Quantum Evolution Kernel Overview

• Quantum Evolution Kernel (QEK) is a method that uses quantum evolution to embed data into quantum states and compute similarity measures for classical models.
• It leverages controlled quantum dynamics like Hamiltonian evolution and state interference to generate robust feature maps for tasks in classification and regression.
• Experimental implementations across NMR, neutral-atom arrays, and ion-trap circuits demonstrate QEK’s practical effectiveness and its unique inductive bias.

Searching arXiv for recent and foundational papers on Quantum Evolution Kernel (QEK). Quantum Evolution Kernel (QEK) denotes a class of quantum kernel constructions in which classical inputs are encoded through quantum dynamics, and similarity is extracted from either state overlaps or measurement distributions generated by the evolution. In quantum machine learning, this idea appears in several experimentally and numerically studied forms: input-dependent Hamiltonian evolution in solid-state NMR, graph-dependent evolution on programmable neutral-atom arrays, and feature-map circuits whose kernel is the return probability $|\langle 0|U^\dagger(x')U(x)|0\rangle|^2$. The resulting kernel matrix is then used by classical kernel methods such as SVMs or Gaussian-process models rather than by explicitly constructing the underlying feature vectors (Kusumoto et al., 2019, Henry et al., 2021, Adams et al., 16 Jan 2026).

1. Definition and scope

Within the kernel-method view of quantum machine learning, data are embedded into quantum states or density operators, and the kernel is the Hilbert-space similarity between those embeddings. A general formulation is

$$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$

with predictors of the form $f(x)=\operatorname{tr}(M\rho(x))$; in this sense, supervised quantum machine learning models can be rephrased as kernel methods (Schuld, 2021). A QEK is a specialization in which the embedding map is generated by quantum evolution, typically through an input-dependent unitary $U(x)$ or through graph-dependent Hamiltonian dynamics (Kusumoto et al., 2019, Henry et al., 2021).

The terminology is not fully uniform across the literature. In some QSVM papers, the acronym “QEK” denotes “Quantum Embedding Kernel,” i.e., a trainable quantum embedding used as a kernel baseline rather than a kernel built explicitly from physical time evolution (Phalak et al., 2024). Outside quantum machine learning, “quantum evolution kernel” can denote conceptually distinct objects, such as the memory kernel in generalized semi-Markov quantum evolution or the evolution kernel for transversity operators in QCD (Chruściński et al., 2017, Manashov et al., 2024). In current QML usage, however, QEK most often refers to kernels whose feature map is instantiated by quantum dynamics.

2. Mathematical constructions

A foundational experimental realization is the NMR kernel introduced in “Experimental quantum kernel machine learning with nuclear spins in a solid” (Kusumoto et al., 2019). There the kernel is

$$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$

with

$$A(\bm{x})=U(\bm{x})I_zU^\dagger(\bm{x}), \qquad I_z=\sum_{\mu=1}^n I_{z,\mu},$$

and

$U(\bm{x})=e^{-iH(x_D)\tau}\cdots e^{-iH(x_1)\tau}.$

Each slice Hamiltonian is

$$H(x_j)=e^{-ix_j I_z}\sum_{\mu<\nu}d_{\mu\nu}(I_{y,\mu}I_{y,\nu}-I_{x,\mu}I_{x,\nu})e^{ix_j I_z}.$$

This defines a feature map by input-dependent Hamiltonian evolution, with the kernel obtained from interference between two such evolutions (Kusumoto et al., 2019).

A second formulation, emphasized in hyperparameter studies of Hamiltonian-evolution feature maps, encodes a datapoint $\mathbf{x}_i$ as

$$\ket{\mathbf{x}_i} = \left( \prod_{j=1}^n \exp\left(-i\frac{t}{T}x_{ij}H_j^{XYZ}\right) \right)^T \bigotimes_{j=1}^{n+1}\ket{\psi_j},$$

where $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$0 is the total evolution time and $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$1 the number of Trotter steps. The associated kernel can be the pure-state fidelity $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$2, the mixed-state inner product $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$3, or projected kernels based on reduced density matrices (Egginger et al., 2023).

Graph QEKs use a different but closely related construction. Rather than measuring a direct overlap, they evolve a graph-dependent quantum system, convert measurement outcomes into a probability distribution $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$4, and define the kernel through Jensen–Shannon divergence: $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$5 This is the kernel introduced for graph-structured data on programmable qubit arrays and later extended to attributed graphs on neutral-atom platforms (Henry et al., 2021, Djellabi et al., 11 Sep 2025). The unifying theme across these constructions is that quantum evolution generates the feature representation, while kernel evaluation avoids explicit reconstruction of the full feature space.

