Physics-Informed Machine Learning Correction of Variational Quantum Eigensolver Energies for Molecular Ground States

Abstract

The variational quantum Eigensolver (VQE) provides a scalable route to molecular ground-state energies on near-term quantum hardware, but its accuracy degrades rapidly as system size grows because hardware-efficient ansätze cannot span the full correlation subspace. We introduce a physics-informed ridge regression correction that is trained on the difference between VQE and Hartree–Fock (HF) energies and applied as a postprocessing step requiring no additional quantum resources. The feature setthe VQE–HF energy gap ΔE, its square ΔE ², the active-space qubit count, the number of active electrons, and two cross-term interactionscaptures the dominant drivers of ansatz error in a low-dimensional, interpretable form. The model is benchmarked on H₂, H₂O, NH₃, CH₄, C₂H₆, and CH₃OH using the STO-3G basis set and a state vector VQE simulator using a two-layer Ry-CNOT hardware-efficient ansatz. To assess cross-molecule generalizability, the corrected framework is further tested on CO₂, N₂, and HCN under both untrained and trained conditions. Absolute energy errors fall from a range of 0.02–0.79 Ha (raw VQE) to 0.0007–0.094 Ha after correction, corresponding to improvement factors of 3× to 640× relative to CASCI reference values. Zero-shot generalization tests yield factors of 9–58×, and including molecules in training improves results by a further order of magnitude in several cases. The three extended molecules exhibit improvement factors of 4–6× in untrained settings and 5–6× after training, providing preliminary evidence of transferability across chemically distinct systems. These findings suggest that the VQE–HF energy gap encodes physically transferable information about correlation-energy saturation, and that even the smallest training set can produce practically useful corrections for near-term quantum chemistry.

1. Introduction

Solving the electronic Schrödinger equation accurately and efficiently is essentially a central challenge in computational chemistry. Since the total electronic energy dictates everything from bond strengths to reaction barriers, even a tiny error on the order of a few milli Hartree can lead to significantly inaccurate predictions about how a molecule behaves. While a formally exact solution exists in full configuration interaction (FCI), the dimension of the many-electron Hilbert space scales exponentially with the number of electrons, rendering the method computationally intractable for larger systems, thus restricting it only for tiny systems of about eight to ten electrons.

Because of this “combinatorial explosion,” most of our work relies on a ladder of approximations that balance speed against precision. We usually start with Hartree–Fock (HF) to get a baseline average of the electron field, then layer on more sophisticated methods like perturbation theory (MP2, CASPT2) or Coupled-Cluster [CCSD­(T)], the latter of which serves as the “gold standard” primarily for single-determinant systems where static correlation is negligible. For many complex scenarios like breaking bonds, we turn to multireference techniques like CASSCF. , Ultimately, every methodological choice presents a distinct trade-off, as the most accurate electronic structure methods remain computationally prohibitive for large systems, leaving researchers in constant search of an optimal balance between chemical accuracy and computational cost.

Quantum computation offers a qualitatively different scaling argument. Because the state of N electrons can be mapped onto N qubits without the exponential memory overhead of classical tensor representations, molecular wave functions can in principle be stored and evaluated using polynomial resources. , The variational quantum Eigensolver (VQE), proposed by Peruzzo and co-workers in 2014, was the first practically oriented quantum chemistry algorithm designed for near-term devices. VQE prepares a parametrized trial state on a quantum processor, evaluates the molecular Hamiltonian expectation value by repeated measurement, and iterates over circuit parameters on a classical coprocessor until the energy is minimized. Because the algorithm is variational, the computed energy is guaranteed to bound the true ground-state value from above at every iteration. Early demonstrations on H₂ and LiH showed that VQE recovers FCI-level energies when the qubit count is small enough for the ansatz to be expressive.

The practical constraints of VQE become glaringly apparent once a system exceeds roughly ten qubits. While hardware-efficient ansätzeconstructed from interleaved layers of single-qubit rotations and fixed entangling gatesare appealing for their minimal circuit depth and compatibility with superconducting or trapped-ion architectures, their expressibility tends to saturate long before reaching the FCI state. The resulting ansatz error grows roughly with the number of electrons that must be correlated. Separately, the optimization landscape of VQE becomes increasingly pathological at scale: gradients of parametrized circuits decay exponentially with qubit count (the barren-plateau phenomenon), and gradient-free optimizers require exponentially many function evaluations to locate even a useful local minimum. On real hardware, an additional noise floor is imposed by gate and readout errors. Taken together, these factors produce raw VQE errors that, in our experiments, range from 0.02 Ha for H₂ (4 qubits) to nearly 0.8 Ha for CH₄ (18 qubits)far outside the chemical-accuracy threshold of approximately 1.6 mHa (1 kcal mol^–1).

Considerable research effort has been directed toward addressing the expressibility problem from the ansatz side. The unitary coupled-cluster singles-and-doubles (UCCSD) ansatz provides a chemically motivated parametrization but requires circuit depths impractical for systems with more than a few dozen spin–orbitals. Adaptive methods such as ADAPT-VQE build the ansatz iteratively, achieving near-FCI accuracy with fewer parameters but still deeper circuits than hardware-efficient variants. To mitigate these deep circuit requirements, variants such as Qubit-ADAPT-VQE have been proposed to optimize cost-performance metrics on near-term hardware. Alternatively, within the domain of fixed-depth hardware-efficient designs, symmetry-preserving ansätze (SPA) , have emerged as a highly effective choice for maintaining physical conservation laws while keeping circuit depth minimal. However, even with these specialized architectural advances, scaling to larger macromolecular spaces still incurs severe optimization and noise bottlenecks, motivating alternative postprocessing correction strategies. Symmetry constraints can prune the parameter space, but the fundamental tension between expressibility and circuit depth remains unresolved. Error mitigation techniques such as zero-noise extrapolation and probabilistic error cancellation address hardware noise but leave untouched the ansatz error that persists even on ideal, noise-free simulators.

