Fidelity-Based Quantum Kernels in QML

• Fidelity-based quantum kernels are defined as the squared overlap or Hilbert–Schmidt inner product between quantum states produced by data-dependent feature maps.
• They serve as precomputed similarity measures for classical learners like support vector machines, balancing quantum expressivity with convex optimization.
• Bandwidth tuning and local kernel variants mitigate issues such as exponential concentration and affect the trade-off between quantum advantages and classical emulation.

Fidelity-based quantum kernels are quantum kernel functions that quantify similarity by the overlap of quantum states produced by a data-dependent feature map. In the standard pure-state setting, classical data $x$ are embedded as $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$ and the kernel is the squared overlap $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$; for mixed states, some works use the Uhlmann fidelity, while others instantiate the kernel as the Hilbert–Schmidt inner product $\operatorname{Tr}[\rho(x)\rho(x')]$, which coincides with the squared overlap on pure states (Tanner et al., 9 Apr 2026, Schnabel et al., 2024). These kernels are used as precomputed similarities for classical learners such as support vector machines and kernel ridge regression, and they occupy a central position in non-variational quantum machine learning because they separate quantum feature embedding from classical convex optimization (Tanner et al., 9 Apr 2026).

1. Formal definition and kernel-theoretic setting

The basic object is a data-encoding map $x \mapsto \rho(x)$, usually realized by a unitary $U(x)$ acting on a reference state. For pure states,

$$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$

For mixed states, one common definition is the Uhlmann fidelity

$$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$

whereas several benchmarking and implementation studies instead use the Hilbert–Schmidt kernel

$$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$

explicitly noting that this equals the pure-state fidelity kernel when $\rho(x)=|\psi(x)\rangle\langle\psi(x)|$ (Martínez-Peña et al., 2023, Schnabel et al., 2024, Tanner et al., 9 Apr 2026).

For a dataset $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$0, the kernel matrix is $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$1. In the Hilbert–Schmidt formulation, the Gram matrix is PSD because it derives from an inner product (Schnabel et al., 2024). The resulting matrix is passed to a classical learner. In the SVM setting, one uses the standard soft-margin dual

$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$2

with constraints $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$3 and $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$4, and prediction takes the form

$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$5

(Martínez-Peña et al., 2023). In kernel ridge regression, the predictor is $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$6 (Schnabel et al., 2024, Tanner et al., 9 Apr 2026).

A broader structural perspective is provided by the unified framework of trace-induced quantum kernels, where the global fidelity kernel

$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$7

appears as a special case of a generalized trace-induced kernel built from positive combinations of “Lego” kernels. In that framework, global fidelity kernels, subsystem projected kernels, and newly introduced $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$8-body local projected kernels are all instances of the same trace-induced construction (Gan et al., 2023).

2. Feature maps and kernel estimation protocols

Fidelity-based kernels are defined by the choice of quantum feature map. Large-scale benchmarking studies have examined nine feature-map ansätze, including separable circuits such as SeparableRxEncoding and ZFeatureMap, and entangling constructions such as YZ_CX_EncodingCircuit, HZY_CZ_EncodingCircuit, HardwareEfficientEmbeddingRx, HighDimEncodingCircuit, ZZFeatureMap, ParamZFeatureMap, and ChebyshevPQC. In that benchmark, depth $|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}$9, qubit number up to $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$0, and an embedding-width parameter $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$1 were tuned as hyperparameters (Schnabel et al., 2024).

Several hardware-compatible estimators are used in practice. For pure states, the Loschmidt echo test prepares $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$2, applies $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$3, and estimates the all-zero probability; the SWAP test uses two state registers and an ancilla; the Hadamard test estimates real and imaginary parts of $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$4; and randomized measurements or classical shadows can be used for projected or reduced-state variants (Tanner et al., 9 Apr 2026). These procedures differ in qubit count, circuit depth, and shot variance, but they all realize the same basic task: overlap estimation.

