Invariance Audits for Quantum Kernels and Variational Rewinding: A Real-to-Hermitian Taxonomy of Projector, Flag, Anchor, and Density Geometry

Abstract: Machine-learning models often replace vectors by normalized directions, projectors, covariances, subspaces, ordered flags, quantum states, or density operators before any classifier is fitted. This replacement is an invariance decision: it determines which distinctions are kept and which are quotiented out. We develop a self-contained real-to-Hermitian taxonomy for auditing such representations in quantum machine learning. On the real side, we formalize Grassmann and flag projector kernels, prove positive semidefiniteness and block-gauge invariance of a weighted flag kernel, and give a same-span block-swap witness showing when whole-span Grassmann geometry must fail while ordered flags succeed. On the quantum side, we prove that a noiseless fidelity kernel is exactly the Hilbert--Schmidt inner product between the associated rank-one Hermitian projectors, and that a QVR-style return probability is exactly an overlap score between the input projector and a learned anchor operator. Rank-constrained returns correspond to complex Grassmann anchors, while mixed or multimodal class models are naturally represented by density or positive-semidefinite anchors. Controlled vector, subspace, statevector, anomaly, finite-shot, and quotient-witness experiments support the same conclusion: quantum and geometric lifts are useful when their invariances match the task, and fail correctly when discarded information is label-bearing. The paper makes no hardware-speedup or quantum-advantage claim.

• The paper presents an explicit taxonomy unifying classical and quantum geometric lifts used in machine learning, clarifying which data distinctions are preserved or discarded.
• It rigorously proves kernel properties and invariance conditions, showing quantum fidelity kernels equate to Hermitian projection methods.
• Empirical diagnostics highlight that aligning representation invariances with task-specific labels is crucial for model effectiveness and accurate performance.

Invariance Audits for Quantum Kernels and Variational Rewinding: A Real-to-Hermitian Taxonomy

Problem Statement and Objectives

The paper "Invariance Audits for Quantum Kernels and Variational Rewinding: A Real-to-Hermitian Taxonomy of Projector, Flag, Anchor, and Density Geometry" (2607.07927) investigates the geometric invariance structures imposed by classical and quantum data representations in machine learning, particularly in quantum kernel methods and QVR-style return models. The central thesis is that the initial representation or lift—vector normalization, projectivization, subspace/flag encoding, quantum state preparation, density averaging—is a scientific invariance decision that dictates what distinctions are kept and which are removed prior to classifier fitting. The study develops an explicit and self-contained taxonomy of these lifts, proves key properties of their associated kernels, and positions quantum kernel methods and variational rewinding within this principled geometric framework.

Real-to-Hermitian Representation Taxonomy

The paper delivers an extensive taxonomy that unifies classical and quantum geometric representations in ML:

• Classical real-side: vectors, normalized directions, projectors, covariances, subspaces (Grassmannian), ordered flags.
• Quantum side: pure quantum states, Hermitian projectors (removing global phase), rank-$r$ anchors (complex Grassmannian), density anchors (mixed states), and general Hermitian decision operators.

Each representation imposes distinct invariance constraints—for example, unit normalization removes scale, real projectors remove sign, Grassmann points quotient out basis information, and quantum projectors eliminate global phase. The taxonomy provides an exact dictionary for these geometric objects, their invariances, and their induced scores, allowing for rigorous audit of whether discarded information (e.g., sign, magnitude, block ordering, phase) is truly nuisance or is label-bearing.

Mathematical Results: Kernel Properties and Invariance Witnesses

The paper provides several rigorous results:

• Positive semidefiniteness and invariance: Proves that weighted flag projection kernels are PSD and block-gauge invariant; spectral activation followed by top-$p$ reprojection is identity for exact projectors.
• Failing span separation: Whole-span Grassmannians cannot distinguish same-span ordered block swaps, which flags can.
• Quantum fidelity kernels: Shows that a noiseless fidelity kernel is literally the Hilbert–Schmidt inner product between rank-one Hermitian projectors, $$K(x, y) = |\langle\phi(x)|\phi(y)\rangle|^2 = \operatorname{tr}(P_x P_y)$$.
• Anchor-overlap scores for QVR: Establishes that QVR-style return probabilities are Hermitian anchor-overlap scores, $p_\theta(x) = \operatorname{tr}(P_x A_\theta)$, where the anchor can be rank-one, rank-$r$, or a general density operator.