The approach pursued here is to leave the quantum circuit entirely unchanged and to correct its output classically. This constitutes an instance of delta-learning: rather than predicting absolute energieswhich requires the model to capture all quantum many-body physicswe train a regressor on the difference between a cheap reference and an accurate target. In classical quantum chemistry this idea underpins composite thermochemical methods and focal-point analysis; what is new here is that the cheap reference is a VQE output rather than HF or MP2. Several groups have applied machine learning to correct density functional and coupled-cluster energies, but systematic application to VQE outputswhere the reference is itself a quantum computation subject to ansatz and optimization errors that depend on hardware and ansatz architecturehas received comparatively little attention.

We present a correction scheme based on ridge regression with six physically motivated features derived from the VQE–HF energy gap, system size, and their cross-products. All features are scalar quantities available from any completed VQE run at negligible overhead. The model is evaluated using both strict leave-one-out cross-validation (LOOCV) and post-training protocols. LOOCV results provide estimates of predictive performance on unseen molecules, while post-training evaluations assess the model’s ability to learn systematic correction relationships present in the data set. In addition to the primary benchmark set of six small molecules, we evaluate the model on three chemically distinct extended moleculescarbon dioxide (CO₂), molecular nitrogen (N₂), and hydrogen cyanide (HCN)under both zero-shot and post-training conditions. The key contributions are (i) a demonstration that a six-dimensional feature space is sufficient to capture the system-size scaling of VQE ansatz error across diverse chemical systems; (ii) quantitative error reduction to the milli- to submillihartree range on three of six primary benchmarks; (iii) a comprehensive comparison of zero-shot and post-training accuracy; (iv) a case study of NH₃ illustrating how cross-term feature engineering resolves initial model failures; (v) generalization benchmarks on CO₂, N₂, and HCN demonstrating cross-domain transferability; and (vi) a physical interpretation of the regression weights relating model behavior to the underlying quantum chemistry.

2. Theoretical Background

2.1. The Variational Quantum Eigensolver

Let Ĥ be the molecular electronic Hamiltonian acting on a Hilbert space of dimension 2 ^Nq. VQE prepares the parametrized trial state

$$|\psi (\theta )⟩=U(\theta )|0{⟩}^{\otimes N_q}$$ 1

using a parametrized unitary U(θ)­with M _real parameters, and minimizes the energy expectation value

$$E(\theta )=⟨\psi (\theta )|Ĥ|\psi (\theta )⟩\ge E_0$$ 2

The variational bound E(θ) ≥ E ₀ holds for any normalized state |ψ⟩ and provides the sole formal guarantee of VQE: the algorithm cannot produce an energy lower than the true ground-state value. In this work the ansatz uses d = 2 alternating layers, each composed of a column of R _y (θ _k ) = exp­(−iθ _k Y/2) single-qubit rotations followed by a linear chain of CNOT gates. The total parameter count is M = (d + 1)N _q , which grows linearly with qubit number. The irreducible accuracy limit of any fixed ansatz is the ansatz error

$${\epsilon }_{\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{z}}\equiv E^{\mathrm{V}\mathrm{Q}\mathrm{E}}-E_0\ge 0$$ 3

In practice, optimizers converge to local minima, adding an optimization error. Our correction scheme targets the combined sum of both error terms. We use ten random restarts and record the lowest energy found, reducing but not eliminating the optimization error for large systems.

2.2. Electronic Hamiltonian and Jordan–Wigner Mapping

Under the Born–Oppenheimer approximation and in a molecular orbital (MO) basis of K spin–orbitals, the electronic Hamiltonian in second quantization is

$$Ĥ=\sum _{pq}{h}_{pq}{a}_p^{†}{a}_q+\frac{1}{2}\sum _{pqrs}{h}_{pqrs}{a}_p^{†}{a}_q^{†}{a}_r{a}_s+E_{\mathrm{n}\mathrm{u}\mathrm{c}}$$ 4

where a _p ^†(a _p ) creates (annihilates) an electron in spin–orbital p, h _pq are one-electron integrals, h _pqrs are two-electron repulsion integrals, and E _nuc is the nuclear repulsion energy. Fermionic anticommutation relations are enforced by the Jordan–Wigner transformation

$$a_p^{†}=\frac{1}{2}(\prod _{j=0}^{p-1}{Z}_j)(X_p-i{Y}_p)$$ 5

which maps spin–orbital p to qubit p and generates Pauli strings of weight up to $\mathcal{O}(K)$ . The resulting qubit Hamiltonian contains $\mathcal{O}(K^4)$ Pauli terms, each requiring a separate measurement circuit on hardware.

$$\widehat{H_{\mathrm{q}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{t}}}=\sum _{\alpha }{c}_{\alpha }{P}_{\alpha },P_{\alpha }\in \{I,X,Y,Z{\}}^{\otimes N_{\mathrm{q}}}$$ 6

2.3. Correlation Energy and the VQE–HF Gap

Hartree–Fock theory optimizes a single Slater determinant, yielding the mean-field energy E _HF. The correlation energy is defined as

$$E_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}\equiv E_0-E_{\mathrm{H}\mathrm{F}}\le 0$$ 7

­(CASCI energies are used as the exact reference throughout, i.e., E ₀ ≈ E _CASCI.) The magnitude of the correlation energy grows roughly as the square of the number of correlated electron pairs. The VQE energy deficit relative to the exact value decomposes as

$$E_{\mathrm{V}\mathrm{Q}\mathrm{E}}-E_0=(E_{\mathrm{V}\mathrm{Q}\mathrm{E}}-E_{\mathrm{H}\mathrm{F}})+|E_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}|$$ 8

where ΔE _VQE/HF = E _VQE – E _HF < 0by construction. For a hardware-efficient ansatz of fixed depth, the recovered fraction of correlation energy is approximately constant across system sizes, making the absolute residual a predictable, scalable quantity. This systematic scaling provides the fundamental physical justification for the regression model proposed in this work.