A concrete analog implementation appears on trapped-ion simulation platforms. There the feature map is generated by time evolution under an input-dependent transverse-field Ising Hamiltonian

$k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$5

with long-range XX interactions realized via Mølmer–Sørensen entangling interactions and site-dependent Stark-shift $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$6 fields. The overlap is estimated by initializing $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$7, applying $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$8 followed by $k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2$9, and measuring the frequency of the $\operatorname{Tr}[\rho(x)\rho(x')]$0 outcome (Martínez-Peña et al., 2023). In that setting, shot noise scales as $\operatorname{Tr}[\rho(x)\rho(x')]$1 for $\operatorname{Tr}[\rho(x)\rho(x')]$2 repetitions, and total runtime scales with the number of pairs, number of ions, and shots (Martínez-Peña et al., 2023).

3. Inductive bias, bandwidth control, and exponential concentration

The dominant theoretical issue for fidelity-based kernels is exponential concentration. Multiple studies describe the same pathology in slightly different language: spectrum flattening, vanishing similarity, or concentration toward an identity-like Gram matrix. In the fidelity-kernel setting, increasing qubit count or circuit expressivity can make off-diagonal overlaps shrink rapidly, so that kernel values become nearly constant or nearly diagonal, harming trainability and generalization (Slattery et al., 2022, Tanner et al., 9 Apr 2026). A closely related large-scale study reports that for global fidelity kernels on tabular data, the median and upper-tail off-diagonal entries decrease with dimension while effective rank declines (Zendejas-Morales et al., 18 Feb 2026).

Bandwidth tuning is the principal mitigation strategy. In the trapped-ion implementation, bandwidth is controlled by physical hyperparameters $\operatorname{Tr}[\rho(x)\rho(x')]$3, with $\operatorname{Tr}[\rho(x)\rho(x')]$4 and $\operatorname{Tr}[\rho(x)\rho(x')]$5 determining how sensitively the feature map depends on the input; larger values yield higher-frequency feature maps, whereas smaller values act as a smoother kernel (Martínez-Peña et al., 2023). In broader benchmarks, the analogous control variable is the embedding width $\operatorname{Tr}[\rho(x)\rho(x')]$6, and statistically significant correlations between performance and $\operatorname{Tr}[\rho(x)\rho(x')]$7 were observed across classification and regression tasks (Schnabel et al., 2024). In hyperspectral classification, the same role is played by a global scaling parameter $\operatorname{Tr}[\rho(x)\rho(x')]$8 in $\operatorname{Tr}[\rho(x)\rho(x')]$9, with smaller $x \mapsto \rho(x)$0 increasing typical overlaps and reducing concentration (Delilbasic et al., 17 May 2026).

This tuning is usually interpreted as inductive-bias control. Large Hilbert spaces can enlarge expressivity while destabilizing generalization, and bandwidth restricts the effective function class or spectral bias (Martínez-Peña et al., 2023, Schnabel et al., 2024). Empirically, validation-optimal regions often occur in an intermediate regime: enough nonlinearity to separate data, but not so much that the Gram matrix becomes ill-conditioned or concentrated (Martínez-Peña et al., 2023, Schnabel et al., 2024).

A central controversy follows from this same mechanism. Numerical studies on classical data argue that once fidelity kernels are tuned strongly enough to avoid concentration, they become close to classical kernels. One study shows that keeping the largest eigenvalue from decaying with qubit count makes tuned fidelity kernels well-approximated by classical comparators, with geometric difference dropping below the regime needed by a necessary condition for quantum advantage (Slattery et al., 2022). A later study sharpened this claim by showing that bandwidth-tuned fidelity kernels closely resemble RBF kernels and, at small optimal bandwidths, low-order Taylor approximations of RBF kernels (Flórez-Ablan et al., 7 Mar 2025). This does not amount to a universal impossibility result, but it frames the main debate: the same tuning that restores learnability may also reduce classical–quantum separation.

