Quantum Probabilistic Label Refining

• The paper introduces a quantum framework that refines data labels by combining quantum measurement, entropy-based methods, and ensemble techniques to address label noise.
• It utilizes operator algebra and probabilistic kernel refinement to generate soft, structure-aware labels that capture input-specific uncertainty and high-order correlations.
• The approach demonstrates improved robustness, calibration, and interpretability in noisy environments, outperforming traditional label smoothing and Bayesian methods.

Quantum Probabilistic Label Refining is a framework and family of methodologies aiming to enhance the quality, informativeness, and reliability of data labels in both classical and quantum machine learning by leveraging quantum probabilistic representations, probabilistic kernel refinement, quantum uncertainty, and advanced ensemble or optimization schemes. Motivated by the limitations of rigid one-hot labels, poor robustness under noise, and artifacts in standard smoothing or Bayesian approaches, Quantum Probabilistic Label Refining introduces label representations derived from quantum processes—most notably, quantum measurement via the Born rule and structure-enforcing techniques from operator algebra. By integrating principles from quantum measurement theory, information geometry, and non-commutative probability, these methods generate input- and structure-aware probabilistic labels that enable more robust, interpretable, and accurate learning, particularly under label noise, uncertainty, or adversarial conditions.

1. Conceptual Foundations: Quasi-Probability Kernel Refinement and Quantum Analogues

The refinement of quasi-probability kernels, as formulated over a measurable space $(X,\mathcal{F})$ with a sub-$\sigma$-algebra $\mathcal{E}\subset\mathcal{F}$, provides the foundational classical theory for label refining (Preston, 2010). A quasi-probability kernel $T: X\times\mathcal{F}\to\mathbb{R}_+$ specifies, via the set $JE(T)$, a family of measures for which $T$ is a conditional expectation operator: $$JE(T) := \{ p\in\mathcal{P}(\mathcal{F}) : p(gf) = p(g\,T(f)),\;\forall f\in B(\mathcal{F}),\,g\in B(\mathcal{E}) \}.$$ Refining $T$ is achieved by constructing a restriction to a measurable set $D\in\mathcal{E}$: $o(x,F) = 1_D(x) \cdot T(x,F)$ such that $JE(o) = JE(T)$ and desirable properties—properness (factorization), adaptedness, or normality—are achieved. Properness, for instance, requires $T(gf)(x) = g(x)T(f)(x)$ for all $g\in B(\mathcal{E})$, $f\in B(\mathcal{F})$. The choice of $D$ is determined by the set where such properties hold, e.g.,

$$D = \bigcap_{g\in G', f\in G} \{x\in S_T : T(gf)(x) = g(x)T(f)(x)\}$$

with $S_T$ the support of $T$ and $G,G'$ countable determining sets.

In the quantum setting, this theory is carried over to operator-valued kernels, such as positive operator-valued measures (POVMs) on a non-commutative probability space (C*- or von Neumann algebra). Here, label refining may target the restriction of operator-valued kernels $\hat{T}$ to subspaces $D$, aiming for analogues of properness and normality—such as enforcing that measurement operators act as projections (with spectrum $\{0,1\}$) on relevant commutative subalgebras, or that conditional expectations correspond to quantum analogues of supporting states (Preston, 2010).

2. Quantum Entropy, Probabilistic Similarity, and Entropy-Based Refining

Quantum entropy-based methods furnish a means for label refinement by quantifying the integration of a query into a class structure via entropy changes. In Quantum Low Entropy based Associative Reasoning (QLEAR learning) (Jankovic, 2017), data are mapped into density matrices $p$ (positive semidefinite, trace one) representing quantum states, and the quantum Tsallis entropy,

$$S_q(p) = \frac{1}{1-q} [\operatorname{Tr}(p^q) - 1], \quad q > 0,\, q\neq 1,$$

is used to capture the "natural structure" of a data class. For a query $x$, the relative change in entropy—computed before and after adding $x$ to the density matrix representing a class—serves as an indicator of the sample's fit, facilitating a probabilistic label assignment. The refinement operates by minimizing terms of the form $dE_s - \alpha dE_{ns}$, with $dE_s$ and $dE_{ns}$ the entropy changes in same-class and non-same-class prototypes.

These approaches circumvent assumptions of linear separability and reduce high-variance errors typical in nearest-neighbor methods, as the entropy-based label refinement is sensitive to the quantum-informational geometry of data clusters. Experimental validation on nonlinearly separable problems (e.g., XOR, AND, IRIS) demonstrates robustness and generalization advantages.

3. Quantum Measurement, Soft Label Generation, and Hybrid Quantum-Classical Supervision

Quantum Probabilistic Label Refining frameworks integrate quantum non-determinism directly in the label creation procedure (Qi et al., 1 Oct 2025). A variational quantum circuit (VQC) encodes input data $x\in\mathbb{R}^d$ via angle or amplitude encoding into an $n$-qubit quantum state,

$$|\psi(x;\theta)\rangle = U(\theta) V(x) |0\rangle^{\otimes n},$$

where $V(x)$ depends on the chosen encoding. The layered VQC, through entangling and rotation gates, incorporates high-order and multipartite feature correlations. Measurement of the final state yields a distribution over outcomes according to the Born rule,

$$P(y\mid x) = |\langle y | \psi(x;\theta)\rangle|^2.$$

Retaining the relevant $K$ output classes and repeating measurements ("shots"), the framework constructs a label distribution $P^{\text{quantum}}(y|x)$ encoding input-specific uncertainty stemming from both quantum indeterminacy and data structure.