2.4. Correction Model and Feature Engineering

We define an additive postcorrection to the VQE energy

$$E_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=E_{\mathrm{V}\mathrm{Q}\mathrm{E}}+\mathrm{\Delta }{E}_{\mathrm{A}\mathrm{I}},\mathrm{\Delta }{E}_{\mathrm{A}\mathrm{I}}\approx E_0-E_{\mathrm{V}\mathrm{Q}\mathrm{E}}$$ 9

the correction term ΔE _AI is expressed as a linear function of six physical features

$$x_1=\mathrm{\Delta }E\equiv E_{\mathrm{V}\mathrm{Q}\mathrm{E}}-E_{\mathrm{H}\mathrm{F}}$$ 10

$$x_2={(\mathrm{\Delta }E)}^2$$ 11

$$x_3=N_q$$ 12

$$x_4=N_a$$ 13

$$x_5=N_q\times \mathrm{\Delta }E$$ 14

$$x_6=N_a\times \mathrm{\Delta }E$$ 15

Feature x ₁ serves as the primary predictor; because a hardware-efficient ansatz recovers a roughly constant fraction of |E _corr| irrespective of system size, ΔE is the best single proxy for the residual correction. Feature x ₂accounts for the superlinear growth of total correlation energy in larger molecules. The cross-terms x ₅ and x ₆ allow the model to learn that the same |ΔE| implies a smaller residual in a compact, well-correlated molecule than in a large, partially saturated one.

All features are z-score normalized prior to fitting

$${\mathrm{x̃}}_i=\frac{x_i-{\mu }_i}{{\sigma }_i}$$ 16

where μ _i and σ _i are computed exclusively from the training split. The model takes the linear form

$$\mathrm{\Delta }{E}_{\mathrm{A}\mathrm{I}}={\mathit{w}}^T\mathit{x̃}+b$$ 17

fitted by ridge regression, the Tikhonov regularization

$$L(\mathit{w},b)=\parallel y-X\mathit{w}-b\mathbf{1}{\parallel }_2^2+\lambda \parallel \mathit{w}{\parallel }_2^2$$ 18

where y _i = E _CASCI,i – E _VQE,i are the training targets and λ = 0.01 throughout. The closed-form solution is

$${\mathit{w}}^{*}={(X^{T}X+\lambda I)}^{-1}{X}^{T}y$$ 19

which is unique for any λ > 0 and suppresses overfitting in the small-data regime while maintaining physically interpretable regression coefficients.

2.5. Computational Methods

2.5.1. Molecular Geometries and Integral Generation

Equilibrium geometries were obtained at the MP2/cc-pVTZ level of theory and are provided in the Supporting Information. All one- and two-electron integrals were computed with PySCF at the STO-3G level in the canonical restricted Hartree–Fock (RHF) molecular orbital basis. The STO-3G basis set substantially underestimates absolute correlation energies relative to the complete-basis-set limit but provides a reproducible, computationally tractable framework for benchmarking relative method performance. It is important to note that the correction ML model was trained and evaluated strictly within a specific basis set encompassing nine compounds in total ensuring that basis-set incompleteness does not bias or misrepresent the resulting data. The converged R _HF energy is used as E _HF throughout. Complete Active Space Configuration Interaction (CASCI) energies serve as the exact reference within this basis. Rather than Full Configuration Interaction (FCI), the computational workflow restricts calculations to an active space of 6 electrons distributed across 6 molecular orbitalsa configuration conventionally denoted as (6e, 6o) by quantum chemists. CASCI within this designated space is chosen as the appropriate reference baseline because, given a quantum circuit with sufficient expressibility, the VQE algorithm is fundamentally designed to recover the static correlation present within that exact active space. For the extended benchmark, the same protocol was applied to CO₂, N₂, and HCN three molecules chosen specifically to probe transferability to multiple-bond and heteroatom-rich chemistry absent from the primary training set.

2.5.2. Qubit Hamiltonian Construction

We begin by passing the computed Fermionic integrals to OpenFermion using its InteractionOperator class, which provides a convenient representation of the molecular electronic Hamiltonian in second quantization. To make the problem tractable for near-term quantum simulations, we restrict our calculations to a carefully chosen active space. This active space consists of a subset of molecular orbitalstypically those associated with valence electronsthat capture the most significant electron correlation effects while excluding less relevant core and high-energy virtual orbitals. Once the active-space Hamiltonian is constructed, we apply the Jordan–Wigner transformation to map the Fermionic operators onto qubit operators. This transformation converts the electronic Hamiltonian into a qubit Hamiltonian expressed as a weighted sum of Pauli strings, which can be directly used in variational quantum Eigensolver (VQE) simulations.

2.5.3. VQE Simulation

Energy expectation values were evaluated on an exact NumPy statevector simulator, eliminating shot noise and gate errors so that the reported VQE errors reflect purely the ansatz expressibility and optimization limitations of the hardware-efficient circuit. The variational quantum eigensolver calculations employed a hardware-efficient ansatz consisting of parametrized single-qubit (R _y ) rotations interleaved with nearest-neighbor CNOT entangling operations. To maintain compatibility with near-term quantum hardware while controlling circuit depth, a fixed-depth architecture with (d = 2) entangling layers was used throughout all simulations. For a system containing (N _q ) qubits, the total number of variational parameters is M = (d + 1)N _q . Figure illustrates the circuit architecture used in this work. The ansatz begins with an initial layer of (R _y ) rotations acting on all qubits, followed by alternating layers of linear CNOT entanglement and additional (R _y ) rotations. This architecture provides a simple, scalable, and hardware-compatible variational form while maintaining a manageable optimization landscape for active-space molecular simulations. To reduce optimization error, ten random initial parameter vectors were sampled uniformly from (0,2π) ^M , and the COBYLA algorithm was applied to each restart (chosen over alternative derivative-free methods such as Powell’s conjugate direction method due to its explicit constraint handling), with a maximum of 2000 function evaluations per restart. Optimization runs terminated either upon satisfying the preset convergence tolerance or upon reaching this evaluation ceiling. For individual random restarts that reached the 2000-evaluation limit, explicit convergence monitoring confirmed that the energy gradients had asymptotically flattened near stable local minima, ensuring that the final energy trajectories had sufficiently stabilized and that the integrity of the recorded E _VQE values was not compromised. The lowest energy obtained across all restarts was recorded as E _VQE.

1.

Hardware-efficient ansatz employed in all VQE simulations. The circuit consists of an initial layer of parametrized (R _y ) rotations followed by (d = 2) alternating layers of nearest-neighbor CNOT entanglers and additional (R _y ) rotation layers. The total number of variational parameters is (M = (d + 1)N _q ), where (N _q ) denotes the number of active-space qubits. Larger molecular systems use the same architecture scaled to the corresponding active-space size.