4. Variants, localizations, and responses to concentration

One response has been to reinterpret fidelity kernels within broader kernel families. In the trace-induced framework, projected kernels arise by restricting the operator set or subsystem support, and $x \mapsto \rho(x)$1-body local projected kernels provide a systematic way to bias learning toward lower-order correlations while reducing measurement costs relative to the global fidelity kernel (Gan et al., 2023). This suggests that the standard global fidelity kernel is only one point in a larger design space.

A second response localizes the overlap itself. Patch-wise local kernels average subsystem similarities,

$x \mapsto \rho(x)$2

while multi-scale kernels mix several patch granularities,

$x \mapsto \rho(x)$3

In Qiskit-based experiments, these constructions consistently mitigated concentration and produced larger off-diagonal percentiles and higher effective rank than the global fidelity baseline, although accuracy gains were dataset-dependent rather than universal (Zendejas-Morales et al., 18 Feb 2026).

A third line of work studies “benign overfitting” constructions. Local-global quantum kernels combine a local subsystem kernel with a global full-system fidelity kernel,

$x \mapsto \rho(x)$4

and under separable structure reduce to a “spiky-smooth” form $x \mapsto \rho(x)$5. Numerical experiments indicate that increasing the spiky global exponent can transform catastrophic overfitting into benign overfitting while retaining interpolation (Tomasi et al., 21 Mar 2025). This suggests that controlled mixtures of local and global fidelity structure can regularize the spectrum without discarding overlap information altogether.

A fourth response alters the classical post-processing while reusing the same quantum measurements. The Hamming quantum kernel uses the measurement statistics of the overlap circuit,

$x \mapsto \rho(x)$6

rather than the single all-zero probability used by the fidelity kernel. Simulations from $x \mapsto \rho(x)$7 to $x \mapsto \rho(x)$8 qubits found that it outperformed the fidelity quantum kernel whenever $x \mapsto \rho(x)$9 or more qubits were used, without requiring additional quantum resources (Agnihotri et al., 29 May 2026). Another, more radical, alternative replaces fidelity entirely: the anti-symmetric-logarithmic-derivative quantum Fisher kernel was shown analytically and numerically to avoid the vanishing similarity issue for alternating layered ansätze, while fidelity kernels did not (Suzuki et al., 2022).

5. Empirical behavior across tasks and platforms

Benchmarking work now spans synthetic classification, regression, many-body physics, hardware execution, and large-scale classical simulation. A comprehensive study trained and optimized over $U(x)$0 models across five dataset families and $U(x)$1 datasets, comparing fidelity and projected quantum kernels in QSVM and kernel ridge regression. Up to $U(x)$2 qubits, fidelity and projected kernels delivered broadly comparable test performance; projected kernels showed only a slight overall edge, with a clear difference appearing only for the most complex regression instance QFMNIST with $U(x)$3 (Schnabel et al., 2024).

On trapped-ion analog simulations, fidelity kernels achieved competitive binary-classification performance with few qubits. For the Circles dataset, the trapped-ion kernel reached test accuracy $U(x)$4 at $U(x)$5, matching the optimized classical RBF comparator; for Moons, $U(x)$6 and $U(x)$7 both reached $U(x)$8; for Ad Hoc, $U(x)$9 achieved $$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$0 in the noiseless case, and the best noisy result was $$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$1, exceeding the classical RBF’s $$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$2 in that setting (Martínez-Peña et al., 2023). These experiments also reported robustness under depolarizing and statistical noise, with some cases slightly improving under small depolarization (Martínez-Peña et al., 2023).

