Quantum-Inspired SMOTE: Innovative Oversampling

• Quantum-Inspired SMOTE (QI-SMOTE) is a family of oversampling methods that leverages quantum-inspired concepts like fidelity metrics, swap tests, and entanglement to generate synthetic minority samples.
• It improves on classical SMOTE by replacing local Euclidean interpolation with techniques such as centroid-based movement, angular distance measures, and energy-based sampling to better capture global data structure.
• Its design effectively blends quantum-inspired geometry with practical classical implementations, yielding enhanced performance across diverse applications from telecom churn to medical diagnostics.

Quantum-Inspired SMOTE (QI-SMOTE) denotes a class of oversampling methods for imbalanced learning that borrow concepts from quantum information, quantum circuits, or quantum-inspired geometry to generate synthetic minority samples. Across the recent literature, the term covers several distinct mechanisms: classical feature-space oversamplers guided by fidelity-like similarity and margins, simulated quantum-circuit methods based on swap tests and rotations, entanglement- and VQE-based feature augmentation followed by SMOTE, and adjacent energy-based generative schemes such as Quantum Restricted Boltzmann Machines (QRBMs) that fill the same conceptual role as a principled minority-class oversampler (Behera et al., 18 Dec 2025, Patel et al., 16 Jan 2026, Kashtriya et al., 2 Sep 2025, Mohanty et al., 2024, Mohanty et al., 31 Jan 2025, Sinno et al., 5 Feb 2025).

1. Terminology and conceptual scope

The recent literature uses the label “Quantum SMOTE” or “Quantum-Inspired SMOTE” in more than one sense. In one line of work, QSMOTE is explicitly “a family of oversampling algorithms defined and run entirely in the classical feature space,” with the quantum inspiration residing in fidelity-like similarity, Hilbert-space geometry, and probability-based margin filtering (Behera et al., 18 Dec 2025). In another, QSMOTE is also a classical oversampling algorithm, but its similarity measure is obtained through a simulated compact SWAP test, so the method remains classical in execution while borrowing directly from quantum-circuit primitives (Patel et al., 16 Jan 2026). A third formulation defines QI-SMOTE as a classical algorithm that “is not a quantum algorithm running on quantum hardware,” but that uses superposition, layered entanglement, and VQE-style optimization before returning to classical SMOTE in an augmented feature space (Kashtriya et al., 2 Sep 2025).

A more circuit-centric branch, introduced as Quantum-SMOTE, amplitude-encodes minority samples and centroids, estimates angular distance with a compact swap test, and generates new data through quantum rotations rather than neighbor interpolation (Mohanty et al., 2024). Quantum-SMOTEV2 removes $K$-means clustering, replaces multiple centroids with a single global centroid, and introduces Angular Outliers (AOL) as a criterion for additional synthetic generation in sparse angular regions (Mohanty et al., 31 Jan 2025). In parallel, QRBM-based balancing on D-Wave Pegasus is not named QI-SMOTE, but is described as occupying “exactly in the role that ‘Quantum-Inspired SMOTE (QI-SMOTE)’ aims to fill”: a global generative mechanism for minority oversampling driven by energy-based modeling and quantum sampling rather than local interpolation (Sinno et al., 5 Feb 2025).

This suggests that QI-SMOTE is best understood as a methodological family rather than a single canonical algorithm. The family is unified by one objective—synthetic minority generation for class-imbalance mitigation—but differs in whether the “quantum” ingredient appears as a similarity measure, a state-encoding formalism, a circuit-simulated transformation, a measurement-based classifier interface, or a generative energy model.

2. Relation to classical SMOTE

Classical SMOTE generates synthetic minority instances by interpolation between minority neighbors. In its standard form,

$\tilde{x} = x_i + \lambda (x_j - x_i),$

with $x_i$ a minority sample, $x_j$ a selected minority neighbor, and $\lambda \sim U(0,1)$ or an analogous interpolation factor (Behera et al., 18 Dec 2025, Mohanty et al., 2024). The QI-SMOTE literature retains the same data-level purpose as SMOTE but replaces or supplements this local Euclidean interpolation with alternative geometry.