3. Encoding, evolution, and measurement protocols

In the solid-state NMR implementation, the protocol begins from the equilibrium ensemble state

$$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$6

with small polarization $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$7. One then applies $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$8, followed by $$\kappa(x,x')=\operatorname{tr}(\rho(x')\rho(x))=|\langle \phi(x')|\phi(x)\rangle|^2,$$9, and finally measures the ensemble-averaged value of $f(x)=\operatorname{tr}(M\rho(x))$0. The measured signal is proportional to

$f(x)=\operatorname{tr}(M\rho(x))$1

which is closely related to a Loschmidt echo and estimates the kernel through interference of two quantum trajectories. The hardware is solid-state NMR on polycrystalline adamantane, using proton spins with dynamics involving upwards of 10 spins (Kusumoto et al., 2019).

In neutral-atom graph QEKs, each graph node is mapped to a qubit or atom, and the graph topology is encoded into the interaction structure of the Hamiltonian. A representative formulation alternates a graph Hamiltonian,

$f(x)=\operatorname{tr}(M\rho(x))$2

with a mixing Hamiltonian,

$f(x)=\operatorname{tr}(M\rho(x))$3

Repeated measurements in the computational basis are then postprocessed into histograms, such as energy distributions or excitation-number distributions, from which the kernel is computed via Jensen–Shannon divergence (Henry et al., 2021, Giusto et al., 6 May 2026). On QuEra’s Aquila platform, physical constraints require embeddings compatible with unit disk graph connectivity and feasible atom placement (Giusto et al., 6 May 2026).

A third experimentally studied workflow appears in autonomous materials science. There the kernel is

$f(x)=\operatorname{tr}(M\rho(x))$4

implemented with a hardware-efficient circuit using Hadamard gates, $f(x)=\operatorname{tr}(M\rho(x))$5 gates, and three-axis rotation gates. For x-ray diffraction data, 25 qubits encode 150 XRD features, and each kernel entry is estimated on IonQ’s Aria hardware from 1,024 circuit repetitions (Adams et al., 16 Jan 2026). This suggests that QEK methodology is not tied to one physical architecture; what remains invariant is the use of controlled quantum evolution to generate a similarity measure for downstream classical learning.

4. Expressivity, hyperparameters, and feature-map design

In Hamiltonian-evolution kernels, evolution time is a central control parameter. In the NMR study, the kernel becomes sharper as the evolution time or pulse cycle number increases, and the performance of the trained model tends to increase with longer evolution time for certain tasks (Kusumoto et al., 2019). A broader hyperparameter study across 11 datasets likewise identifies the total evolution time $f(x)=\operatorname{tr}(M\rho(x))$6 as the dominant hyperparameter, while the RBF bandwidth $f(x)=\operatorname{tr}(M\rho(x))$7 and the choice between distance and inner-product kernels are also highly influential; by contrast, the number of Trotter steps $f(x)=\operatorname{tr}(M\rho(x))$8 and the reduced-density-matrix size $f(x)=\operatorname{tr}(M\rho(x))$9 have comparatively minor effects in the studied regime (Egginger et al., 2023).

Projected kernels are introduced partly to mitigate exponential concentration in large Hilbert spaces. In that setting, the projected inner-product kernel

$U(x)$0

and the projected distance kernel

$U(x)$1

provide alternatives to full-state fidelities (Egginger et al., 2023). Empirically, the distance kernel tends to outperform the inner-product kernel when bandwidth is well tuned, but configurations with maximal geometric difference from a classical kernel do not necessarily coincide with maximal classification accuracy (Egginger et al., 2023).

For graph QEKs on neutral-atom systems, expressiveness can be increased by enriching the graph-to-Hamiltonian map. One extension embeds node features into local detuning fields while retaining edge information in atomic positions and couplings. The local-detuning version is reported to be genuinely more expressive than the global-detuning version for attributed graphs, and the rank of the Gram matrix increases under local detuning. Pooling kernels across multiple time steps further increases expressiveness and improves F1 scores by 2–4 percentage points in the reported experiments (Djellabi et al., 11 Sep 2025). A plausible implication is that the expressive power of QEKs is controlled at least as much by the structure of the physical encoding and the observables as by circuit depth alone.

5. Experimental performance and application domains

The first large-system experimental demonstration of QEK-style learning in the provided corpus is the NMR study using proton spins in a solid (Kusumoto et al., 2019). The reported tasks are one-dimensional regression on sinusoids and sinc functions, and two-dimensional classification on circle and moon datasets. Experimentally, regression accuracy increases with longer evolution time, or equivalently with a larger number of spins involved in the dynamics, up to the point allowed by signal-to-noise. For classification, no clear dependence on evolution time is observed. Spectral analysis indicates dynamics involving at least 10 spins, and numerical simulations were performed up to 20 spins (Kusumoto et al., 2019).

