We propose an explicit feature representation for supervised learning derived from short quantum dynamical trajectories composed of generally noncommuting operations, conceptually inspired by reservoir computing in that fixed, untrained dynamics are coupled to a simple trained readout. For each input, a fixed-depth circuit generates a sequence of prefix states, and features are formed by concatenating expectation values of a predetermined set of single- and two-qubit Pauli observables recorded across depths; multiple randomized but fixed circuit instantiations are aggregated to improve stability. A classicall2- regularized logistic-regression readout is trained on the resulting explicit features. To isolate representational effects, all methods are evaluated under identical preprocessing, repeated stratified holdout splits, and a unified model-selection protocol on a controlled subset of 11 tabular classification datasets, and are compared against classical baselines, including random Fourier features and second-degree polynomial expansions, as well as reservoir-style quantum baselines, with feature dimensionality matched when feasible. We further examine controlled variants targeting commutativity, entangling structure, history concatenation, and observable choice. A staged evaluation contrasts an idealized exact-expectation regime with finite-shot estimation under fixed and larger measurement budgets, thereby assessing sensitivity to sampling noise after re-tuning regularization. Beyond predictive performance, we analyze the induced geometry via empirical kernel spectra and effective-dimension diagnostics to contextualize the representations from a regularization-aware capacity perspective. Empirically, the strongest support concerns noncommutativity and history concatenation, whereas the contribution of the entangling block is mixed; the trajectory-based features induce empirically distinct kernel geometries, while finite-shot robustness remains strongly task-and configuration-dependent.
We propose an explicit feature representation for supervised learning derived from short quantum dynamical trajectories composed of generally noncommuting operations, conceptually inspired by reservoir computing in that fixed, untrained dynamics are coupled to a simple trained readout. For each input, a fixed-depth circuit generates a sequence of prefix states, and features are formed by concatenating expectation values of a predetermined set of single- and two-qubit Pauli observables recorded across depths; multiple randomized but fixed circuit instantiations are aggregated to improve stability. A classicall2- regularized logistic-regression readout is trained on the resulting explicit features. To isolate representational effects, all methods are evaluated under identical preprocessing, repeated stratified holdout splits, and a unified model-selection protocol on a controlled subset of 11 tabular classification datasets, and are compared against classical baselines, including random Fourier features and second-degree polynomial expansions, as well as reservoir-style quantum baselines, with feature dimensionality matched when feasible. We further examine controlled variants targeting commutativity, entangling structure, history concatenation, and observable choice. A staged evaluation contrasts an idealized exact-expectation regime with finite-shot estimation under fixed and larger measurement budgets, thereby assessing sensitivity to sampling noise after re-tuning regularization. Beyond predictive performance, we analyze the induced geometry via empirical kernel spectra and effective-dimension diagnostics to contextualize the representations from a regularization-aware capacity perspective. Empirically, the strongest support concerns noncommutativity and history concatenation, whereas the contribution of the entangling block is mixed; the trajectory-based features induce empirically distinct kernel geometries, while finite-shot robustness remains strongly task-and configuration-dependent.
Quantum reservoir computing offers a promising route for time series learning by modelling sequential data via rich quantum dynamics while the only training required happens at the level of a lightweight classical readout. However, studies featuring efficient and implementable quantum reservoir architectures along with model learning guarantees remain scarce in the literature. To close this gap, we introduce QuaRK, an end-to-end framework that couples a hardware-realistic quantum reservoir featurizer with a kernel-based readout scheme. Given a sequence of sample points, the reservoir injects the points one after the other to yield a compact feature vector from efficiently measured k-local observables using classical shadow tomography, after which a classical kernel-based readout learns the target mapping with explicit regularization and fast optimization. The resulting pipeline exposes clear computational knobs -- circuit width and depth as well as the measurement budget -- while preserving the flexibility of kernel methods to model nonlinear temporal functionals and being scalable to high-dimensional data. We further provide learning-theoretic generalization guarantees for dependent temporal data, linking design and resource choices to finite-sample performance, thereby offering principled guidance for building reliable temporal learners. Empirical experiments validate QuaRK and illustrate the predicted interpolation and generalization behaviours on synthetic beta-mixing time series tasks.
Quantum reservoir computing (QRC) uses fixed quantum dynamics to encode a time series and trains only a classical readout, but a benchmark capacity alone does not reveal whether task information is lost in the reservoir, the measurement, feature compression, or finite sampling. We introduce \emph{task-resolved Fisher spectroscopy}, in which prediction targets define orthonormal score coordinates on the stationary distribution of input histories. Reweighting labeled histories along these scores generates an exactly affine family of reservoir states and measurement outcomes. In the same coordinates, we obtain a many-body Fisher-information hierarchy relating the state quantum Fisher information, the Fisher information of the complete measurement record, and moment matrices retained through many-body order r. The quadratic form of each moment matrix is exactly the stationary capacity of the optimal linear readout, while a finite-measurement extension predicts the approach from one-shot records to ideal expectation-value features. The measurement-level quantities require only stationary labeled records and measured outcome strings, not an input model or quantum-state tomography. In a five-spin open reservoir, interactions route fourth-order temporal information into higher-body correlations, so low-order compression can incur orders-of-magnitude sampling overhead even when a conventional parity target remains accessible. For correlated inputs and outputs, record-defined task scores predict held-out capacities, the required feature order, and measurement-budget dependence; optimizing the local measurement axis recovers otherwise hidden task information. The framework connects benchmark performance to experimentally actionable diagnoses of reservoir encoding, measurement choice, classical representation, and shot allocation.