Several recurring criticisms of classical SMOTE motivate these variants. In medical tabular data, linear interpolation may create “physiologically impossible combinations,” may increase class overlap, and may ignore global structure and high-dimensional feature dependencies (Kashtriya et al., 2 Sep 2025). In industrial fault classification, centroid-directed interpolation can also distort boundaries for linear or probabilistic models when classes overlap strongly, a phenomenon described as “interpolation-induced boundary distortion” (Patel et al., 16 Jan 2026). In telecom churn, the original Quantum-SMOTE paper explicitly frames its method as avoiding dependence on $k$-nearest neighbors and Euclidean distance, and replacing them with cluster centroids, angular relations, and quantum rotation (Mohanty et al., 2024).

QI-SMOTE variants therefore differ from classical SMOTE along three main axes. First, they often replace neighbor-based local interpolation with centroid-based or manifold-aware movement, either toward cluster centroids, a global centroid, or regions of low energy (Behera et al., 18 Dec 2025, Mohanty et al., 31 Jan 2025, Sinno et al., 5 Feb 2025). Second, they replace Euclidean similarity with cosine-squared “fidelity,” swap-test-derived angular distance, or energy-based sampling (Behera et al., 18 Dec 2025, Patel et al., 16 Jan 2026, Mohanty et al., 2024). Third, some formulations add classifier-aware filtering or optimization layers before final sample generation, such as margin-based retention of high-confidence synthetic points or VQE-based evolution toward low-energy representations (Behera et al., 18 Dec 2025, Kashtriya et al., 2 Sep 2025).

The contrast is especially sharp in the QRBM formulation. There, SMOTE is characterized as a local geometric oversampler, whereas the QRBM learns an energy function over binary feature vectors and samples from the global Boltzmann distribution, with quantum annealing approximating low-energy configurations (Sinno et al., 5 Feb 2025). A plausible implication is that QI-SMOTE, in its broader sense, marks a transition from interpolation-centric oversampling to geometry-aware or generative oversampling.

3. Principal algorithmic families

The present literature contains several distinct QI-SMOTE families.

Family	Characteristic mechanism	Representative works
Fidelity-/margin-based QSMOTE	Centroid guidance, cosine-squared fidelity, margin filtering	(Behera et al., 18 Dec 2025)
Swap-test angular QSMOTE	Amplitude encoding, compact SWAP test, angle-controlled synthesis	(Patel et al., 16 Jan 2026, Mohanty et al., 2024, Mohanty et al., 31 Jan 2025)
Entanglement/VQE-assisted QI-SMOTE	Hadamard + $RY$, layered entanglement, Ising-like Hamiltonian, VQE, then SMOTE	(Kashtriya et al., 2 Sep 2025)
Energy-based quantum oversampling	QRBM learning and quantum-assisted sampling for minority generation	(Sinno et al., 5 Feb 2025)

In fidelity-based QSMOTE, the minority sample is moved toward a cluster centroid with a step scaled by

$$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$

which is exactly the squared cosine similarity and, under normalized pure-state interpretation, the pure-state fidelity (Behera et al., 18 Dec 2025). The same paper also defines a margin-based variant in which synthetic samples produced by a base oversampler are retained only if

$\left|P(y=1\mid x)-0.5\right| > \text{margin},$

with logistic regression supplying the minority-class probability and margin set to $0.1$ in the reported experiments (Behera et al., 18 Dec 2025). These variants are fully classical in feature space, but they are explicitly aligned with later PGM and kPGM classification in Hilbert-space geometry.

