculations addressed both the ground state $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$2 and the first singlet excited state $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$3 at Frank-Condon and conical intersection geometries (Gocho et al., 2021). Ethylene served as a benchmark system, while phenol blue was described as a larger, more application-relevant dye molecule and as a nonfluorescent dye relevant to fast internal conversion and conical intersections. Supplementary scaling demonstrations were also carried out for formaldehyde and trans-butadiene.

For the main calculations, the active space was CASSCF(2,2), that is, 2 electrons in 2 orbitals, typically HOMO and LUMO. The supplementary larger-scale tests used CASSCF(4,3) for formaldehyde and CASSCF(4,4) for trans-butadiene. The basis sets were STO-3G for ethylene and 6-31G(d) for phenol blue. Ethylene Frank-Condon and conical intersection geometries were optimized classically at the CASSCF level using MOLPRO and GRRM, whereas the phenol blue geometries were taken from a previous study.

The quantum-software stack combined Qiskit for circuit construction and simulation with SciPy’s COBYLA for classical optimization. The reported backends were a statevector simulator, a noisy-QASM simulator using a noise model based on ibmq_belem, and real hardware on ibm_kawasaki. For noisy-QASM and real hardware, the measurement count was 8192 shots. Measurement error mitigation was applied for noisy-QASM but not on the real device because calibration updates affected the result.

The electronic-structure workflow was embedded in a full SA-CASSCF / SS-CASSCF outer loop. The reported loop was: run VQE, VQD, or VQE-AC for energies; measure the 1-RDM and 2-RDM; update orbitals; and repeat until convergence. State-averaged CASSCF was used when both $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$4 and $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$5 were optimized together, whereas SS-CASSCF was used for ground-state-only calculations.

5. Reported numerical behavior

For ground-state calculations in ethylene at the Frank-Condon geometry, the paper compared RealAmplitudes RA(2), RA(6), EfficientSU2, and the spin-restricted ansatz (Gocho et al., 2021). On the statevector simulator, all converged essentially exactly. On noisy-QASM, the spin-restricted ansatz gave the smallest or comparable error, with lower parameter count and shorter circuits, while the heuristic ansätze had larger errors, especially the more complex ones.

For excited states, the key comparison was between VQD and VQE-AC. In ethylene, the performance of VQD depended strongly on $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$6: $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$7 was not sufficient at the conical intersection geometry and converged to the wrong state, $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$8 worked better, and larger $$C_1(\boldsymbol{\theta})= \langle\Psi(\boldsymbol{\theta})|\hat{H}|\Psi(\boldsymbol{\theta})\rangle +\beta\,\big|\langle\Psi(\boldsymbol{\theta})|\Psi_0\rangle\big|^2,$$9 values became too strong and caused inaccuracies even in ideal simulation. By contrast, VQE-AC gave small errors at both Frank-Condon and conical intersection geometries without tuning $\Psi_0$0. In ethylene, the noisy-QASM errors for the spin-restricted VQE-AC case were reported as up to about 0.45 kcal/mol.

For phenol blue, the reported pattern was similar. VQE-AC remained stable and accurate without tuning, while VQD again required geometry-dependent tuning and its best $\Psi_0$1 differed from ethylene. On noisy-QASM, the VQE-AC errors were reported as at most about 0.14 kcal/mol in the main results. The authors state that VQE-AC yielded smooth potential-energy-surface behavior across geometries, which they identify as crucial for photochemistry.

The real-device results on ibm_kawasaki were reported for phenol blue. The calculations converged smoothly, the number of orbital update iterations was less than 10, and the deviations from exact solutions were reported as up to about 2 kcal/mol for state energies and up to about 0.5 kcal/mol for excitation energies. At the conical intersection geometries, the errors were reported to be as small as 0.5 kcal/mol. A plausible implication is that the method’s principal empirical advantage is not only low absolute error under the reported conditions, but also stable behavior in the vicinity of near-degeneracies where geometry-dependent penalty tuning is otherwise required.

