Projected Quantum Kernel (PQK)

• PQK is a hybrid quantum-classical kernel that projects quantum state measurements onto low-dimensional, classically accessible features for practical machine learning.
• It achieves computational efficiency by reducing exponential state overlap measurements to scalable local tomography, making it suitable for NISQ devices.
• By combining quantum feature maps with classical positive-definite kernels, PQK balances expressivity with resource constraints through careful hyperparameter tuning.

A projected quantum kernel (PQK) is a family of hybrid quantum-classical kernels in which classical inputs are first embedded into quantum states via a parameterized quantum circuit, then “projected” onto a low-dimensional set of classically accessible features—typically reduced density matrices (RDMs) of small subsystems—and finally paired with a conventional positive definite kernel to form a similarity measure suitable for machine learning tasks such as classification and regression. This construction aims to combine the expressivity of quantum feature maps with practical tractability on near-term (NISQ) quantum hardware, addressing limitations of full-fidelity quantum kernels such as exponential concentration and resource scaling.

1. Mathematical Construction and Core Definitions

The PQK framework begins with a quantum feature map

$$x \in \mathbb{R}^d \mapsto |\psi(x)\rangle = U(x)\,|0\rangle^{\otimes n}$$

where $U(x)$ is a data-dependent parameterized quantum circuit on $n$ qubits. The global quantum kernel or fidelity kernel is the squared overlap

$K_F(x,x') = |\langle\psi(x) | \psi(x')\rangle|^2.$

Projected quantum kernels replace the global overlap with a comparison of local (few-qubit) features. The canonical construction is to compute reduced density matrices (RDMs) for a subset $S$ of qubits: $$\rho_S(x) = \operatorname{Tr}_{\overline{S}} \left[ |\psi(x)\rangle\langle\psi(x)| \right].$$ Two canonical PQK forms are:

• Hilbert–Schmidt projected kernel:

$$K_{PQK}(x,x') = \mathrm{Tr}\left[\rho_S(x)\, \rho_S(x')\right].$$

• Gaussian projected kernel (RBF style):

$$K_{PQK}(x,x') = \exp\bigg(-\gamma\sum_{\kappa} \|\rho_{S_\kappa}(x) - \rho_{S_\kappa}(x')\|_F^2\bigg).$$

Here, $\{S_\kappa\}$ denotes the collection of subsystems (e.g., all single-qubit RDMs), and $\gamma$ is a kernel bandwidth parameter. In practice, the feature vector for classical processing is $U(x)$0 for Pauli operators $U(x)$1 and subsystems $U(x)$2 (Egginger et al., 2023, Schnabel et al., 2024, Miyabe et al., 2023, d'Amore et al., 20 May 2025, Tanner et al., 9 Apr 2026).

By combining projective measurements and a classical outer kernel,

$U(x)$3

the PQK bridges the quantum Hilbert space embedding and classical kernel theory.

2. Resource Efficiency and Scalability

Resource efficiency is a chief motivation for PQK approaches. Measuring global fidelities or full quantum state overlaps rapidly becomes intractable due to exponential scaling ($U(x)$4). By contrast, PQKs only require state tomography or expectation values on small (usually one- or two-qubit) subsystems, giving $U(x)$5–$U(x)$6 measurement scaling depending on the projection scheme.

As demonstrated in hardware experiments (up to 61 qubits (Utro et al., 30 Jul 2025)) or noisy simulators (up to 20 qubits (Miyabe et al., 2023)), PQK allows classification tasks and other kernel algorithms to operate within the limitations of coherence and error rates typical for current NISQ devices (Schnabel et al., 2024, Tanner et al., 9 Apr 2026). The linear-in-$U(x)$7 cost for local measurements stands in contrast to the quadratic or exponential cost for direct Gram matrix construction via fidelity overlaps.

Benchmarks across machine learning tasks (classification, regression, bandit optimization) confirm that PQK can achieve comparable accuracy to classical baselines, and in some regimes modestly surpass them, while remaining computationally tractable (Egginger et al., 2023, Miyabe et al., 2023, Schnabel et al., 2024, Perciavalle et al., 18 Apr 2025, Alagiyawanna et al., 6 Jan 2026, d'Amore et al., 20 May 2025, Utro et al., 30 Jul 2025).

