Covariant Quantum Kernels

• Covariant quantum kernels are mathematical constructs that intertwine group symmetries with quantum states, enabling invariant transition amplitudes and practical operational benefits.
• They underpin advances in quantum gravity and cosmology by discretizing path integrals and linking canonical and spinfoam models through symmetry-respecting techniques.
• In quantum machine learning and operator algebras, covariant kernels ensure robust, noise-resilient data representation and efficient channel classification even in high-dimensional systems.

Covariant quantum kernels are mathematical objects—primarily transition amplitudes, operator kernels, or positive-definite maps—constructed to respect symmetry, covariance, or invariance under a specified group action. They arise in diverse fields, including quantum gravity, open-system dynamics, quantum field theory, quantum machine learning, and quantum information theory. Covariant quantum kernels are distinguished by their ability to intertwine dynamical evolution, measurement, or data representation with the underlying symmetry constraints, leading to physically meaningful, often operationally advantageous, structures.

1. Mathematical Structure and Defining Principles

Covariant quantum kernels are defined to intertwine group actions with the space of quantum states, operators, or functionals.

• Canonical Example in Quantum Learning:

For data $x$ on a group $G$, and a unitary representation $D(x)$ with fiducial state $|\psi\rangle$, the kernel

$$K(x, x') = |\langle \psi | D(x)^\dagger D(x') | \psi \rangle|^2$$

depends only on the relative group element and is invariant under simultaneous group action: $K(gx, gx') = K(x, x')$ (Glick et al., 2021, Agliardi et al., 10 Dec 2024, Henderson et al., 17 Sep 2025).

• Frameworks in Mathematical Physics and Quantum Gravity:

In quantum cosmology and gravity, covariant kernels appear as transition amplitudes summed over geometric data “covariantly”—for example, in the spinfoam formalism,

$$Z = \sum_{j_f, i_e} \prod_f d(j_f) \prod_v A_v(j_e, i_v),$$

producing a fully diffeomorphism-invariant path integral kernel over discrete geometries (Vidotto, 2011).

• CP Maps and Measurement Theory:

In operator algebra and quantum measurement, a covariant completely positive (CP) kernel $S$ intertwines the group $G$’s action in both domain and codomain—formally, for all $g$,

$$S(b) = J^* T(b) J,\quad J U(g) = \tilde U(g) J, \quad \tilde U(g) T(b) = T(\beta_g(b)) \tilde U(g)$$

where $J$ is an intertwiner and $T$ a *-representation (Haapasalo et al., 2015).

Covariance ensures that under a group transformation, the kernel or its image transforms in a specified, consistent way, enforcing physical or data symmetries at the level of the kernel object itself.

2. Covariant Kernels in Quantum Gravity and Cosmology

The construction and interpretation of covariant quantum kernels are central in non-perturbative quantum gravity, especially in the path-integral and canonical-quantization approaches:

• Spinfoam Cosmology:

The covariant path integral is discretized into a sum over spinfoams, with transition amplitudes defined as

$$Z = \sum_{j_f, i_e} \prod_f d(j_f) \prod_v A_v(j_e, i_v)$$

where amplitudes $A_v$ have a semiclassical limit reproducing Regge action, and coherent states are chosen to enforce cosmological homogeneity and isotropy. The resulting quantum kernel captures quantum fluctuations and is manifestly diffeomorphism-invariant (Vidotto, 2011).

• Canonical–Covariant Link:

In Loop Quantum Cosmology (LQC), the Hamiltonian constraint is exponentiated and group-averaged to yield a covariant transition kernel amenable to spinfoam-like expansion:

$$\langle \lambda_F | \lambda_0 \rangle_{\text{phy}} = (1/2\pi) \int_{-\infty}^{\infty} d\alpha \langle \lambda_N | e^{i\alpha C_H} | \lambda_0 \rangle$$

thus directly linking the canonical and covariant (spinfoam) formalisms.

These approaches maintain the dynamical and symmetry structure of general relativity at the quantum level, allowing ultraviolet-finite and non-perturbative formulations of cosmological transition kernels (Vidotto, 2011).

3. Covariant Kernels in Operator Algebras, Measurement, and Quantum Channels

The structure of covariant quantum kernels plays a foundational role in quantum information, measurement theory, and the general analysis of CP maps:

• Covariant KSGNS Construction:

The Kolmogorov–Stinespring–Gel′fand–Naĭmark–Segal (KSGNS) theory extends to covariant settings by associating every CP map or positive kernel with a minimal dilation intertwining the group symmetry. The minimal dilation $(\mathcal{M}, T, J, \tilde U)$ ensures that

$S(b) = J^* T(b) J, \qquad J U(g) = \tilde U(g) J$

enabling the classification of observables and instruments under group action (Haapasalo et al., 2015).

• Extremality Conditions:

To characterize “pure” covariant kernels (non-mixed, noise-free measurements), the extremal condition is:

$$[D, \tilde U(g)] = 0\,\,\forall g, \quad (Jv | D Jv) = 0\,\,\forall v \implies D = 0$$

enabling operational tests for the extremality of covariant quantum devices.

• Covariant Quantum Channels:

In the analysis of random covariant quantum channels, group twirling operations impose covariance (e.g., unitary, orthogonal, hyperoctahedral, or diagonal). These channels can be classified by scalar and matrix parameters reflecting their symmetry, with threshold phenomena determining properties such as positivity of partial transpose (PPT) and entanglement-breaking behavior (Nechita et al., 6 Mar 2024).

Such kernels underlie the structure of “covariant quantum relations and graphs” on finite-dimensional $C^*$-algebras equipped with compact quantum group action, enabling rigorous combinatorial and categorical analysis of symmetry-constrained quantum communication protocols (Verdon, 2023).