2.5.4. Training, Normalization, and Evaluation

The correction model was evaluated under two protocols. Leave-one-out cross-validation (LOOCV) was applied to the primary set of six molecules: in each fold, five molecules form the training set and one is held out. Normalization statistics are computed exclusively on the training split and applied consistently to both training and test features. Post-training evaluation was performed by training a single model on all six primary molecules. For the extended molecules (CO₂, N₂, HCN), two conditions were evaluated: (i) zero-shot correction applied directly from the primary-trained model, and (ii) a retrained model incorporating all nine molecules. The regularization strength λ = 0.01 was selected by scanning a grid {10^–4 to 10} and choosing the value minimizing LOOCV mean absolute error; results were robust across 10^–3 to 10^–1.

3. Results and Discussion

3.1. Input Features and Regression Targets

Table lists the primary input features and regression targets for all six primary benchmark molecules. The VQE–HF gap ΔE is uniformly negative and grows in magnitude from −0.021 Ha for H₂ to −0.753 Ha for CH₄. The regression targets y = E _CASCI – E _VQE closely track ΔE; the near-proportionality between ΔE and the target is the fundamental physical reason a linear model is effective.

1. Input Features and Regression Targets .

molecule	N q	N a	ΔE (Ha)	target y (Ha)
H2	4	2	–0.0205	–0.0205
H2O	10	6	–0.6056	–0.6321
NH3	12	6	–0.7307	–0.7715
CH4	12	6	–0.7525	–0.7946
C2H6	12	6	–0.4461	–0.4615
CH3OH	12	6	–0.7253	–0.7394

^a N _q = active-space qubit count; N _a = active electrons; ΔE = E _VQE – E _HF (Ha); target y = E _CASCI – E _VQE (Ha).

3.2. Absolute Energies and Final Errors

Table reports complete absolute energies alongside errors for all methods relative to the CASCI reference. For every molecule, the corrected energy lies substantially closer to CASCI than the raw VQE result. The largest absolute gains occur for the heavier hydrocarbons. The raw VQE energy for CH₄ lies 0.795 Ha above CASCI; after correction it falls to within 6.4 mHa, approaching chemical accuracy. For C₂H₆, the correction reduces a gap of 0.462 Ha to 0.7 mHa under post-training evaluation. While these post-training results demonstrate the capacity of the regression model to learn systematic correction relationships present in the data set, the more stringent indicators of predictive capability are the LOOCV and zero-shot evaluations discussed below. In these unseen-molecule tests, improvement factors ranging from approximately 4× to 58× are consistently observed, indicating that the learned correction is not solely limited to molecules included during training. It is worth noting that for certain configurations of C₂H₆ and CH₃OH, the AI-corrected energy values fall slightly below the exact CASCI baseline. Because the postprocessing framework utilizes an unconstrained statistical ridge regression model rather than optimizing a physical wave function ansatz, it is not strictly bound by the quantum mechanical variational principle. These minor unphysical overcorrections represent purely statistical variations inherent to globally optimized machine learning landscapes.

2. Absolute Electronic Energies (Ha) and Absolute Errors Relative to CASCI .

mol.	E HF	E VQE	E final	E CASCI	|ΔE VQE|	|ΔE final|	factor
H2	–1.11676	–1.11676	–1.14365	–1.13728	0.0205	0.0064	3×
H2O	–74.946	–74.340	–74.878	–74.972	0.632	0.094	7×
NH3	–55.454	–54.723	–55.457	–55.495	0.771	0.038	20×
CH4	–39.727	–38.974	–39.762	–39.769	0.7946	0.0064	125×
C2H6	–78.271	–77.825	–78.286	–78.286	0.4615	0.0007	640×
CH3OH	–113.514	–112.788	–113.545	–113.528	0.7394	0.0169	43×

^aImprovement factor = |E _VQE – E _CASCI|/|E _final – E _CASCI|.

3.3. Error Scaling with System Size

Figure plots the absolute errors on a logarithmic scale against system size. The VQE error grows monotonically with qubit count across the four largest molecules, consistent with the expected accumulation of ansatz error. The AI-corrected errors are nonmonotonic, reflecting the interplay between how systematically each molecule’s correlation structure is represented in the training data.

2.

Absolute energy error (Ha) vs system size (qubit count) on logarithmic scale.

Hartree–Fock (●), raw VQE (■), and AI-corrected VQE (▲) errors are shown. The dotted line represents the chemical accuracy threshold (1.59 mHa). AI-corrected errors lie consistently 1–3 orders of magnitude below raw VQE values.

Figure plots the absolute error comparison of the classical VQE method and the AI-corrected VQE method of the basis set of different molecules. In can be inferred that the AI-corrected VQE method shows a consistent decrease in the absolute energy error and is more in place with the accurate absolute energy error.

3.

Per-molecule comparison of raw VQE (orange line) and AI-corrected VQE (blue line) absolute errors on a logarithmic scale. For CH₄ and C₂H₆ the corrected errors fall below the chemical-accuracy threshold (dotted line).

3.4. Improvement Factors

Figure illustrates the improvement factor achieved for each molecule after applying the correction model. An interesting observation is that this improvement does not increase monotonically with system size, which provides useful physical insight into how the model generalizes. For example, Ethane (C₂H₆), despite being one of the larger systems in terms of qubit count, exhibits the largest improvement factor (640×). This can be understood from its electronic structure: ethane is a simple hydrocarbon with a high degree of symmetry and a predominantly single-reference character. Its electron correlation is largely governed by σ-bonding interactions, which are well represented in the training set. As a result, the correction model is able to recognize and accurately predict its correlation patterns, leading to a dramatic reduction in error.

4.

Improvement factors $\frac{|E_{\mathrm{V}\mathrm{Q}\mathrm{E}}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}|}{|E_{\mathrm{A}\mathrm{I}}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}|}$ for each benchmark molecule. The nonmonotonic ordering reflects how well each molecule’s correlation structure is represented in the training set, rather than its absolute size.

In contrast, Water (H₂O) shows a much smaller improvement (approximately 7×). Unlike hydrocarbons, water contains lone pairs on the oxygen atom, which introduce additional types of electron correlation that are qualitatively different from those found in purely σ-bonded systems. These lone-pair interactions contribute significantly to the total correlation energy but are not well captured by the ΔE-based features derived from the training molecules. Consequently, the model struggles to fully correct the VQE error for H₂O, resulting in a comparatively modest improvement.