3. Hyperparameters, Variants, and Model Selection

PQK performance is highly sensitive to several key hyperparameters:

• Feature map parameters: Circuit depth, entangling structure, choice of data encoding, and bandwidth (scaling of features) are decisive for expressivity and avoidance of exponential concentration (Egginger et al., 2023, Schnabel et al., 2024, Tanner et al., 9 Apr 2026).
• Projection dimension: The number and nature of subsystems (number of traced-out qubits or size of blocks) control the tradeoff between information retention and susceptibility to concentration (Egginger et al., 2023, Suzuki et al., 2023, d'Amore et al., 20 May 2025).
• Outer kernel: Choice of the classical positive-definite function (e.g., RBF, Matérn, polynomial) and its parameters directly impacts the model capacity and regularization (Schnabel et al., 2024, Miyabe et al., 2023).
• Regularization: Classical kernel parameters such as SVM $U(x)$8 or ridge regression penalty must be tuned jointly with quantum-specific hyperparameters to avoid overfitting or ill-conditioning (Egginger et al., 2023, Schnabel et al., 2024).

Systematic grid or fANOVA hyperparameter studies (Egginger et al., 2023, Schnabel et al., 2024) indicate that the feature map scaling (“quantum bandwidth”) is typically the dominant lever, with projection dimension and outer kernel bandwidth as secondary but critical factors. Circuit depth influences expressivity and Fourier components encoded, but excessive depth can induce concentration and/or "barren plateau" issues (Suzuki et al., 2023, Tanner et al., 9 Apr 2026).

A summary of common PQK parameterizations is presented below:

Hyperparameter	Example Choices	Primary Effect
Projection size $U(x)$9	1–2 qubits (RDM), up to $n$0	Tradeoff: generalization vs variance
Feature map type	RotX, 3D, ZZ, Hamiltonian-Evolution	Encoded correlation structure
Outer kernel	RBF ($n$1), polynomial, Matérn	Smoothness, regularization
SVM/$n$2/ridge-$n$3	$n$4 to $n$5	Overfit/underfit control

Recommended pipelines perform guided line searches or joint tuning over the most sensitive variables, ensuring computationally efficient model selection (Egginger et al., 2023).

4. Empirical Performance and Applications

PQK-based kernel methods have been validated on a range of classical and physics-motivated learning tasks:

• Classical datasets: On UCI and real-world IoT data, PQK-SVMs with shallow circuit embeddings can match or slightly exceed classical RBF kernel accuracy, although the advantage over classical models is often marginal without careful hyperparameter tuning (d'Amore et al., 20 May 2025, Schnabel et al., 2024).
• Physical prediction tasks: For regression problems in quantum chemistry and many-body physics, PQK-based support vector regressors outperform linear models and approach RBF kernel performance when measured at optimal times, exploiting the underlying quantum dynamical structure (Perciavalle et al., 18 Apr 2025).
• Data-scarce regimes and feature extraction: In small-data scenarios (e.g., binary classification on limited MNIST/CIFAR samples or complex bioinformatics data), PQK-enhanced neural networks—especially CNN-PQK hybrids—show substantial accuracy gains over purely classical baselines, attributed to the richer geometry of the quantum feature map (Alagiyawanna et al., 6 Jan 2026, Utro et al., 30 Jul 2025).
• Multiple kernel learning & financial classification: PQK, integrated within quantum multiple kernel aggregation schemes, delivers the strongest and most robust classification performance across diverse financial datasets, including high-qubit-count hardware runs up to 20 qubits with full error mitigation (Miyabe et al., 2023).
• Control and Bandit optimization: PQK subspace approximations with rigorously controlled projection dimension can balance information gain and regret in kernelized bandit frameworks, outperforming both highly expressive but overcomplex full kernels and extremely reduced proxies (Huang et al., 1 Jul 2026).

However, PQK alone will not yield a generic quantum advantage unless the data structure induces a large geometric difference to any classical kernel (high $n$6), e.g., for hidden subgroup structure or quantum-native pattern recognition (Naguleswaran, 2024, Tanner et al., 9 Apr 2026).

5. Theoretical Guarantees, Limitations, and Concentration

Theoretical insights into PQK fall into three categories:

• Concentration behavior: Full fidelity quantum kernels suffer exponential decay of both mean and variance with $n$7 (vanishing similarity), making distinct inputs indistinguishable. PQKs, by projecting onto low-dimensional local RDMs, slow down this effect—but do not eliminate it. Variance decays as $n$8 for $n$9-qubit blocks and circuit depth $K_F(x,x') = |\langle\psi(x) | \psi(x')\rangle|^2.$0; shallow, local circuits and low-entanglement seed states retain polynomial variance (Suzuki et al., 2023, Miroszewski et al., 2024, Tanner et al., 9 Apr 2026).
• Sample complexity and geometric difference: PQK enables a regime where the model complexity for learning can be much lower than for any classical kernel (under complexity and geometric difference diagnostics), but achieving provable quantum sample advantage requires specific data structures or feature maps (Egginger et al., 2023, Naguleswaran, 2024, Utro et al., 30 Jul 2025, Tanner et al., 9 Apr 2026).
• Resource lower bounds: Even with PQK, the number of quantum measurement “shots” required to resolve kernel matrix entries grows exponentially with system size (though with a smaller exponent than for fidelity kernels), especially above 8–10 qubits. Accurate empirical rule-of-thumb formulas tie the required shot count to spread and concentration effects in the kernel distribution (Miroszewski et al., 2024).

