Table of Contents

Deep theory meets hands-on practice: quantum computing, quantum machine learning, and quantum AI from first principles to the research frontier.

First Edition · 2026

Chapter 0 + 69 numbered chapters · 11 parts · 6 appendices · Capstone projects

Front Matter

9 entries
  1. F1
    Why This Book ExistsThe evaluation-first stance, the seven-forms method, and why quantum AI is not magic AI.
  2. F2
    What This Book CoversThe eleven-part arc from qubits to production-grade quantum AI software.
  3. F3
    Who Should Read This BookAnyone who wants both the theory and the build: AI/ML engineers, researchers, and practitioners moving from classical to quantum.
  4. F4
    Diagnostic Self-TestShort checks on Python, linear algebra, probability, and ML to pick your onramp path.
  5. F5
    Notation GuideDirac notation, state vectors, operators, and conventions, all in one place.
  6. F6
    Chapter-Dependency GraphA DAG of prerequisites with suggested paths for the three course modes.
  7. F7
    Running the Code and Coding With a CopilotOne-click environment setup and the Build-With-AI discipline that runs throughout.
  8. F8
    About the AuthorsWho wrote this book and how.
  9. F9
    About the Hands-On AI Science SeriesThe nine-book series and where this volume fits.

Foundations · Mathematical Onramp

1 chapter

The mathematics the rest of the book uses: complex numbers, vectors, matrices, tensor products, probability, and Dirac notation.

  1. 0
    Mathematical Foundations You Actually Need NewComplex numbers, inner products, unitarity, tensor products, eigenvalues, probability, and gentle Dirac notation.

Part I · The Rules of the Quantum Computing Game

6 chapters

Quantum computing as a rule-based computational model: qubits, measurement, gates, rotation gates, and circuits.

  1. 1
    Quantum Computing Without PhysicsThe computational model, not the physics story. Why it matters for AI, and what "Quantum AI" can and cannot mean.
  2. 2
    Qubits: The Smallest Quantum Data ObjectThe qubit as a vector of amplitudes. Amplitude vs probability. Normalization. Why amplitudes can be negative or complex.
  3. 3
    Measurement: From Quantum State to Classical OutputThe Born rule, sampling, shots, and collapse. Measuring in a chosen basis and expectation values.
  4. 4
    Quantum Gates: Legal Moves in the GameUnitarity, reversibility, and the common one-qubit gates: X, Y, Z, H, S, T.
  5. 5
    The Bloch Sphere and Rotation Gates NewThe Bloch sphere, RX/RY/RZ as parameterized rotations, and the trainable-weight interpretation that all of Part VI needs.
  6. 6
    Quantum Circuits: Programs Made of GatesCircuit notation, translating diagrams to code, circuits as model architectures. PennyLane and Qiskit first builds.

Part II · Essential Quantum Ideas for AI

5 chapters

Superposition, interference, tensor products, multi-qubit gates, entanglement, and reversibility.

  1. 7
    Superposition: Representation Before ComputationSuperposition without exaggeration. Not "all answers at once"; a distributed representation with careful output extraction.
  2. 8
    Interference: Making Some Answers LouderAmplitudes adding and canceling. Phase kickback as the recurring mechanism behind algorithms.
  3. 9
    Multiple Qubits, Tensor Products, and Multi-Qubit GatesExponential state spaces, tensor products, CNOT, controlled-U, SWAP, and Toffoli.
  4. 10
    Entanglement: Dependencies Beyond Classical CorrelationProduct vs entangled states, Bell pairs, measurement dependency, and what distinguishes entanglement from correlation.
  5. 11
    Reversibility, No-Cloning, and Information FlowWhy quantum gates are reversible, why unknown states cannot be copied, and what that means for algorithm design.

Part III · Building Quantum Algorithms

7 chapters

The structure of quantum algorithms and their AI/ML relevance: search, optimization, feature extraction, and objective minimization.