The swap-test family represents the most literal quantum interpretation of oversampling. In the industrial-condition-monitoring formulation, the full dataset is clustered with $\tilde{x} = x_i + \lambda (x_j - x_i),$0-means, each minority sample is paired with its assigned centroid $\tilde{x} = x_i + \lambda (x_j - x_i),$1, both are amplitude-encoded as normalized state vectors, and a compact SWAP test estimates an overlap that defines the angle

$\tilde{x} = x_i + \lambda (x_j - x_i),$2

A synthetic sample is then constructed as

$\tilde{x} = x_i + \lambda (x_j - x_i),$3

where $\tilde{x} = x_i + \lambda (x_j - x_i),$4 unless $\tilde{x} = x_i + \lambda (x_j - x_i),$5 (Patel et al., 16 Jan 2026). The original Quantum-SMOTE paper follows a similar angular logic but uses compact swap-test states and applies $\tilde{x} = x_i + \lambda (x_j - x_i),$6, $\tilde{x} = x_i + \lambda (x_j - x_i),$7, or $\tilde{x} = x_i + \lambda (x_j - x_i),$8 rotations to the encoded minority point, with X rotations used in the main Telecom Churn experiment (Mohanty et al., 2024). Quantum-SMOTEV2 simplifies this design by replacing cluster centroids with a single global centroid and by adding AOL-based boosting in low-density angular regions (Mohanty et al., 31 Jan 2025).

The entanglement/VQE-based formulation is structurally different. QI-SMOTE there first maps each minority sample to a quantum-inspired circuit with one qubit per feature, applies a Hadamard gate to each qubit, encodes normalized features via $\tilde{x} = x_i + \lambda (x_j - x_i),$9, applies layered entanglement using CNOT, CZ, and Toffoli gates, and then evolves the resulting state under an Ising-like Hamiltonian optimized by VQE (Kashtriya et al., 2 Sep 2025). The real part of the evolved amplitudes is appended to the dataset, and standard SMOTE is then applied in this “quantum-enhanced” feature space (Kashtriya et al., 2 Sep 2025). The oversampling step remains classical, but the representation on which SMOTE operates has already been transformed by quantum-inspired structure.

Finally, the QRBM line moves beyond interpolation altogether. The data are binarized, a $x_i$0 QRBM is trained on minority attack samples only, and synthetic attack records are obtained from quantum-assisted sampling on D-Wave Pegasus hardware (Sinno et al., 5 Feb 2025). Because the method learns $x_i$1 globally and samples new binary feature configurations rather than interpolating between examples, it functions as an energy-based minority generator rather than a SMOTE variant in the narrow sense. Nevertheless, the paper explicitly positions this mechanism as filling the role QI-SMOTE aims to fill (Sinno et al., 5 Feb 2025).

4. Mathematical structure

The mathematical formulations of QI-SMOTE mirror its algorithmic diversity. The simplest member is the KNN-based QSMOTE variant, which is mathematically identical to classical SMOTE: $x_i$2 Its main role in the literature is comparative, not foundational (Behera et al., 18 Dec 2025).

Fidelity-based QSMOTE replaces neighbor interpolation with centroid-directed movement weighted by a fidelity-inspired factor. For a minority sample $x_i$3 and its centroid $x_i$4, with direction vector $x_i$5,

$x_i$6

If $x_i$7 and $x_i$8 are already similar, the step is large; if they are dissimilar, the step is small (Behera et al., 18 Dec 2025). The margin-based variant adds a probabilistic filter,

$x_i$9

to reject ambiguous synthetic points near the decision surface (Behera et al., 18 Dec 2025).

The swap-test formulations treat classical vectors as amplitude-encoded states. In the industrial QSMOTE benchmark, each non-zero vector $x_j$0 becomes

$x_j$1

For minority sample $x_j$2 and centroid $x_j$3, the overlap

$x_j$4

is estimated via compact SWAP test, and the angle $x_j$5 determines the sampling range (Patel et al., 16 Jan 2026). In the original Quantum-SMOTE formulation, the swap test yields a “swap test probability,” from which the angular distance is defined as

$x_j$6

and this angle is then converted into a rotation angle modulated by the splitting factor and an angle increment (Mohanty et al., 2024). Quantum-SMOTEV2 preserves this rotation-angle logic while using angular distributions around a single centroid and IQR-based angular outlier detection (Mohanty et al., 31 Jan 2025).