These identities clarify that the algebraic distinction between "quantum" and "classical" models is a choice of invariance quotient, not an inherent quantum property at the kernel level.

Empirical Diagnostics and Numerical Results

The experiments are conducted on controlled toy datasets from scikit-learn and synthetic time-series anomaly benchmarks, using deterministic splits and rigorous validation-selection discipline. The empirical findings highlight the following:

• Classical baselines: Vector RBF remains consistently strong; non-vector geometric lifts (Grassmann, flag, density) are useful when their invariances match the data, and fail when they erase label-bearing information.
• Flag vs. Grassmann witness: Flag models perfectly resolve positive-control block-swap tasks where Grassmannians fail due to erased block order.
• Quantum kernel comparisons: Product-angle quantum kernels achieve strong performance on tabular data via classical closed form. Entangled circuits only perform well when designed/trained to match data structure.
• QVR anchor models: Pure-center anchors perform poorly in multiclass classification; rank-$r$ and density anchors significantly improve accuracy.
• Trace-identity audits: Statevector quantum returns and Hermitian trace scores are equivalent to machine precision, confirming the mathematical identity.
• Synthetic invariance witnesses: Models fail (as intended) when the label violates imposed invariance (e.g., global phase as label).

Key numerical results include balanced accuracy exceeding 0.95 with classical RBF and quantum kernel variants on several datasets, but only when the representation matches the task's invariances. The native anomaly audit achieves ROC–AUC ~0.86 with exact trace/return probability and demonstrates finite-shot robustness at practical shot counts.

Practical and Theoretical Implications

The research implies that the efficacy of geometric and quantum lifts hinges critically on their imposed invariances relative to the scientific object (task label structure). For practical quantum ML:

• Representation matching: Selecting a lift aligned with scientific invariances—whether radius, block structure, or phase—is essential; mismatched invariance leads to correct failure (audited by synthetic witnesses).
• Anchor generalization: Rank-$r$ or density anchors are advisable for noisy, multimodal, or multiclass data, whereas pure-state anchors are restrictive.
• Kernel method equivalence: Quantum kernel methods are not inherently “quantum” beyond projective geometry; they represent a Hilbert–Schmidt kernel that is mathematically accessible classically.
• Finite-shot effects: Shot noise adds layer of uncertainty but does not break trace consistency.

Theoretically, the paper advances the understanding that quantum and geometric ML methods must be interpreted through their quotient geometry rather than procedural or hardware implementation. This provides a precise framework for benchmarking, model selection, and diagnosis of when quantum lifts "fail correctly" and when they succeed.

Limitations and Future Directions

The scope is intentionally conservative:

• Simulations are statevector-based; no hardware claims are made.
• Dataset complexity and circuit depth are restricted.
• Circuit entanglement is not trained.
• Numerical outcomes should be read as diagnostic summaries, not operational benchmarks.

Future directions include matched studies on real time-series anomaly data, with identical encoder, measurement, shot, noise, and training budget for both circuit and Hermitian models. The theory further suggests that quantum-advantage claims should be tightly coupled to invariance audit, not merely to hardware acceleration.

Conclusion

The paper establishes a rigorous real-to-Hermitian taxonomy for geometric representations in ML and quantum ML, providing mathematical identities, diagnostic experiments, and invariance audits which clarify the distinction and equivalence between classical and quantum kernel methods, and the conditions under which variational rewinding and anchor-based models are appropriate. The choice of representation and imposed invariance is central; quantum and geometric lifts are justified when their invariances correctly quotient out nuisance variation, and they fail reliably when discarded information is label-bearing. This framework enables principled selection and audit of ML models across both classical and quantum domains.