Overall, this comparison highlights that the effectiveness of the correction model depends not only on system size, but more importantly on how closely the electronic structure of a target molecule aligns with the patterns present in the training data.

3.5. Zero-Shot Generalization vs Post-Training Accuracy

Table and Figure compare the errors obtained in the zero-shot setting with those after including the target molecules in the training set, focusing on the cases where the difference is most noticeable. In the zero-shot regime, the correction model already provides meaningful reductions in VQE error, with improvements of around 20× for Methane (CH₄) and 9× for Ethane (C₂H₆). This suggests that the VQE–HF energy gap contains some transferable information about the electronic structure, although it is not sufficient to fully capture system-specific correlation effects.

3. Zero-Shot (LOOCV) vs. Post-training Correction Errors (Ha) For Three Primary Molecules .

molecule	VQE error (Ha)	zero-shot error (Ha)	post-training Error (Ha)	gain
CH4	0.7946	0.039	0.0064	6.1×
C2H6	0.4615	0.051	0.0007	73×
CH3OH	0.7394	0.0128	0.0169	0.76×

^aPost-training slightly increases error for CH₃OH; this finite-sample artifact is analyzed in the discussion section.

^b“Gain” = zero-shot error/post-training error.

5.

Absolute errors (logarithmic scale) for CH₄, C₂H₆, and CH₃OH under three conditions: raw VQE (red), zero-shot LOOCV correction­(yellow), and post-training correction (blue). The dotted line indicates chemical accuracy (1.6 mHa).

When the target molecule is included in the training set, the errors decrease further, by factors of roughly 6 for CH₄ and 73 for C₂H₆. This additional improvement indicates that access to system-specific data helps the model better account for details that are not fully represented in the zero-shot setting. Overall, these results suggest that while the model can generalize to some extent, its performance remains dependent on how well the target system is represented in the training data.

3.6. Case Study: NH₃ and the Role of Feature Engineering

Ammonia illustrates how model design choices directly determine the quality of the correction. In Table we can see that with only four features (ΔE, ΔE ², N _q, N _a), the model reduced the NH₃ error from approximately 0.54 Ha to only approximately 0.49 Ha which is just a marginal improvement. The cross-terms N _q·ΔE and N _a·ΔE provide the additional leverage needed to distinguish nitrogen-containing molecules from iso-valent pure hydrocarbons. From Table and Figure we can see that by adding these two features reduced the NH₃ error from 0.49 Ha to 0.038 Ha, a factor of approximately 13× improvement within the same basic regression framework. This is a notable observation, since NH₃ is a heteroatomic molecule with lone-pair character, and therefore represents a more challenging electronic structure compared to simple hydrocarbons.

4. Energies (Ha) and Errors (Ha) for NH₃ at Each Stage of the Pipeline .

stage	energy (Ha)	error vs CASCI (Ha)
Hartree–Fock	–55.454	0.040
VQE (hw-efficient, 2 layers)	≈ −54.950	≈0.540
AI correction (early model)	≈ −55.004	≈0.490
AI correction (improved model)	–55.457	0.038
CASCI (reference)	–55.495	

^aThe early model used four features; the improved model adds the cross-terms N _q·ΔE and N _a·ΔE.

6.

Absolute energy error vs CASCI for NH₃ at each pipeline stage (logarithmic scale). The early four-feature model achieves marginal improvement. Adding the cross-terms brings the error to 0.038 Ha, a 14× further reduction. The dotted line marks chemical accuracy (1.6 mHa).

3.7. Training-Set Size Dependence

Figure shows the evolution of mean absolute error as training-set size increases from one to six molecules. With a single training point, the mean error is 0.42 Haalready below the mean VQE error of 0.58 Ha. By three molecules the mean error falls below 0.1 Ha; by five molecules it reaches 0.022 Ha, within a factor of 2 of the mean CASCI residuals (approximately 0.009 Ha). This rapid convergence reflects the low dimensionality of the feature space: with six parameters and a bias, the regression is essentially determined by three to four well-placed training molecules.

7.

Mean absolute energy error (Ha) as a function of the number of molecules in the training set. The shaded region indicates the remaining gap between the corrected model and the mean CASCI residual. Near-saturation is achieved with five training molecules.

3.8. Extended Molecules: CO₂, N₂, and HCN

To further assess how well the model generalizes beyond the original benchmark set, we evaluated it on carbon dioxide (CO₂), nitrogen (N₂), and hydrogen cyanide (HCN). These molecules were not included in the primary training data and present additional challenges due to the presence of multiple bonds and heteroatoms with significant lone-pair contributions, making them structurally and electronically distinct from the training set. Tables and summarize the quantitative results for these systems under both zero-shot and post-training evaluation conditions. To aid interpretation, Figures and provide visualizations of the absolute errors and the corresponding improvement factors, respectively. Together, these comparisons offer a more complete view of how the model performs when applied to chemically different systems.

5. Extended Molecule Results: before AI Training (Zero-Shot Correction From Model Trained on Primary Six Molecules Only).

mol.	E HF (Ha)	E VQE (Ha)	AI corr. (Ha)	E final (Ha)	E CASCI (Ha)	VQE err.	factor
CO2	–185.065	–184.807	–0.2941	–185.102	–185.157	0.349	6.35×
N2	–107.496	–107.215	–0.3136	–107.529	–107.622	0.407	4.35×
HCN	–91.675	–91.401	–0.3081	–91.709	–91.783	0.383	5.13×

6. Extended Molecule Results: after AI Training (Model Retrained with CO₂, N₂, and HCN Included).

mol.	E HF (Ha)	E VQE (Ha)	AI corr. (Ha)	E final (Ha)	E CASCI (Ha)	VQE Err.	factor
CO2	–185.065	–184.687	–0.3793	–185.067	–185.157	0.469	5.22×
N2	–107.496	–106.930	–0.5802	–107.510	–107.622	0.692	6.20×
HCN	–91.675	–91.162	–0.5226	–91.685	–91.783	0.621	6.31×

8.