The VQE-assisted QI-SMOTE formulation introduces a more explicitly quantum mechanical encoding. Feature normalization is defined by

$x_j$7

and the corresponding single-qubit encoding gate is

$x_j$8

After layered entanglement, the state is evolved under an Ising-like Hamiltonian

$x_j$9

with VQE objective

$\lambda \sim U(0,1)$0

The evolved state $\lambda \sim U(0,1)$1 is converted to a real vector by

$\lambda \sim U(0,1)$2

and standard SMOTE is then run on the augmented data (Kashtriya et al., 2 Sep 2025).

The QRBM line uses an energy-based probabilistic model rather than interpolation. Its RBM energy is

$\lambda \sim U(0,1)$3

which induces a Boltzmann distribution

$\lambda \sim U(0,1)$4

For quantum annealing, the model is mapped to an Ising Hamiltonian

$\lambda \sim U(0,1)$5

and quantum sampling replaces classical Gibbs sampling when estimating model statistics during training (Sinno et al., 5 Feb 2025). Although this is not SMOTE in the literal sense, it provides the most explicit example of global minority-distribution modeling in the QI-SMOTE design space.

5. Empirical performance and application domains

Empirical evaluations span telecom churn, clinical mortality prediction, industrial condition monitoring, intrusion detection, and fraud detection. The results are heterogeneous, but several patterns recur: the strongest gains usually appear for non-linear or ensemble classifiers; purely local KNN-style QSMOTE is often weak; and geometry-aware or structure-aware variants tend to dominate basic interpolation.

In the Telco Customer Churn study coupling QSMOTE variants with quantum-inspired classifiers, Random Forest with Fidelity-based QSMOTE achieved Accuracy $\lambda \sim U(0,1)$6, Recall $\lambda \sim U(0,1)$7, and F1 $\lambda \sim U(0,1)$8, the best Random Forest F1 among the reported QSMOTE variants (Behera et al., 18 Dec 2025). More strikingly, PGM with stereographic encoding and $\lambda \sim U(0,1)$9 under standard QSMOTE achieved Accuracy $k$0 and F1 $k$1, the best overall result reported in that work, while kPGM with stereo encoding and two copies reached Accuracy $k$2 and F1 $k$3 (Behera et al., 18 Dec 2025). KNN-based QSMOTE was consistently poor across RF, PGM, and kPGM, with markedly lower recall and F1 than baseline QSMOTE, Fidelity-QSMOTE, or Margin-QSMOTE (Behera et al., 18 Dec 2025).

In industrial and energy-system condition monitoring, QSMOTE was benchmarked on SPID, CWRUBD, EFDD, and IFDD under multi-class settings (Patel et al., 16 Jan 2026). The reported aggregate conclusion was that QSMOTE “consistently corrects distributional skew and significantly enhances the performance of non-linear classifiers such as Random Forests (RF), Support Vector Machines (SVM), and Decision Trees (DT), yielding improvements of up to 170% on EFDD and achieving near-perfect accuracy ($k$4) on IFDD” (Patel et al., 16 Jan 2026). The same study also showed that Logistic Regression and Naive Bayes could degrade substantially on overlapping feature spaces, with EFDD and IFDD providing explicit examples of deterioration after oversampling (Patel et al., 16 Jan 2026).

In medical mortality prediction on MIMIC-III and MIMIC-IV, QI-SMOTE was evaluated against SMOTE, ADASYN, Borderline-SMOTE, ROS, RUS, SMOTE-ENN, SMOTE-TOMEK, and SVM-SMOTE (Kashtriya et al., 2 Sep 2025). Improvement in F1 over the original dataset was reported as $k$5 on MIMIC-III, $k$6 on MIMIC-IV, $k$7 on MIMIC-IV(10), and $k$8 on MIMIC-IV(20) (Kashtriya et al., 2 Sep 2025). On MIMIC-III, the original F1 of $k$9 increased to $RY$0 under QI-SMOTE, while on MIMIC-IV the original F1 of $RY$1 increased to $RY$2 (Kashtriya et al., 2 Sep 2025). The study also reported a Wilcoxon signed-rank comparison against SMOTE-ENN on MIMIC-IV(10), with $RY$3, $RY$4, $RY$5, and effect size $RY$6, indicating a highly significant and consistent advantage across the 10 folds (Kashtriya et al., 2 Sep 2025).