Finding optimal measurement operators is crucial for the performance of quantum reservoir computers (QRCs), since they employ a fixed quantum feature map. We formulate the training of both stateless (quantum extreme learning machines, QELMs) and stateful (memory dependent) QRCs in the framework of kernel ridge regression. We thus extend the kernel viewpoint of supervised quantum models to recurrent QRCs by deriving an exact Hilbert--Schmidt kernel representation of the optimal readout observable on history space. This approach renders an optimal measurement operator that minimizes prediction error for a given reservoir and training dataset. For large qubit numbers, this method is more efficient than the conventional training of QRCs. We discuss efficiency and practical implementation strategies, including Pauli basis decomposition and operator diagonalization, to adapt the optimal observable to hardware constraints. To demonstrate the effectiveness of this approach, we present numerical experiments on image classification and time series prediction tasks, including chaotic and strongly non-Markovian systems. The developed method can also be applied to other quantum machine learning models.
Quantum machine learning (QML) promises new representational and computational capabilities, yet practical deployment on near-term hardware is hampered by resource overheads, depth constraints, and fragile trainability. This thesis advances resource-aware QML by proposing architectures and kernels that retain expressivity while sharply reducing qubit counts, circuit depth, and entangling-gate budgets. The work is presented in an article-based format with three core contributions. First, I introduce a Coherent Feed-Forward Quantum Neural Network (CFF-QNN) that preserves quantum coherence across all layers, mirrors the flexibility of classical feed-forward networks (adjustable hidden layers and nodes), and decouples qubit requirements from input feature dimension. Compared to prevailing QNN baselines, the CFF-QNN reduces both depth and CNOT count by over 50% while achieving strong performance on standard benchmarks (e.g., 91% accuracy on Wisconsin breast cancer and 85% on credit-card fraud). This contribution is accompanied by an international patent filing (WO2025050205A1). Second, I develop a resource-efficient quantum kernel that enables high-dimensional embeddings with substantially fewer qubits and entanglers, achieving linear scaling of entangling gates in the number of qubits. Empirically, the kernel delivers competitive or superior performance to widely used classical kernels and to popular quantum feature maps (e.g., on the Parkinson's disease dataset), with noisy simulations and small-scale runs on superconducting hardware indicating suitability for near-term devices. This contribution is covered by a companion patent filing (WO2025073041A1). Third, I propose a quantum reservoir computing (QRC) scheme that reuses a fixed quantum feature-map circuit as the reservoir while injecting temporal memory via an explicit feedback loop. The register is recycled across time, so quantum resources remain constant, no quantum parameters are trained, and only a lightweight classical readout is fitted. Experiments on chaotic time series (e.g., Mackey-Glass) show competitive predictive accuracy with strict resource efficiency, and illuminate how feedback delay and entangling structure affect memory and error. Collectively, these results chart a path toward practical QML: coherent architectures and kernels that are trainable, scalable in input dimension, frugal in quantum resources, and viable on noisy intermediate-scale devices—while providing design guidelines for future, larger-scale implementations.
Quantum machine learning is often motivated by the idea that quantum systems can expose useful high-dimensional structure that is difficult to access with classical models. We isolate one central component of this claim: the fixed data-encoding map. Amplitude, angle, and basis encoding are evaluated as deterministic feature maps for classical supervised learning under matched output dimensionality and strong classical controls. The benchmark compares these encodings against raw linear models, random Fourier features, polynomial features, PCA, RBF SVMs, and shallow neural networks across diverse classical datasets. Rather than treating performance as a single endpoint, we analyze the geometry of each representation through effective rank, condition number, centered kernel alignment, predictive performance, and practical overhead. The resulting picture is mechanistic: amplitude encoding can remove magnitude information through unit-sphere normalization, angle encoding can become geometrically redundant with raw linear features, and basis encoding can impose a binary Hamming geometry that is poorly aligned with smooth decision structure. These findings do not argue against quantum computation, however, they show that fixed quantum-inspired encoding geometry alone is not a reliable source of machine-learning advantage on classical data.
Reservoir computing promises a fast method for handling large amounts of temporal data. This hinges on constructing a good reservoir--a dynamical system capable of transforming inputs into a high-dimensional representation while remembering properties of earlier data. In this work, we introduce a reservoir based on recurrent quantum feature maps where a fixed quantum circuit is reused to encode both current inputs and a classical feedback signal derived from previous outputs. We evaluate the model on the Mackey-Glass time-series prediction task using our recently introduced CP feature map, and find that it achieves lower mean squared error than standard classical baselines, including echo state networks and multilayer perceptrons, while maintaining compact circuit depth and qubit requirements. We further analyze memory capacity and show that the model effectively retains temporal information, consistent with its forecasting accuracy. Finally, we study the impact of realistic noise and find that performance is robust to several noise channels but remains sensitive to two-qubit gate errors, identifying a key limitation for near-term implementations.