Graph classification is another major application. The original graph QEK paper reports that QEK matched or outperformed standard graph kernels such as graphlet subsampling and random walk kernels on typical benchmark datasets, and that increasing the number of pulse layers improved classification accuracy (Henry et al., 2021). A later hardware-oriented study used the 256-qubit Aquila neutral-atom simulator on the PROTEINS dataset and reported the following results (Giusto et al., 6 May 2026):

Setting	QEK result	Classical reference
PROTEINS12, Aquila, $U(x)$2	F1 $U(x)$3, Accuracy $U(x)$4	SPK F1 $U(x)$5, Accuracy $U(x)$6
PROTEINS256, Aquila, $U(x)$7	F1 $U(x)$8, Accuracy $U(x)$9	SPK F1 $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$0, Accuracy $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$1

These results were presented as evidence that the method remains effective even in the case of a noisy quantum simulator (Giusto et al., 6 May 2026).

In autonomous materials science, quantum kernels were compared with classical kernels for sequential phase-space navigation using x-ray diffraction patterns from an Fe-Ga-Pd ternary composition spread library (Adams et al., 16 Jan 2026). The reported geometric differences are $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$2 for the simulated quantum kernel and $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$3 for the measured quantum kernel, while the model complexities are $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$4 and $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$5, compared with $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$6 for the classical RBF kernel and $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$7 for cosine similarity. Empirically, for $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$8–15 training examples, the quantum kernel outperforms the RBF kernel; for the natural labeling setup, the cosine similarity kernel performed surprisingly well and could exceed the quantum kernel, whereas synthetic label engineering could substantially favor the quantum kernel (Adams et al., 16 Jan 2026). This establishes a recurrent theme in QEK research: empirical advantage is strongly dependent on how well the kernel’s inductive bias matches the data.

6. Limitations, misconceptions, and ongoing directions

Several limitations recur across QEK implementations. In NMR, the polarization is tiny, $$k_{\mathrm{NMR}}(\bm{x}_i,\bm{x}_j) = \bm{\phi}_{\mathrm{NMR}}(\bm{x}_i)^T \bm{\phi}_{\mathrm{NMR}}(\bm{x}_j) = \operatorname{Tr}[A(\bm{x}_i)A(\bm{x}_j)],$$9, and longer evolutions induce more decoherence, limiting the usable signal despite the tendency of sharper kernels to improve some regression tasks (Kusumoto et al., 2019). In Hamiltonian-evolution kernels on classical datasets, optimal quantum configurations generally match but seldom exceed the best classical baselines, and hyperparameters that maximize geometric difference do not necessarily maximize accuracy (Egginger et al., 2023). In graph-QEK hardware studies, current performance gains over classical baselines are typically moderate, and practical deployment is constrained by graph embeddability, hardware geometry, shot costs, and noise (Giusto et al., 6 May 2026, Djellabi et al., 11 Sep 2025).

A common misconception is that more quantum complexity automatically yields better learning. The reported evidence is more qualified. Longer evolution can improve regression in NMR, but classification may plateau; larger projected subsystems can slightly improve accuracy yet may worsen concentration; and greater expressiveness through local observables or local detuning does not always yield the best downstream task performance (Kusumoto et al., 2019, Egginger et al., 2023, Djellabi et al., 11 Sep 2025). Another misconception is terminological: in part of the literature “QEK” refers to Quantum Embedding Kernel rather than Quantum Evolution Kernel, and in still other fields “evolution kernel” refers to non-ML objects such as memory kernels in open quantum dynamics or renormalization kernels in QCD (Phalak et al., 2024, Chruściński et al., 2017).

Current directions indicate two complementary trajectories. One is methodological: projected kernels, local-observable variants such as GDQC, multi-time pooling, and structure-optimized quantum kernels supported by software such as QuASK, which implements projected kernels, trainable kernels, and structure-optimized kernels via simulated annealing and genetic algorithms (Marcantonio et al., 2022, Djellabi et al., 11 Sep 2025). The other is hardware-oriented: Kerr-based kernels, multimode bulk acoustic resonators, and resource-efficient quantum kernels all aim to retain non-classical expressivity while reducing circuit width, depth, or entangling-gate counts (Liu et al., 2022, Frink et al., 12 Dec 2025, Singh et al., 4 Jul 2025). This suggests that the long-term development of QEKs is likely to depend not only on proofs of classical hardness, but also on physically aligned encodings, carefully tuned observables, and application domains in which the induced quantum similarity measure offers a genuinely useful inductive bias.