48 concepts,
one connected field.
01 · Chapter 01Linear algebra for QMLComplex vector spaces, inner products, unitary matrices, tensor products and spectral decomposition.02 · Chapter 01Qubits and Bloch spherePure qubit states are points on a sphere, while mixed states lie inside it. Global phase is unobservable.03 · Chapter 01Measurement and expectation valuesMeasurements turn amplitudes into sampled classical outcomes; observable expectations are statistical averages.04 · Chapter 01Entanglement and tensor productsEntangled states cannot be expressed as products of single-qubit states and may encode correlations unavailable to product states.05 · Chapter 02Supervised learning and riskA model learns a mapping from inputs to labels by minimizing empirical risk with appropriate validation.06 · Chapter 02Optimization and gradientsGradient descent, stochastic updates and adaptive optimizers tune trainable parameters.07 · Chapter 02Generalization and model selectionCapacity, regularization, bias–variance trade-offs and cross-validation determine out-of-sample reliability.08 · Chapter 02Kernel methods and feature spacesKernel methods learn linear predictors in implicit feature spaces; their quality depends on geometry and regularization.09 · Chapter 03Angle encodingMap numeric features to rotation angles on qubits; normalization and periodicity affect information retention.10 · Chapter 03Amplitude encodingRepresent normalized vectors as amplitudes; compact qubit counts can hide expensive state preparation.11 · Chapter 03Basis and computational encodingEncode discrete bit strings as computational basis states, useful for combinatorial and binary inputs.12 · Chapter 03Data re-uploading and expressivityInterleave input-dependent rotations and trainable layers to increase representational flexibility.13 · Chapter 04Parameterized quantum circuitsA variational circuit combines data embedding, trainable gates and measured observables.14 · Chapter 04Variational quantum classifierConvert expectation values into binary scores and optimize a classification loss.15 · Chapter 04Quantum neural networksQNNs are compositional trainable quantum models; architecture and readout define their hypothesis class.16 · Chapter 04Parameter-shift gradientsFor certain gates, analytic derivatives are obtained from shifted circuit evaluations.17 · Chapter 05Fidelity quantum kernelsEncode data as states and measure state overlap to construct a positive semidefinite kernel in ideal conditions.18 · Chapter 05Quantum support vector machinesUse quantum-derived Gram matrices in classical margin-based SVM optimization.19 · Chapter 05Kernel concentrationIn high-dimensional Hilbert spaces, many generic state overlaps concentrate and lose discriminative power.20 · Chapter 05Generalization and inductive biasQuantum learning depends on task-aligned structure, effective dimension and the geometry induced by the feature map.21 · Chapter 06Barren plateausGradients of certain deep or highly expressive random circuits may vanish exponentially with qubit number.22 · Chapter 06Local costs and layerwise trainingLocal observables, problem-inspired ansätze and incremental depth may improve trainability.23 · Chapter 06Shot noise and measurement budgetsEstimating observables requires repeated measurements; uncertainty shrinks as the inverse square root of shots.24 · Chapter 06Hardware noise and error mitigationReadout errors, decoherence and gate infidelity perturb the learned model and its gradients.25 · Chapter 07Born machinesA parameterized circuit defines a probability distribution through computational-basis measurement.26 · Chapter 07Quantum circuit GANsA quantum generator can be trained adversarially against a classical or quantum discriminator.27 · Chapter 07Quantum autoencodersLearn a compression unitary that moves relevant information into a smaller latent subsystem.28 · Chapter 07Quantum Boltzmann modelsThermal-state-based models describe distributions using parameterized Hamiltonians.29 · Chapter 08Learning from quantum statesWhen inputs are quantum states rather than classical vectors, quantum measurements can preserve task-relevant information.30 · Chapter 08Quantum state tomographyInfer a density matrix from measurements, with costs that can scale poorly for general many-qubit states.31 · Chapter 08Quantum phase recognitionUse measurement statistics or wavefunction features to identify phases and phase transitions.32 · Chapter 08Quantum sensing with MLApply learning algorithms to quantum sensor signals for calibration, denoising and parameter estimation.33 · Chapter 09Quantum reinforcement learningQuantum circuits can parameterize policies or value functions in sequential decision-making.34 · Chapter 09Quantum policy gradientsDifferentiate a quantum policy through measured action probabilities and reward objectives.35 · Chapter 09Quantum reservoir computingDrive a fixed quantum dynamical system with input signals and train a classical readout.36 · Chapter 09Time series and quantum recurrent ideasRecurrent and repeated-upload models attempt to represent temporal dependencies.37 · Chapter 10HHL and quantum linear systemsQuantum linear-systems algorithms can encode solutions in quantum states under restrictive input and conditioning assumptions.38 · Chapter 10Quantum PCA and linear algebraQuantum routines can access spectral information of suitable density operators with assumptions on state access.39 · Chapter 10Quantum nearest neighborsDistance and overlap estimation can support similarity search in specified access models.40 · Chapter 10What counts as quantum advantage?A meaningful advantage requires a precisely defined task, access model, resources and strong classical baselines.41 · Chapter 11Chemistry and materialsUse quantum data and physics-informed models to estimate molecular observables or identify material regimes.42 · Chapter 11Finance and anomaly detectionStudy feature-map classifiers for fraud or anomaly detection under severe class imbalance.43 · Chapter 11Medical and biological signalsExplore small-dimensional signal classification with careful governance and patient-level splits.44 · Chapter 11Computer vision and hybrid modelsUse classical feature extraction followed by a small quantum head, or patch-wise quantum filters.45 · Chapter 12Benchmark design and reproducibilityUse matched splits, nested tuning, multiple seeds, confidence intervals and transparent compute budgets.46 · Chapter 12Software stack and simulationCommon workflows combine Python, NumPy, scikit-learn and frameworks such as Qiskit or PennyLane.47 · Chapter 12Cloud architecture and MLOpsPackage datasets, circuits, job queues, experiments and result provenance as versioned services.48 · Chapter 12Open problems and research roadmapResearch directions include inductive bias, trainability, quantum-data tasks, resource-aware compilation and reliable advantage tests.