The original Quantum-SMOTE on Telecom Churn reported substantial downstream gains as the synthetic minority proportion increased from no synthetic data to 50% minority (Mohanty et al., 2024). For Random Forest, test accuracy rose from $RY$7 to $RY$8, F1 from $RY$9 to $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$0, PR-AUC from $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$1 to $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$2, and ROC-AUC from $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$3 to $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$4 (Mohanty et al., 2024). Logistic Regression also improved in F1 from $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$5 to $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$6 and PR-AUC from $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$7 to $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$8, although test accuracy was non-monotonic across oversampling levels (Mohanty et al., 2024). Quantum-SMOTEV2 on the Cell-to-Cell Telecom dataset further reported that Random Forest at 50% minority with AOL achieved test accuracy $$F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2,$$9, F1 $\left|P(y=1\mid x)-0.5\right| > \text{margin},$0, AUC-ROC $\left|P(y=1\mid x)-0.5\right| > \text{margin},$1, and PR-AUC $\left|P(y=1\mid x)-0.5\right| > \text{margin},$2; the neural network improved from baseline test accuracy $\left|P(y=1\mid x)-0.5\right| > \text{margin},$3, F1 $\left|P(y=1\mid x)-0.5\right| > \text{margin},$4, AUC-ROC $\left|P(y=1\mid x)-0.5\right| > \text{margin},$5, PR-AUC $\left|P(y=1\mid x)-0.5\right| > \text{margin},$6 to approximately $\left|P(y=1\mid x)-0.5\right| > \text{margin},$7, $\left|P(y=1\mid x)-0.5\right| > \text{margin},$8, $\left|P(y=1\mid x)-0.5\right| > \text{margin},$9, and $0.1$0, respectively (Mohanty et al., 31 Jan 2025). The same paper argued that moderate synthetic levels of 30–36% with AOL could deliver performance previously associated with about 50% synthetic data in the original method (Mohanty et al., 31 Jan 2025).

The QRBM-based intrusion-detection study is not framed as QI-SMOTE, but it provides a direct comparison with SMOTE and RandomOversampler in the same oversampling role (Sinno et al., 5 Feb 2025). The QRBM generated 1,683,771 usable synthetic attack samples from about 1,700 anneals, producing a balanced dataset of 4,208,618 samples (Sinno et al., 5 Feb 2025). Runtime for balancing the full dataset was $0.1$1 s for QRBM, compared with $0.1$2 s for SMOTE and $0.1$3 s for RandomOversampler (Sinno et al., 5 Feb 2025). With Random Forest, the QRBM-balanced dataset yielded Precision $0.1$4, Recall $0.1$5, and F1 $0.1$6, compared with SMOTE’s F1 $0.1$7 and RandomOversampler’s F1 $0.1$8 (Sinno et al., 5 Feb 2025). These results strengthen the broader claim that global generative oversampling can outperform local interpolation when feature dependencies are strong.

A related but distinct comparison comes from the hybrid quantum-classical adversarial framework Q-SYNTH for fraud detection (Innan et al., 20 May 2026). Although Q-SYNTH is not presented as QI-SMOTE, it is evaluated directly against SMOTE. SMOTE achieved the lowest KS median $0.1$9 and Wasserstein median $\tilde{x} = x_i + \lambda (x_j - x_i),$00, the classical GAN showed the highest downstream detection in several settings, and Q-SYNTH occupied an intermediate position: better marginal similarity than the GAN, downstream performance close to the GAN, and real-vs-synthetic detectability AUC $\tilde{x} = x_i + \lambda (x_j - x_i),$01 (Innan et al., 20 May 2026). A plausible implication is that future QI-SMOTE designs may increasingly blend SMOTE-like locality constraints with stochastic generative modeling.