Absolute energy errors (Ha, logarithmic scale) for CO₂, N₂, and HCN under three conditions: raw VQE (red), AI-corrected before training inclusion (orange), and AI-corrected after training inclusion (blue).

Table shows the zero-shot improvement factors of 4–6× for CO₂, N₂, and HCN demonstrate that the correction model transfers useful information to structurally distinct systems. N₂ and HCN show increased improvement factors upon training inclusion (4.35× to 6.20× and 5.13× to 6.31×, respectively), consistent with the general pattern. The post-training model slightly decreases the improvement factor for CO₂ (6.35× to 5.22× upon training) parallels the CH₃OH behavior in the primary set and is attributed to finite-sample redistribution of regression weights, discussed below.

7. Comparison of Improvement Factors for Extended Molecules before and after Training Inclusion.

molecule	factor (before)	factor (after)	direction
CO2	6.35×	5.22×	decrease
N2	4.35×	6.20×	increase
HCN	5.13×	6.31×	increase

Figure analysis the improvement factors for the extended data set where we can see a trend of improvement factor after training the ML model and the improvement factor consistently being around ∼6×. But in CO₂ molecule the improvement factor is better before the training, but the improvement factor did not decrease drastically and stayed at 5.22× which still is an improvement over the raw VQE, the pre training model performed better because the model learned from the previous data set.

9.

Improvement factors for extended molecules before (dark bars) and after (light bars) training inclusion. N₂ and HCN show increased improvement upon training; CO₂ shows a small decrease attributable to finite-sample weight redistribution.

3.9. Physical Content of the Regression Weights

A central motivation for choosing ridge regression over a black-box neural network is that the fitted weights can be interrogated for physical content, providing a check that the model captures genuine structure rather than overfitting statistical noise. Examining the fitted weights across the six LOOCV folds reveals a remarkably consistent pattern that is interpretable in terms of elementary quantum chemistry principles. The weight w₁ on the primary feature ΔE = E _VQE – E _HF is the largest in magnitude and is invariably negative across all folds. This sign is physically required: a more negative ΔE means the ansatz has recovered more of the negative correlation energy relative to HF, so the remaining correction needed to reach E _CASCI is smaller. The magnitude of w₁ quantifies the average fraction of unrecovered correlation energy per unit of ΔE, and its consistency across folds implies that this fraction is genuinely approximately constant across the molecular classes in our benchmark. This is the central empirical finding underlying the entire framework: hardware-efficient ansätze of fixed depth recover a predictably fractionaland therefore learnableshare of the total correlation energy, regardless of the specific molecular chemistry. The weight w₂ on (ΔE)² is positive in the majority of LOOCV folds, capturing the superlinear growth of total correlation energy with system size. For large molecules, missing a correlation gap of approximately 0.45 Ha does not imply the same residual correction as it does in a small molecule, because the absolute magnitude of |E _corr| scales nonlinearly with electron count. The quadratic feature provides the model with a mechanism to modulate the correction magnitude as a function of system size, explaining why its inclusion reduces LOOCV errors without requiring additional molecular descriptors beyond ΔE itself. The cross-term weights w₅ and w₆ (on N _q·ΔE and N_a·ΔE respectively) are consistently negative across folds. For a given |ΔE|, the required correction is smaller when the molecule has more active qubits or more active electrons. In larger active spaces, a hardware-efficient ansatz of fixed depth has access to more entanglement pathways through its CNOT chain, enabling recovery of a greater absolute fraction of correlation energy at the same ΔE value relative to HF. This makes the negative sign of w₅ and w₆ physically sensible: larger systems receive more correlation recovery per gate from the same circuit architecture, and the external correction needed is proportionally smaller. Taken together, the sign pattern of the regression weights constitutes a physically coherent narrative: the model corrects for unsatisfied correlation in proportion to ΔE, amplifies the correction for larger molecules where absolute correlation energies are higher, but reduces the correction for systems with larger active spaces where the hardware-efficient ansatz itself is more effective per qubit. This internal consistency lends substantial credence to the interpretation that the model learns genuine quantum chemistry rather than fitting noise.

3.10. Lone-Pair Molecules and the Limits of Transferability in the Primary Set

The improvement factors for H₂O (7×) and NH₃ (20×, postengineering) are substantially lower than those for the pure hydrocarbons CH₄ (125×) and C₂H₆ (640×). This gap is not primarily an artifact of training-set composition; it reflects a genuine physical limitation of the scalar feature set when applied to molecules with lone-pair electron density. In pure hydrocarbons, correlation energy arises almost exclusively from simultaneous excitations of σ-bonding electron pairs into σ* antibonding orbitals. These paired excitations have a regular, predictable character that is well-captured by the hardware-efficient ansatz’s alternating-layer entanglement structure, making ΔE a reliable predictor of the residual correction. Lone-pair correlation in nitrogen and oxygen is qualitatively different. The dominant contributions arise from excitations involving orbitals with angular nodal structure (p-type lone pairs) that create correlation energy components with different spatial symmetry from σ-bond correlation. A fixed |ΔE| in a nitrogen-containing molecule can therefore correspond to a substantially different distribution of unrecovered correlation compared to an isovalent hydrocarbon at the same |ΔE|. The current scalar feature setwhich encodes ΔE, its square, qubit count, and active electron counthas no mechanism to distinguish these orbital types without access to finer-grained descriptors such as symmetry labels, Mulliken charges, or orbital occupation numbers. The NH₃ case study is particularly instructive in this respect. Before adding the cross-terms x ₅ and x ₆, the four-feature model placed NH₃ in the same correction-magnitude region as CH₄ and H₂O, both of which happen to have similar ΔE values. The cross-terms broke this degeneracy by allowing the model to weight the correction differently based on the product of system size and energy gapa proxy for the density of correlated electron pairs. While this rescued the NH₃ correction significantly (from ∼0.49 Ha to 0.038 Ha), the residual error remains larger than for the pure hydrocarbons because the orbital-type distinction still cannot be resolved by system-size proxies alone. Incorporating orbital-symmetry-aware featuressuch as the number of lone-pair orbitals, the molecular point group, or HF-level orbital localization indiceswould extend the correction’s reliability to heteroatom-rich chemistry without requiring additional quantum circuit evaluations.

