Quantum Feature Extraction Methods

• Quantum feature extraction is a process that exploits quantum protocols to identify and embed informative representations from quantum states or classical data using methods like overlap-based protocols and variational circuits.
• It employs techniques such as kernel PCA, SWAP and SWITCH tests, and Hamiltonian dynamics to extract state amplitudes and optimize feature maps for applications in sensing, spectroscopy, and machine learning.
• The approach offers exponential measurement cost reductions and improved classification performance, while requiring careful management of circuit depth and noise on current NISQ devices.

Quantum feature extraction refers to the process of identifying, selecting, or constructing informative representations—features—from quantum states or classical data via quantum protocols that explicitly exploit the structure and computational power of quantum systems. Quantum feature extraction plays a foundational role in quantum information processing, quantum machine learning, hybrid quantum-classical algorithms, and quantum-enhanced sensing. Techniques span direct extraction of state amplitudes, variational quantum circuits, Hamiltonian dynamics, kernel and kernel-PCA frameworks grounded in quantum information theory, and quantum autoencoding. Approaches differ substantially in their algorithmic paradigm (supervised, unsupervised, hybrid), operational regime (digital, analog, adiabatic, nonadiabatic), and modality (state readout, data embedding, feature selection, or transformation).

1. Foundational Principles and Mathematical Formalism

A prototypical mathematical model for quantum feature extraction is the overlap-based protocol (Nishi et al., 13 May 2025), which targets quantum state readout via measured overlaps between an unknown "target" state $|\psi\rangle$ and a family of parameterized basis states %%%%1%%%%. The core equations are:

• Quantum overlap:

$$\langle\psi|\phi(\theta)\rangle \equiv \langle\psi|U(\theta)|0\rangle,$$

where $U(\theta)$ prepares the basis function state.

• Linear-combination expansion (feature projection):

$$|\psi\rangle \approx \sum_{j=1}^M \alpha_j(\theta) |\phi_j(\theta)\rangle,$$

with coefficients $\alpha_j$ and basis parameter vector $\theta$.

• Fidelity-based cost function (feature fitting):

$$F(\theta,\alpha) = \left|\left\langle\psi\left|\sum_j \alpha_j(\theta) |\phi_j(\theta)\rangle\right.\right\rangle\right|^2, \quad C(\theta,\alpha) = 1 - F(\theta,\alpha).$$

This formalism directly quantifies the extraction of salient features—e.g., dominant amplitudes, peak locations, widths—encoded in $|\psi\rangle$ via measurement-accessible overlaps.

Quantum versions of kernel PCA (Bény, 2018) recast the feature extraction problem as finding perturbations (features) $X$ of a state $\rho$ that maximize post-coarse-graining distinguishability. The maximal-relevance features are the leading eigenvectors of the operator $\mathcal{R} = \mathcal{E}_\rho^* \mathcal{E}$, where $\mathcal{E}$ models coarse-graining or noise: $$\eta(X) = \frac{\langle X, \mathcal{R}(X)\rangle_\rho}{\langle X,X\rangle_\rho}, \quad \mathcal{R}(X^*) = \eta_{\rm max} X^*.$$ Features are thus mathematically characterized as eigenmodes optimally preserved under quantum channels.

2. Quantum Circuit Implementations and Measurement Protocols

Realizing quantum feature extraction requires quantum circuits that probe desired properties of quantum states or processed data. Overlap estimation utilizes:

• SWITCH test (phase-sensitive): An ancilla-mediated interferometric protocol yielding both real and imaginary parts of $\langle\psi|\phi\rangle$ (Nishi et al., 13 May 2025).
• SWAP test (magnitude-only): Produces $|\langle\psi|\phi\rangle|^2$ via ancilla measurement, central to profile fitting and quantum kernel estimation.

Parametrized quantum circuit (PQC) ansätze act as feature maps, variational convolutional blocks, or autoencoder encoders. Typical architecture elements (Jain et al., 22 Jan 2025, Dou et al., 2022, Yang et al., 2020, Dou et al., 2021):

• Angle encoding via single-qubit $R_y$ rotations per input dimension.
• Entangling layers (CNOT, CZ, controlled-$Z$ gates) generating nonlinear feature mixing.
• Measurement observables: expectation values of $Z_j$ or higher-order correlators $\sigma_i^z \sigma_j^z \cdots$ yield feature vectors.
• Gradient estimation via parameter-shift rule ensures trainability for hybrid quantum-classical stacks.

More advanced protocols employ Hamiltonian evolution and measurement:

• Counterdiabatic/quench dynamics: Data is embedded into many-body spin-glass Hamiltonians whose nonadiabatic/critical evolution amplifies complex correlations, with features read out as multi-Pauli expectation values (Simen et al., 15 Oct 2025, Simen et al., 28 Aug 2025).

3. Feature Selection, Dimensionality Reduction, and Expressivity

Quantum feature extraction can enable both selection of relevant subsets and nonlinear transformation to higher- or lower-dimensional spaces.

• Analog quantum feature selection (QFS): Relevance (via mutual information with labels) and redundancy (pairwise classical mutual information) are mapped onto site-dependent detunings and van der Waals couplings in a Rydberg atom array. Low-energy measurement bitstrings correspond to optimal feature subsets (Orquin-Marques et al., 23 Oct 2025).
• Quantum autoencoders (QAE): Parameterized unitary encoders discard (trace out) "trash" qubits, compressing quantum data to a reduced latent qubit space. Features extracted correspond to the remaining register (Bloch vector), suitable for classification or further analysis (Lo et al., 11 Feb 2025).
• Expressibility: Circuit families are benchmarked via frame potential—distance to Haar measure (Dou et al., 2022). Greater expressibility correlates with improved feature discrimination up to a saturation depth beyond which barren plateaus set in. K-means clustering over random PQC outputs is used to select diverse filter banks in unsupervised quanvolutional networks (Dou et al., 2021).