6. Limitations, misconceptions, and open directions

A recurrent misconception is that QI-SMOTE necessarily implies execution on quantum hardware. The literature does not support that reading. Several papers state explicitly that their methods are classical oversampling algorithms or quantum-inspired classical pipelines, even when quantum concepts, simulated circuits, or quantum notation are used (Behera et al., 18 Dec 2025, Patel et al., 16 Jan 2026, Kashtriya et al., 2 Sep 2025). Conversely, some adjacent work does run on quantum or quantum-annealing hardware, as in the Pegasus QRBM study (Sinno et al., 5 Feb 2025). The term therefore describes inspiration and mechanism more reliably than hardware substrate.

Computational cost varies sharply by formulation. The entanglement/VQE-based medical QI-SMOTE reported, on a dataset with 700 samples, 3 features, minority count $\tilde{x} = x_i + \lambda (x_j - x_i),$02, and 100 additional minority samples, classical SMOTE requiring less than $\tilde{x} = x_i + \lambda (x_j - x_i),$03 s and less than 1 MB, whereas QI-SMOTE required approximately 50–100 s and up to approximately 200 MB on an Intel i7 CPU with 32 GB RAM (Kashtriya et al., 2 Sep 2025). The original Quantum-SMOTE and Quantum-SMOTEV2 also depend on state preparation, swap-test simulation, and repeated rotation circuits, even though both emphasize low-depth circuits and compact swap tests (Mohanty et al., 2024, Mohanty et al., 31 Jan 2025). The QRBM study, in contrast, reported balancing tasks in milliseconds once the hardware and embedding infrastructure were in place, but also documented hardware constraints, embedding complexity, chain breakage, and noise as practical limitations (Sinno et al., 5 Feb 2025).

Performance is also model-dependent. The industrial robustness study showed that Random Forest and SVM often preserved over 95% of baseline accuracy even under maximum quantum-inspired noise, whereas Naive Bayes and Decision Trees could become unstable on high-variance datasets, and Logistic Regression and Naive Bayes could degrade after oversampling in overlapping feature spaces (Patel et al., 16 Jan 2026). In the Telco churn QSMOTE-plus-PGM study, KNN-based QSMOTE was consistently poor, indicating that merely rebranding local interpolation without changing its geometry may not provide the advertised benefit (Behera et al., 18 Dec 2025).

The literature also remains narrow in benchmark scope. One paper reports on Telco Customer Churn restricted to the first 1000 samples (Behera et al., 18 Dec 2025); another focuses on MIMIC-III and MIMIC-IV mortality prediction with a specific curated feature set (Kashtriya et al., 2 Sep 2025); Quantum-SMOTEV2 is evaluated on a single large telecom churn dataset (Mohanty et al., 31 Jan 2025). This suggests caution in treating any one variant as settled across domains.

Open directions are explicit. The QSMOTE-PGM/kPGM study proposes new encodings, more efficient kPGM algorithms such as Nyström approximations and random features, extension to multi-class and multi-label imbalanced settings, and more principled quantum-inspired metrics beyond fidelity and cosine similarity (Behera et al., 18 Dec 2025). The medical QI-SMOTE paper highlights hybrid quantum-classical implementations, hyperparameter robustness studies, and extension to multimodal and time-series data (Kashtriya et al., 2 Sep 2025). The QRBM study points toward larger and deeper energy-based models, improved classical embedding and sampling algorithms such as simulated quantum annealing, and broader applications in healthcare, finance, cybersecurity, and anomaly detection (Sinno et al., 5 Feb 2025). Across these lines of work, the overarching trajectory is clear: QI-SMOTE is evolving from a metaphor for quantum-flavored interpolation into a broader framework for structure-aware minority generation, in which the most consequential design choice is not whether the method is quantum-labeled, but whether it captures the geometry, dependence structure, and decision-relevant support of the minority class.