These quantum-derived soft labels are then used to train classical deep networks (such as CNNs) via a soft-target cross-entropy loss,

$$\mathcal{L}'_{\text{ce}} = - \sum_{k=1}^K y^{(\text{quantum})}_k \log p_k,$$

with $y^{(\text{quantum})}_k$ the quantum-generated probability for class $k$ and $p_k$ the model's prediction.

Empirical results on vision datasets (MNIST, Fashion-MNIST) demonstrate that this hybrid process increases robustness under noise (up to 50% higher accuracy under heavy perturbations), improves calibration (agreement between predicted and true uncertainties), and yields better interpretability, as refined labels are more aligned with human or foundational model assessments of ambiguous examples (Qi et al., 1 Oct 2025).

4. Probabilistic Decoupling and Variational Inference Frameworks

A complementary class of quantum-inspired probabilistic label refinement methods is based on variational decoupling of observed and true label distributions (Nørregaard et al., 2020). Here, each sample $i$ has a latent class-distribution $Y_i$ (with a Dirichlet prior) and labels $s_i$ are generated via a transition (confusion) matrix $T$ (also with Dirichlet prior),

$p(s_i|Y_i,T) = \sum_y Y_{iy} T_{ys_i}.$

The observed label likelihood is thus a convolution of class and transition probabilities. Variational inference maximizes an evidence lower bound (ELBO) involving nontrivial expectation terms, approximated via Taylor expansions around means. Optimization seeks agreement between reconstructed and observed labels,

$$\hat{S}_{iy} = \mathbb{E}_{q}[Y_{iy}]\;\mathbb{E}_{q}[T_{ys}].$$

Although not strictly quantum, this framework's use of vectors analogous to density matrices and explicit transition operators parallels quantum measurement's treatment of label refining via projection operators. Extensions suggest suitability for quantum machine learning contexts—especially where latent superposition or noisy transition processes play a central role.

5. Geometric Optimization in Probabilistic Quantum State Labeling

The geometric approach to probabilistic quantum cloning and identification directly generalizes label refinement to quantum state spaces (Liu et al., 2019). For $N$ known quantum states, the optimal probabilistic cloning and identification is characterized via a Hermitian matrix,

$$M = X^{(m)} - \sqrt{\Gamma} X^{(n)}_p \sqrt{\Gamma},$$

with $X^{(r)}_{ij} = \langle\Psi_i|\Psi_j\rangle^r$, and the determinant condition

$$\det(X^{(m)} - \sqrt{\Gamma} X^{(n)}_p \sqrt{\Gamma}) = 0$$

defining the achievable parameter space for failure probabilities $\{q_i\}$. The optimization—finding the tangent point between this constraint surface and a hyperplane orthogonal to the prior vector $\eta$—yields the minimum average label refinement (cloning or identification failure) cost. This geometric formalism efficiently unifies cloning and probabilistic state identification, supporting optimal quantum label refinement strategies under arbitrary prior beliefs.

6. Quantum Ensemble, Filtering, and Bagging Approaches to Robust Label Refinement

Quantum-assisted ensemble approaches, such as quantum bagging with unsupervised base learners and quantum annealing for filtering, extend label refining to practical noisy-label scenarios. In Quantum Bagging (Rathi et al., 8 Sep 2025), unsupervised quantum $k$-means (QMeans) is deployed as the base learner within a bootstrapped ensemble; QRAM-based quantum subsampling constructs diverse datasets in superposition, and cluster assignments are post-labeled through majority voting.

Performance evaluations—involving datasets with artificial label noise—show that quantum bagging achieves resilience to label corruption, with ensemble variance diminishing as base learner count increases, and often outperforms classical KMeans bagging under identical noise conditions.

Black-box optimization with quantum annealing (Otsuka et al., 12 Jan 2025) addresses probabilistic label refinement by modeling the data-subset selection problem as a QUBO: a surrogate model of validation loss as a quadratic function of the indicator vector of instances is minimized via D-Wave clique sampling. The process efficiently filters mislabeled data, as evidenced on synthetic majority bit tasks, and delivers improved generalization, reduced loss, and substantially faster sampling than classical simulated annealing methods.

7. Broader Implications, Calibration, and Future Directions

Quantum probabilistic label refining systematically enhances robustness, calibration, and interpretability of supervised learning pipelines. Unlike label smoothing or post-hoc calibration techniques, refined probabilistic labels—especially those derived from quantum measurement—capture sample-specific uncertainty and high-order feature correlations, yielding better performance under distribution shifts and corruptions (Challa et al., 2023, Qi et al., 1 Oct 2025). Quantile-based formulations, as in QuantProb (Challa et al., 2023), further generalize labels via duality between quantiles and probabilities, producing calibrated uncertainty that remains stable under distributional perturbations.

The theoretical generalization from commutative (classical) to non-commutative (quantum) label-refinement strategies (via operator kernel restriction, geometric optimization, and variational inference on transition operators) opens quantitative pathways for high-precision probabilistic learning in quantum information science, quantum-enhanced ensemble learning, and uncertainty-aware artificial intelligence. Potential applications span robust image classification, out-of-distribution detection, medical and financial decision-making, and quantum communication protocols.

A plausible implication is that, as quantum hardware matures, full end-to-end quantum label refinement—encompassing deep feature quantum encoding, measurement-based soft labels, and quantum-calibrated model training—will further close the gap between theoretical optimality and practical performance in noisy or adversarial environments.