4. Applications and Empirical Performance across Modalities

Quantum feature extraction frameworks are extensively validated across quantum and classical tasks:

• Wavefunction readout and spectroscopy: Overlap-based fitting reconstructs grid-based amplitudes and X-ray absorption spectra with $O(M)$ measurements, where $M\ll2^n$, achieving infidelities $O(10^{-3})$ and subexponential scaling (Nishi et al., 13 May 2025).
• Kernel PCA for noisy classification: Learned-noise-adapted kernels dramatically improve digit classification accuracy (e.g., $\sim$3% vs. $\sim$10–15% for RBF with $k=10$ components) and feature robustness against modeled corruption (Bény, 2018).
• Quantum feature selection: Neutral-atom QFS achieves AUC gains of $1.5–2.3\%$ and 75–84% feature reduction relative to classical MI-ranking on real-world binary classification datasets (Orquin-Marques et al., 23 Oct 2025).
• Counterdiabatic and quenched features: Counterdiabatic dynamics on IBM hardware with 156 qubits yields AUC increases up to $+0.12$ (hybrid classical+quantum vs. classical), with quantum-extracted two-body correlators dominating SHAP feature importance (Simen et al., 15 Oct 2025). AQFM protocols deliver up to $210\%$ improvement in precision or recall on domain-specific tasks (e.g., medical diagnosis, drug discovery) by extracting nonclassical statistics inaccessible to classical preprocessing (Simen et al., 28 Aug 2025).
• Hybrid image/speech pipelines: Quantum variational or pre-processing filters improve or match deep classical networks in image classification, THz imaging, and speech recognition, leveraging quantum measurement as feature maps at intermediate pipeline stages (Koike-Akino et al., 2022, Ainelkitane et al., 21 Jun 2025, Slabbert et al., 2024, Yang et al., 2020).

5. Algorithmic and Hardware Considerations, Limitations, and Scaling

Performance and applicability depend on algorithmic and experimental constraints:

• Measurement scaling: Overlap-based feature extraction protocols achieve exponential reductions in measurement cost for states that are compressible in a small set of basis components ($O(M/\epsilon^2)$ instead of $O(2^n/\epsilon^2)$) (Nishi et al., 13 May 2025).
• Ansatz quality and initialization: Effective extraction requires an expressive and well-initialized ansatz basis; poor choices or unfavorable parameter landscapes can trap optimizers in suboptimal minima (Nishi et al., 13 May 2025, Dou et al., 2022).
• Circuit depth and hardware noise: Counterdiabatic and higher-order interaction terms rapidly increase circuit depth, challenging current NISQ devices with noise and connectivity limits (Simen et al., 15 Oct 2025).
• Scalability: Simulation and real-device studies demonstrate viability for $n=4–156$ qubits, but efficient feature extraction for higher $n$ requires ongoing advances in qubit count, coherence times, and error mitigation (Orquin-Marques et al., 23 Oct 2025, Simen et al., 28 Aug 2025).

6. Extensions and Thematic Directions

Quantum feature extraction continues to diversify into:

• Higher-dimensional and tensor-decomposed basis functions: Enabling scalable quantum readout in many-body systems (Nishi et al., 13 May 2025).
• Explicit quantum kernels for SVMs, GP, and other kernel methods: Via feature maps that can be evaluated on-device or classically sampled from quantum measurements (Slabbert et al., 2024).
• Quantum sensing and spectroscopy: Extraction of domain-specific features such as spectral peaks, densities, or echo profiles (Nishi et al., 13 May 2025, Koike-Akino et al., 2022).
• Hybrid quantum-classical autoencoders and deep models: Integration with CNNs, U-Nets, or other deep learning frameworks for classification and segmentation workflows (Jain et al., 22 Jan 2025, Slabbert et al., 2024).

7. Open Questions and Outlook

While empirical results strongly suggest practical utility for quantum feature extraction, several issues are the subject of ongoing investigation:

• Theoretical characterization of when, and under what conditions, "quantum advantage" arises—especially in the context of feature maps that cannot be efficiently approximated classically (Simen et al., 28 Aug 2025, Simen et al., 15 Oct 2025).
• Quantification of entanglement, expressibility, and their correlation (or lack thereof) with downstream discriminative performance (Ainelkitane et al., 21 Jun 2025, Dou et al., 2022).
• Strategies for architecture search, automated ansatz design, and hybrid methodologies that optimally trade off circuit depth, qubit count, and classical pre/post-processing (Dou et al., 2021, Jain et al., 22 Jan 2025).
• Robustness to hardware noise and finite sampling in realistic NISQ-era devices (Orquin-Marques et al., 23 Oct 2025, Simen et al., 15 Oct 2025, Koike-Akino et al., 2022).

Quantum feature extraction thus constitutes a rapidly advancing intersection of quantum information, machine learning, and experimental physics, with protocols grounded in measurable quantities (overlaps, expectation values) and strong performance across modalities from quantum chemistry to machine vision and control (Nishi et al., 13 May 2025, Bény, 2018, Orquin-Marques et al., 23 Oct 2025, Simen et al., 15 Oct 2025).