4. Covariant Kernels in Quantum Machine Learning

Covariant quantum kernels offer intrinsic advantages for structured data in quantum machine learning:

• Fidelity and Group-Covariant Kernels:

Embedding data $x$ from $G$ via $D(x)|\psi\rangle$ yields kernels

$$\kappa(x, x') = |\langle\psi | D(x)^\dagger D(x') | \psi\rangle|^2$$

which are invariant under $G$ and can perfectly distinguish cosets, achieving efficient classification when data is symmetry-structured (Glick et al., 2021, Agliardi et al., 10 Dec 2024, Henderson et al., 17 Sep 2025). Optimization of the fiducial state (kernel alignment) improves performance on real-world data.

• Mitigation of Exponential Concentration and Barren Plateaus:

For generic random quantum kernels, the variance collapses exponentially in the number of qubits, resulting in barren plateaus and loss of trainability. For group-covariant quantum kernels, variance

$$\operatorname{Var}_{x, x'}[\kappa(x, x')] \to \frac{m-1}{m^2},\qquad (N \to \infty)$$

remains $O(1)$, independent of system size, preserving trainability even for $N \gg 1$ (Henderson et al., 17 Sep 2025). This is analytically proven and confirmed numerically.

• Robustness to Coherent Noise:

Explicit error bounds for noise in state preparation, unitary representation, and group element assignment show that covariant kernels retain finite variance and distinguishing power even with substantial noise—variance remains robust up to high error rates (Henderson et al., 17 Sep 2025).

This combination of symmetry compliance, stable landscape, and noise resilience positions covariant quantum kernels as leading candidates for scalable and near-term quantum machine learning tasks, as further reinforced by large-scale experimental demonstrations (e.g., 156-qubit SVC performance on IBM devices) (Agliardi et al., 10 Dec 2024).

5. Covariant Kernels in Quantum Field Theory and Quantum Geometry

Covariant quantum kernels also appear as Green’s functions, propagators, and quantum derivatives that respect background gauge or general coordinate symmetry:

• Covariant Perturbation Expansion:

In quantum field theory, the off-diagonal heat kernel $K(T; x,y)$ encodes the transition amplitude between points $x$ and $y$. Covariant perturbation theory expresses $K(T)$ as an asymptotic expansion that intertwines interaction and geometric symmetries, yielding nonlocal effective actions and accurate scattering amplitudes (Gou et al., 2016).

• Quantum Covariant Derivative:

Geometrically, the quantum covariant derivative $\hat{\nabla}$ unifies coordinate and gauge covariance for parameterized quantum states $|\psi(x)\rangle$. It is constructed to be compatible with the quantum geometric tensor (quantum metric and Berry curvature),

$$\hat{\nabla}_{X}|v\rangle = |D\psi\rangle\bigl[(\partial v/\partial x)+\Upsilon\, v\bigr]\, X$$

enabling adiabatic perturbation theory that is manifestly invariant under gauge and coordinate transformations (Requist, 2022).

Such tools enforce the preservation of physical and information-theoretic structures under dynamical evolution and deformation of quantum states or processes.

6. Zero-Error Information Theory and Quantum Combinatorics

Symmetry-respecting (“covariant”) quantum kernels are critical in modern combinatorial approaches to quantum communication:

• Covariant Quantum Relations and Graphs:

The kernel associated to a covariant channel $f:A\to B$ is determined by a support projection $s(\tilde{f}) \in \operatorname{End}(Y^*\otimes X)$, with the necessary and sufficient condition for a covariant kernel to underlie a channel given by invertibility of the partial trace:

$$\operatorname{Tr}_X[\tilde{p}] \text{ invertible} \implies \text{exists covariant channel}$$

(Verdon, 2023).

• Classifying Zero-Error Source-Channel Coding:

Covariant homomorphisms between confusability graphs provide a categorical framework for zero-error quantum communication with symmetry constraints. For compact (quantum) group $G$, every covariant $G$-graph can be obtained as the confusability graph of some covariant channel.

This categorical/graph-theoretic perspective connects operational channel functions with the algebra of covariant kernels, opening pathways to Lovász-type invariants, SDP relaxations, and robust coding schemes under symmetry.

7. Physical and Mathematical Implications

Covariant quantum kernels unify deep structural requirements (symmetry, invariance, compatibility with physical laws) with practical objectives (robustness, expressivity, and trainability):

• Trainability and Quantum Advantage:

By adapting the kernel construction to group and coset structures, quantum learning models become simultaneously expressive, trainable, and robust, and can be tailored to classically hard (e.g., discrete logarithm) learning problems (Henderson et al., 17 Sep 2025, Glick et al., 2021).

• Experimental Realization and Data Processing:

Large-scale demonstrations (including on 156-qubit devices) validate that, with error mitigation (e.g., Bit Flip Tolerance), covariant quantum kernels can operate in the ‘utility’ scale and match classical SVM accuracy, even for real-world data with approximate group structure (Agliardi et al., 10 Dec 2024, Sabarad et al., 12 Dec 2024).

• Theoretical Extensions:

The methods are extensible to higher-order kernel constructions, non-Abelian group symmetries, tensor network frameworks, and noncommutative geometric or categorical quantum channels. Open problems include time-efficient embedding of all covariant kernels as quantum feature maps and the extension to infinite-dimensional or noncompact group settings (Gil-Fuster et al., 2023).

Covariant quantum kernels thus provide a unifying mathematical and operational foundation for quantized, symmetry-structured data processing, physical evolution, measurement, and machine learning models across quantum sciences.