6. Relation to other constrained VQE strategies

VQE-AC is closely related to, but not equivalent to, other constraint-aware VQE schemes. The distinction is most explicit when comparing it with tailored variational forms for constrained combinatorial optimization and with physics-informed constrained VQE for quasi-degenerate topological systems.

In “Modeling Linear Inequality Constraints in Quadratic Binary Optimization for Variational Quantum Eigensolver” (Quinones et al., 2020), the central idea is a hand-designed constrained ansatz. The circuit $\Psi_0$2 is built so that the set of basis states with nonzero amplitude is exactly the feasible set for a modeled constraint, including $\Psi_0$3, $\Psi_0$4, $\Psi_0$5, and $\Psi_0$6. The paper emphasizes that this differs fundamentally from penalties because infeasible states are not in the support of the ansatz at all. It is therefore strongly related in spirit to constraint-aware VQE, but it is not automatically adjusting constraints during optimization; the constraints are manually designed and hard-coded into specialized circuits.

In “Detecting quasi-degenerate ground states in topological models via variational quantum eigensolver” (Ciaramelletti et al., 2024), the aim is to make VQE reliable in the presence of symmetry- or topology-protected quasi-degeneracy in SSH and Kitaev open chains. The methods include a modified cost function with entanglement constraints,

$\Psi_0$7

warm-start and adiabatic continuation strategies, a problem-inspired ansatz using $\Psi_0$8 rotations and

$\Psi_0$9

and a progressive parameter sweep with acceptance checks. The paper explicitly notes that it does not present a method explicitly called VQE with Automatically-Adjusted Constraints, and it does not implement a formally automatic constraint scheduler in the sense of a dedicated adaptive-constraint algorithm. The modified cost function uses fixed physically motivated constraints; warm-starting is adaptive in initialization rather than in constraint coefficients; and the problem-inspired ansatz is a structural restriction of the variational manifold.

These comparisons clarify a common misconception. VQE-AC is not a generic label for every VQE variant that uses constraints. In the strict sense represented by the excited-state chemistry formulation, VQE-AC denotes direct constrained optimization in which the effective constraint strength is adjusted internally by the classical optimizer rather than supplied as a manually tuned penalty weight. Constraint-aware ansatz design and fixed penalty augmentation are closely related methodologies, but they instantiate different mechanisms for enforcing feasibility or state selectivity.

7. Scope, advantages, and limitations

The central reported advantage of VQE-AC over VQD is the elimination of manual penalty tuning (Gocho et al., 2021). Because the best $\beta$0 depends on molecule, geometry, ansatz, and target excited state, VQD is fragile for workflows that traverse multiple geometries. VQE-AC replaces that geometry-dependent tuning problem with direct overlap constraints handled by COBYLA. The authors further argue that this makes the method better suited to potential-energy-surface scans and suggest that multiple excited-state constraints can be handled without an exponential increase in complexity.

The method’s effectiveness in the reported study also depends on its combination with a low-depth, symmetry-preserving ansatz. The source text therefore presents VQE-AC not as an isolated optimizer trick, but as one component of a broader strategy that includes parity mapping with symmetry reduction, inverse-circuit fidelity measurement for overlaps, and a spin-restricted variational form that remains close to the desired spin sector.

The reported caveats are equally specific. VQE-AC still depends on optimizer behavior and on the feasibility of the constraint threshold. Real-device errors, although improved, remained up to about $\beta$1 in state energies on ibm_kawasaki. Larger active spaces require deeper circuits. The overlap cutoff, such as $\beta$2, is still a design choice even though it is not a tunable penalty weight.

Within these stated limits, VQE-AC occupies a specific place in the constrained-VQE landscape: a direct constrained optimization method for excited-state calculations, introduced to avoid pre-determination of constraint weights, combined with a spin-restricted chemistry-inspired ansatz, and empirically evaluated at Frank-Condon and conical intersection geometries where smooth state tracking is especially demanding.