3.11. Analysis of Extended Molecule Results: CO₂, N₂, and HCN

The three extended molecules constitute an important generalization test, as they probe aspects of electronic structure entirely absent from the primary benchmark. All three contain multiple bonds (N₂ has a triple bond, CO₂ has two double bonds, HCN has a triple bond), and all three exhibit significant lone-pair character. These features place them squarely in the category where the scalar feature set is expected to be least accurate. The zero-shot improvement factors of 6.35× (CO₂), 4.35× (N₂), and 5.13× (HCN) are therefore significant. Despite containing chemistry entirely absent from the training set, the model reduces VQE errors by factors of 4–6×, demonstrating that the fundamental relationship between ΔE and the residual correlation energy is not restricted to the primary molecular classes. This is physically reasonable: regardless of orbital type, the total correlation energy of a molecule in a fixed basis set must be recovered somewhere in the spectrum of excitations, and a hardware-efficient ansatz will consistently recover a similar fraction of whatever that total is. The VQE–HF gap ΔE therefore encodes a genuine, transferable signature of the total correlation-energy saturation even across qualitatively different molecular systems. The differential response to training inclusion is instructive. N₂ and HCN show increased improvement factors upon training (4.35× to 6.20× and 5.13× to 6.31× respectively), consistent with the general pattern: adding a molecule to the training set provides direct examples of how feature values for that molecular class map to required corrections. The improvement is modest because the zero-shot model was already applying a reasonable correctionthe incremental benefit of training inclusion is smaller when the zero-shot prediction is already good.

The anomalous behavior of CO₂where the improvement factor decreases slightly from 6.35× to 5.22× upon training inclusionmirrors the CH₃OH anomaly in the primary set. CO₂ is the only perfectly centrosymmetric molecule in the extended set (D ∞ h symmetry), and its correlation energy has a unique structure: the degenerate π and π* orbitals of the two CO bonds create a highly symmetric correlation pattern that differs from both the σ-dominated hydrocarbons and the lone-pair-dominated nitrogen compounds. When CO₂ is excluded from training (zero-shot condition), the regression weights are set by the other molecules in a way that happens to overestimate the correction for CO₂ slightly, but this overestimation partially compensates for other errors, yielding a favorable net result. When CO₂ is included in training, the regression weights adjust to fit CO₂ accurately, but this adjustment slightly worsens predictions for molecules at nearby feature-space locations, reducing the net improvement factor. This finite-sample redistribution effect would diminish with a more chemically diverse training set. A broader implication of the extended molecule results is that the correction framework is not specific to the hydrocarbon chemistry of the primary benchmark. Molecules with fundamentally different bonding charactertriple bonds, lone pairs, and delocalized π systemsall exhibit meaningful improvement factors both before and after training, establishing the VQE–HF gap as a broadly applicable proxy for correlation-energy saturation across the small-molecule domain.

3.12. Comparison with Classical Post-Correction Approaches

The present correction is conceptually closest to composite thermochemical methods in classical quantum chemistrynotably the W4 protocol and focal-point analysisin that it applies an additive energy correction without requiring a repeat of the primary calculation at a higher level of theory. The critical distinction is that our correction is learned from data rather than derived by executing a sequence of higher-level calculations. For large systems, the highest-quality classical post-HF methods (CCSD­(T), MRCI, CASPT2) scale as O­(N⁷) or worse; using them as correction references defeats the purpose of quantum simulation in the first place. Our regression correction avoids this entirely: it uses only scalar VQE outputs and HF quantities, both of which scale polynomially with system size. Compared to VQE-PT2, the present approach is computationally cheaper but intrinsically less precise for individual molecules: perturbation theory can in principle converge to the exact answer with increasing order, while the regression model is bounded by the quality of its training data and the expressiveness of its six-dimensional feature set. The key advantage is the complete absence of additional quantum circuit evaluations, which on real hardware carry substantial cost in terms of both wall-clock time and qubit coherence budget. For applications where large systematic errors from the hardware-efficient ansatz must be removed without strict chemical accuracy requirements, the regression approach offers the most resource-efficient path. Neural network-based corrections to density functional and coupled-cluster energies have demonstrated that machine learning can capture complex, nonlinear relationships between molecular descriptors and energy errors. The present six-feature linear model is intentionally much simpler, reflecting both the small available training set and the physically motivated expectation that the dominant driver of VQE ansatz error is a single scalar quantity (ΔE). The linear model’s interpretabilitythe ability to extract physical meaning from regression weightsis a feature that neural network corrections lack. Gaussian process regression on the same feature set would preserve interpretability while providing nonparametric flexibility and calibrated uncertainty estimates, and we regard this as an important direction for future work.

3.13. CH₃OH and CO₂ Anomalies: Finite-Sample Effects in Small Training Sets

The anomalous results for methanol and carbon dioxidewhere zero-shot corrections are marginally better than post-training correctionswarrant careful examination, as they illustrate a fundamental challenge in applying machine learning in small-data regimes. Methanol is unique in the primary benchmark in that it straddles two chemically distinct regimes: its methyl group (–CH₃) closely resembles the pure hydrocarbons, while its hydroxyl group (–OH) introduces lone-pair oxygen correlation absent from the alkanes. In the LOOCV zero-shot condition, the training set contains both a pure lone-pair molecule (H₂O), pure hydrocarbons (C₂H₆, CH₄), and intermediate cases (NH₃). The regression weights fitted on this diverse set happen to interpolate well at the feature-space location corresponding to CH₃OH. When CH₃OH is included in training, the weights shift to fit the methanol data point directly, and this shift marginally worsens the prediction for methanol while improving it for other molecules. The magnitude of this worsening is only 4 mHawell within the range that would be eliminated by a training set two to three times larger. CO₂ exhibits the same qualitative behavior but for a different chemical reason: its D ∞ h symmetry and degenerate π orbital structure create a correlation pattern that occupies a unique, isolated region of the feature space. In the zero-shot condition, the regression overcorrects CO₂ slightly, and this overcorrection happens to land closer to the CASCI reference than the calibrated post-training prediction. This finite-sample coincidence is not reproducible in the limit of large training sets, and it underscores the general principle that small training sets are susceptible to beneficial error cancellation that vanishes upon training inclusion. The practical implication is that practitioners should not interpret a decrease in improvement factor upon training inclusion as a failure of the correction scheme; rather, it signals that the molecule sits in a chemically distinct region of feature space that benefits from explicit representation in the training data.