In Gaussian process regression for molecular potential energy surfaces, fidelity kernels were optimized by a compositional circuit search guided by a BIC-like criterion. On six-dimensional PES benchmarks, the best quantum models achieved average interpolation errors of $$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$3 for H$$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$4O$$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$5, $$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$6 for H$$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$7CO, and $$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$8 for HNO$$|\psi(x)\rangle = U(x)|0\rangle^{\otimes n}, \qquad k(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = |\langle 0|U^\dagger(x)U(x')|0\rangle|^2.$$9, but optimized classical compositional kernels converged to the same errors (Guo et al., 2024). This supports the narrower claim that fidelity kernels can match strong classical kernels in regression without exceeding them on those tasks.

In quantum many-body applications, fidelity kernels are often replaced by fidelity per site. For the transverse-field Ising chain, a QSVM built from fidelity-per-site kernels learned the critical point and extracted $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$0 close to the exact value $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$1 (Sancho-Lorente et al., 2021). A later resource-scaling study on Ising, XY, XX, and XXZ models linked shot complexity to symmetry: moving from $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$2-symmetric Ising/XY regimes to $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$3-symmetric XX and XXZ regimes increased kernel concentration and therefore shot costs under finite-shot bounds (Ali et al., 18 Mar 2026).

Hardware-scale demonstrations have also appeared. Covariant fidelity kernels combined with centered alignment and Bit Flip Tolerance were run on IBM hardware up to $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$4 qubits. On real Vehicle-to-Grid data, mitigated accuracies at $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$5 qubits reached $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$6, compared to $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$7 without BFT; on synthetic union-of-subspaces data at $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$8 qubits, mitigated accuracy reached $$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2,$$9, compared to $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$0 for classical models and $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$1 for unmitigated quantum kernels (Agliardi et al., 2024). In hyperspectral classification, tensor-network contraction and GPU acceleration enabled fidelity-kernel evaluation on hundreds of spectral bands; on four $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$2-band Indian Pines splits, the quantum model achieved $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$3 accuracy versus $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$4 for the standard RBF kernel, and on a four-class variant it reached $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$5 (Delilbasic et al., 17 May 2026).

6. Limitations, misconceptions, and open directions

A recurrent misconception is that fidelity-based kernels are defined uniquely. For pure states, the squared overlap, Hilbert–Schmidt inner product, and projector-based overlap estimators coincide. For mixed states, however, papers distinguish Uhlmann fidelity from $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$6, and this distinction matters for noise, metric properties, and implementation (Tanner et al., 9 Apr 2026, Schnabel et al., 2024). A second misconception is that entanglement is always essential. Large-scale benchmarking found that separable circuits such as SeparableRxEncoding and ZFeatureMap often remained competitive, and entanglement was not uniformly decisive across classical-data tasks (Schnabel et al., 2024).

The principal limitations are computational and statistical. Global fidelity kernels require $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$7 pairwise overlaps, and finite-shot estimation can become prohibitive as qubit count rises or overlaps concentrate (Martínez-Peña et al., 2023, Tanner et al., 9 Apr 2026). Hardware constraints include coherence limits, immediate Hamiltonian sign changes in analog protocols, routing overhead, and readout error (Martínez-Peña et al., 2023, Agliardi et al., 2024). On the theory side, concentration, ill-conditioning, and dequantization remain the central obstacles to claims of practical quantum advantage on classical data (Slattery et al., 2022, Tanner et al., 9 Apr 2026, Flórez-Ablan et al., 7 Mar 2025).

Current research directions therefore cluster around controlled inductive bias rather than unconstrained expressivity. The literature points to bandwidth optimization, local and multi-scale constructions, projected and $$k^{\mathrm{FQK}}(x,x')=\operatorname{Tr}[\rho(x)\rho(x')],$$8-body kernels, alternative overlap estimators, task-aware feature-map design, and hardware-aware screening as the most active routes forward (Gan et al., 2023, Zendejas-Morales et al., 18 Feb 2026, Liu et al., 26 Jun 2025). A plausible implication is that the long-term role of fidelity-based quantum kernels may be less as a universal replacement for classical kernels than as a structured overlap primitive whose usefulness depends on matching feature map, measurement scheme, and task geometry.