PQK performance is further degraded by excessive circuit depth, maximally entangled initial states, or strong noise, conditions under which local marginals become maximally mixed and kernel values concentrate near a constant, erasing class information (Suzuki et al., 2023, Tanner et al., 9 Apr 2026). Bandwidth hyperparameter optimization—via input rescaling or circuit design—can mitigate such pathological behavior but at the risk of dequantization (collapsing to a classical kernel regime) if overapplied (Schnabel et al., 2024, Tanner et al., 9 Apr 2026).

6. Practical Implementation and Algorithmic Recipes

PQK implementation proceeds in several coordinated steps (Miyabe et al., 2023, Utro et al., 30 Jul 2025, Schnabel et al., 2024, Tanner et al., 9 Apr 2026):

1. Quantum feature map design: Select a shallow, hardware-friendly data-embedding circuit (e.g., RotX, ZZ, short Trotterized evolutions) that encodes essential data correlations without incurring excessive entanglement.
2. Projection: Perform local tomography (measuring Pauli operators in a fixed basis) on each selected subsystem (typically single qubits). This yields an $K_F(x,x') = |\langle\psi(x) | \psi(x')\rangle|^2.$1-dimensional vector of expectation values.
3. Classical kernel application: Use an outer kernel, usually RBF with tunable bandwidth, to compute the Gram matrix from the classical feature vectors, enabling the use of standard SVM or ridge regression.
4. Hyperparameter tuning: Jointly optimize quantum and classical kernel hyperparameters using cross-validation or guided grid search.
5. Error mitigation and measurement management: On hardware, combine Pauli twirling, readout error correction, and optimized shot allocation to control statistical and hardware-induced error (Miyabe et al., 2023, Utro et al., 30 Jul 2025, Miroszewski et al., 2024).

For advanced settings, PQK can be incorporated into multiple-kernel aggregation (QMKL), neural tangent kernel pipelines, and GP bandit optimization frameworks with principled dimensionality reduction for tradeoffs between approximation error and information gain (Miyabe et al., 2023, Nakaji et al., 2021, Huang et al., 1 Jul 2026).

7. Outlook, Open Questions, and Future Directions

PQK offers a flexible, resource-efficient bridge between quantum state encoding and classical machine learning, affording practical trainability and partial quantum expressivity on present-day hardware (Miyabe et al., 2023, Utro et al., 30 Jul 2025). Qualitative and quantitative gains have been demonstrated for small/structured data, data-scarce regimes, and special datasets with strong quantum-native patterns (Utro et al., 30 Jul 2025, Naguleswaran, 2024, Alagiyawanna et al., 6 Jan 2026, Perciavalle et al., 18 Apr 2025).

Critical open directions include:

• Algorithmic design of feature maps and projection schemes that maximize the geometric difference with respect to classical kernels on real-world data.
• Analytical and empirical identification of problem classes (e.g., group-covariant, hidden subgroup, quantum phase recognition) where PQK pipelines yield a provable or empirically demonstrated advantage (Tanner et al., 9 Apr 2026, Naguleswaran, 2024).
• Advanced error mitigation strategies and scaling shot-efficient estimation to tens or hundreds of qubits (Miyabe et al., 2023, Utro et al., 30 Jul 2025).
• Automated hyperparameter and projection selection to guarantee both generalization and tractable resource consumption (Schnabel et al., 2024, Huang et al., 1 Jul 2026).
• Hybridization with trainable neural architectures and adaptive online learning systems to leverage PQK representations in nonstationary or complex environments (Alagiyawanna et al., 6 Jan 2026, Nakaji et al., 2021).

While generic performance parity with tuned classical baselines is typical, the empirical and theoretical evidence indicates that PQKs remain the best-scaling, experimentally accessible route to quantum kernel learning on currently available quantum devices, especially for applications involving high-order correlations, data-scarce high-dimensional problems, and quantum-native datasets (Miyabe et al., 2023, Utro et al., 30 Jul 2025, Tanner et al., 9 Apr 2026).