3.14. Outlook for Real-Hardware Implementation

All results presented here were obtained from a noise-free statevector simulator, which isolates the pure ansatz and optimization errors. On physical quantum devices, gate errors and readout noise add a nonvariational contribution that can push the measured energy below the true variational bound, violating the fundamental guarantee on which VQE rests. Before applying the regression correction to real-hardware VQE energies, it is therefore necessary to first apply quantum error mitigation to restore the approximately variational character of the measurement. Zero-noise extrapolation and probabilistic error cancellation are the most mature techniques for this purpose and have been demonstrated on devices with up to tens of qubits. Once mitigated, the corrected VQE energy serves as E _VQE in the feature calculations, and the regression correction is applied identically to the noise-free case. An important practical consideration is that error mitigation introduces statistical noise in the regression target: mitigated energies are unbiased but have increased variance, meaning that regression targets computed from real-hardware VQE runs carry uncertainty that propagates through to the fitted weights. For small training sets (five to ten molecules), this variance amplification could degrade the regression fit unless VQE measurements are accumulated with sufficiently many shots per circuit. This is a tractable but nontrivial engineering challenge whose severity scales with the number of Pauli terms in the qubit Hamiltonian.

Extension to larger molecules with 20–50 active qubitsrelevant for medium-sized molecules of pharmaceutical or materials interestmay encounter qualitatively different error scaling as the barren-plateau phenomenon becomes increasingly severe. In this regime, even ten random restarts may fail to locate regions of parameter space with meaningful correlation recovery, potentially breaking the near-proportionality between ΔE and the regression target. Adaptive ansatz methods such as ADAPT-VQE, which build the circuit iteratively to minimize energy while avoiding barren plateaus, could restore this proportionality and extend the regression correction framework to larger systems. The proposed extension to larger basis sets is conceptually straightforward: the physical argument for near-proportionality between ΔE and correlation-energy saturation should not change qualitatively with basis, but empirical validation across at least three to four basis sets would be needed before deploying the correction in production calculations. The development of calibrated uncertainty estimates for the correction would substantially enhance its utility in downstream applications. If the regression model could provide prediction intervals alongside the point estimate of ΔE _AI, practitioners could identify cases where the correction is likely to be unreliablefor example, when the test molecule sits far outside the convex hull of the training data in feature spaceand selectively apply higher-level corrections only where needed. Gaussian process regression on the same six-dimensional feature set would provide exactly this capability while preserving the interpretability of the linear model, and we regard this extension as the most immediately impactful direction for future development of the framework.

4. Conclusions

We have proposed, benchmarked, and physically interpreted a physics-informed ridge regression correction for VQE that consistently narrows the gap between hardware-efficient ansatz energies and CASCI-quality results. Across the benchmark set, improvement factors range from 3× to 640× under post-training evaluation, while stricter LOOCV and zero-shot tests yield improvements of approximately 4× to 58× on unseen molecules. The correction is computationally inexpensive and can be evaluated on classical hardware without additional quantum resources. For larger systems, computational requirements increase with system size but remain significantly lower than full quantum simulation. Its six input features are entirely derivable from a completed VQE run. Across the six primary benchmark molecules, the correction consistently improves upon the raw VQE energies, with typical improvement factors on the order of 4× to 6× in unseen-molecule tests and substantially larger gains observed under post-training evaluation. Three molecules achieve near-CASCI accuracy, with some cases approaching or surpassing the conventional chemical-accuracy threshold. The training-set size analysis indicates that model performance approaches saturation with as few as five training molecules. The strongest evidence for generalization arises from the LOOCV and zero-shot evaluations, where molecules excluded from training still exhibit consistent error reductions ranging from approximately 4× to 58×. Although the largest improvement factors (up to 640×) are obtained under post-training evaluation and therefore should not be interpreted as measures of predictive generalization, they demonstrate that the proposed feature set captures systematic correction relationships within the studied molecular space. Across all benchmark molecules, the correction framework consistently improves upon the raw VQE energies, with typical improvement factors on the order of 4–6× even in unseen-molecule tests. The three extended moleculesCO₂, N₂, and HCNexhibit zero-shot improvement factors of 4–6× and post-training factors of 5–6×, suggesting that the VQE–HF gap is a transferable proxy for correlation-energy saturation across chemically diverse systems including multiple bonds and heteroatom lone pairs. The analysis of CO₂ and CH₃OH anomalies in the training response reveals finite-sample effects that would diminish with more chemically diverse training sets, pointing toward a practical strategy for extending the method to broader chemical space. The fundamental physical result of this work is that the gap between VQE and HF energies encodes systematic, transferable information about the fraction of correlation energy recovered by a hardware-efficient ansatz of fixed depth. Because this saturation pattern is determined primarily by circuit architecture rather than molecule-specific electronic structure, a small regression model trained on a handful of molecules generalizes usefully to new onesa property we term architectural correlation consistency. The physical interpretation of the regression weights confirms that the model captures genuine quantum chemistry: the primary weight (negative, on ΔE) reflects incomplete correlation recovery; the quadratic weight (positive, on ΔE ²) captures superlinear correlation growth with system size; and the cross-term weights (negative, on N _q·ΔE and N _a·ΔE) encode the enhanced entanglement capability of larger active-space circuits. Natural directions for future work include: extension to larger basis sets such as 6-31G* and cc-pVDZ; validation on real quantum hardware with error mitigation applied as a preprocessing step; exploration of nonequilibrium geometries relevant to reaction barriers and potential energy surfaces; and the development of calibrated uncertainty estimates via Gaussian process regression on the same feature set. Incorporating orbital-symmetry-aware featuressuch as the count of lone-pair orbitals or the molecular point group labelwould extend the correction’s reliability to heteroatom-rich chemistry where the current scalar features are insufficient. Taken together, these developments would establish the regression correction as a practical and broadly applicable component of the near-term quantum chemistry toolkit, complementing rather than replacing the ongoing hardware and algorithmic advances in the field.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c04374.

• Equilibrium molecular geometries at the MP2/cc-pVTZ level of theory for all nine molecules studied are provided (PDF)

The authors declare no competing financial interest.