--- title: 'Quantum-Inspired SMOTE: Innovative Oversampling' url: https://www.emergentmind.com/topics/quantum-inspired-smote-qi-smote type: topic --- # Quantum-Inspired SMOTE: Innovative Oversampling 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 [2512.16960] [2601.11423] [2509.02863] [2402.17398] [2501.19001] [2502.03086]. ## 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 [2512.16960]. 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 [2601.11423]. 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 [2509.02863]. 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 [2402.17398]. 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 [2501.19001]. 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 [2502.03086]. 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 [2512.16960] [2402.17398]. 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 [2509.02863]. 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” [2601.11423]. 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 [2402.17398]. 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 [2512.16960] [2501.19001] [2502.03086]. Second, they replace Euclidean similarity with cosine-squared “fidelity,” swap-test-derived angular distance, or energy-based sampling [2512.16960] [2601.11423] [2402.17398]. 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 [2512.16960] [2509.02863]. 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 [2502.03086]. 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 | [2512.16960] | | Swap-test angular QSMOTE | Amplitude encoding, compact SWAP test, angle-controlled synthesis | [2601.11423], [2402.17398], [2501.19001] | | Entanglement/VQE-assisted QI-SMOTE | Hadamard + \(RY\), layered entanglement, Ising-like Hamiltonian, VQE, then SMOTE | [2509.02863] | | Energy-based quantum oversampling | QRBM learning and quantum-assisted sampling for minority generation | [2502.03086] | 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 [2512.16960]. 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 [2512.16960]. 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 \(K\)-means, each minority sample is paired with its assigned centroid \(c(x)\), both are amplitude-encoded as normalized state vectors, and a compact SWAP test estimates an overlap that defines the angle \[ \alpha(x,c)=\cos^{-1}(s(x,c)). \] A synthetic sample is then constructed as \[ \widetilde{x}=x+r\,\hat{d}(x,c), \qquad r\sim \mathrm{Unif}(0,\alpha(x,c)/\kappa), \] where \(\hat{d}(x,c)=(c-x)/\|c-x\|_2\) unless \(c=x\) [2601.11423]. The original Quantum-SMOTE paper follows a similar angular logic but uses compact swap-test states and applies \(R_X(\theta)\), \(R_Y(\theta)\), or \(R_Z(\theta)\) rotations to the encoded minority point, with X rotations used in the main Telecom Churn experiment [2402.17398]. 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 [2501.19001]. 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 \(RY(\theta_i)\), applies layered entanglement using CNOT, CZ, and Toffoli gates, and then evolves the resulting state under an Ising-like Hamiltonian optimized by VQE [2509.02863]. The real part of the evolved amplitudes is appended to the dataset, and standard SMOTE is then applied in this “quantum-enhanced” feature space [2509.02863]. 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 \(120 \times 120\) QRBM is trained on minority attack samples only, and synthetic attack records are obtained from quantum-assisted sampling on D-Wave Pegasus hardware [2502.03086]. Because the method learns \(P(\text{attack features})\) 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 [2502.03086]. ## 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: \[ \tilde{x} = x_i + \lambda (x_j - x_i). \] Its main role in the literature is comparative, not foundational [2512.16960]. Fidelity-based QSMOTE replaces neighbor interpolation with centroid-directed movement weighted by a fidelity-inspired factor. For a minority sample \(x\) and its centroid \(c\), with direction vector \(\vec d = c-x\), \[ F(x,c) = \left( \frac{x \cdot c}{\|x\| \, \|c\|} \right)^2, \qquad \tilde{x} = x + \lambda \cdot F(x,c)\cdot \frac{\vec d}{\|\vec d\|}. \] If \(x\) and \(c\) are already similar, the step is large; if they are dissimilar, the step is small [2512.16960]. The margin-based variant adds a probabilistic filter, \[ \left| P(y=1\mid x) - 0.5 \right| > \text{margin}, \] to reject ambiguous synthetic points near the decision surface [2512.16960]. The swap-test formulations treat classical vectors as amplitude-encoded states. In the industrial QSMOTE benchmark, each non-zero vector \(v \in \mathbb{R}^d\) becomes \[ |\psi_v\rangle = \frac{1}{\|v\|_2}\sum_{m=1}^d v_m |m\rangle. \] For minority sample \(x\) and centroid \(c\), the overlap \[ s(x,c) = \langle \psi_x \mid \psi_c \rangle = \cos \alpha(x,c) \] is estimated via compact SWAP test, and the angle \(\alpha(x,c)\) determines the sampling range [2601.11423]. In the original Quantum-SMOTE formulation, the swap test yields a “swap test probability,” from which the angular distance is defined as \[ \text{angular\_distance} = 2\cos^{-1}\big(\sqrt{\text{swap\_test\_probability}}\big), \] and this angle is then converted into a rotation angle modulated by the splitting factor and an angle increment [2402.17398]. Quantum-SMOTEV2 preserves this rotation-angle logic while using angular distributions around a single centroid and IQR-based angular outlier detection [2501.19001]. The VQE-assisted QI-SMOTE formulation introduces a more explicitly quantum mechanical encoding. Feature normalization is defined by \[ f_i' = \frac{f_i - \min(f)}{\max(f)-\min(f)} \in [0,1], \qquad \theta_i = \pi \cdot f_i', \] and the corresponding single-qubit encoding gate is \[ RY(\theta_i)= \begin{bmatrix} \cos(\theta_i/2) & -\sin(\theta_i/2)\\ \sin(\theta_i/2) & \cos(\theta_i/2) \end{bmatrix}. \] After layered entanglement, the state is evolved under an Ising-like Hamiltonian \[ H = \sum_i b_i Z_i + \sum_{i,j} w_{ij} Z_i Z_j, \] with VQE objective \[ \theta^* = \arg\min_\theta \langle \psi(\theta)|H|\psi(\theta)\rangle. \] The evolved state \(|\psi_{\text{evolved}}\rangle\) is converted to a real vector by \[ |\psi_{\text{real}}\rangle = \Re\big(|\psi_{\text{evolved}}\rangle\big), \] and standard SMOTE is then run on the augmented data [2509.02863]. The QRBM line uses an energy-based probabilistic model rather than interpolation. Its RBM energy is \[ E(\mathbf{v},\mathbf{h}) = -\sum_i b_i v_i - \sum_j c_j h_j - \sum_{i,j} v_i W_{ij} h_j, \] which induces a Boltzmann distribution \[ P(\mathbf{v},\mathbf{h}) = \frac{\exp(-E(\mathbf{v},\mathbf{h}))}{Z}. \] For quantum annealing, the model is mapped to an Ising Hamiltonian \[ H = -\sum_i h_i s_i - \sum_{i