  1. 12
    Quantum Algorithm Design PatternPrepare, transform, interfere, measure, repeat. Oracles, phase kickback revisited, uncomputation.
  2. 13
    Deutsch-Jozsa AlgorithmThe first clean demonstration of algorithmic advantage: constant vs balanced functions in one query.
  3. 14
    Grover Search and Amplitude AmplificationQuadratic speedup for unstructured search. Amplitude amplification as a general primitive.
  4. 15
    Quantum Fourier Transform, Phase Estimation, and Shor in BriefPeriodic structure, phase estimation, and the factoring connection (post-quantum security implication noted).
  5. 16
    Quantum Linear Algebra and the HHL Algorithm NewHHL, QRAM, the fine print on exponential speedup claims, and the dequantization that followed.
  6. 17
    Hamiltonians, Energy, and Objective FunctionsHamiltonians as cost functions. Ground state as optimum. Preparing for VQE and QAOA.
  7. 18
    VQE: Variational Quantum EigensolverThe hybrid quantum-classical loop. Ansatz, energy expectation, classical optimizer, and the ML connection.

Part IV · Classical Machine Learning Refresher

4 chapters

A compact ML onramp focused on concepts that map directly to quantum ML. Explicitly skippable for ML-literate readers.

  1. 19
    Machine Learning as Function LearningData, labels, models, loss, generalization. Classical baselines for toy datasets.
  2. 20
    Feature Spaces and KernelsFeature maps, the kernel trick, SVMs, and kernel matrices. Preparing for quantum kernels.
  3. 21
    Neural Networks and Trainable ModelsParameters, gradients, backpropagation. Preparing for variational circuits.
  4. 22
    Generative Models and SamplingDistribution learning, generative vs discriminative. Preparing for quantum generative models.

Part V · Encoding Data into Quantum Circuits

3 chapters

The data-loading problem, feature maps, QRAM, and measurement as feature extraction.

  1. 23
    The Data Loading ProblemWhy encoding is not free. Basis, angle, amplitude encoding. QRAM and the open feasibility question.
  2. 24
    Quantum Feature MapsData-dependent circuits, data re-uploading, depth, and expressivity.
  3. 25
    Measurement as Feature ExtractionObservables, expectation values, shot noise, and classical post-processing.

Part VI · Quantum Machine Learning Models

13 chapters

The full QML model zoo: kernels, variational circuits, QNNs, gradients, barren plateaus, expressivity, equivariant models, tensor networks, reservoir computing, generative models, and RL.

  1. 26
    Quantum KernelsQuantum feature-space similarity, kernel estimation, SVM, and exponential concentration.
  2. 27
    Variational Quantum CircuitsParameterized circuits as trainable models, ansatz design, hybrid optimization.
  3. 28
    Variational Models Are Kernel Methods NewThe encoding-induced kernel, why ansatz choices matter less than encoding choices, and what this means for advantage.
  4. 29
    Quantum Neural NetworksQNN structure, hybrid models, and comparison to classical neural networks.
  5. 30
    Gradients, Parameter-Shift, and TrainingThe parameter-shift rule, automatic differentiation, shot noise, and optimization difficulty.
  6. 31
    Barren Plateaus and TrainabilityVanishing gradients, the Lie-algebraic DLA criterion (2024), and the barren-plateau/simulability link.
  7. 32
    What Can Quantum Models Learn? Expressivity and Generalization NewCircuits as truncated Fourier series. ICLR 2025 generalization bounds vs classical RFF. Sample complexity.
  8. 33
    Quantum Convolutional and Structured ModelsQCNNs, local inductive bias, barren-plateau-free design, and the simulability tradeoff (PRX Quantum 2025).
  9. 34
    Geometric and Equivariant Quantum Machine Learning NewSymmetry-aware ansatze, equivariant gates, inductive bias, and improved trainability.
  10. 35
    Tensor Networks: The Bridge Between Quantum and Classical ML NewMPS, bond dimension, classical tractability, and the tensor-network classifier.
  11. 36
    Quantum Reservoir Computing NewFixed quantum reservoirs sidestep barren plateaus. Temporal tasks and linear readouts.
  12. 37
    Quantum Generative ModelsBorn machines, quantum Boltzmann machines, distribution learning, and the 68-qubit advantage claim.
  13. 38
    Quantum Reinforcement LearningQuantum parameterized policies and optimization inside RL loops.

Part VII · Quantum Optimization for AI

4 chapters

QAOA, QUBO, annealing, and optimization pipelines for AI tasks.

  1. 39
    QAOA: Quantum Approximate Optimization AlgorithmCost and mixer Hamiltonians, alternating layers, the classical loop, and a MaxCut example.
  2. 40
    QUBO and Ising FormulationsBinary objectives, QUBO, Ising models, and mapping AI problems to energy functions.
  3. 41
    Quantum Annealing and Quantum-Inspired OptimizationAnnealing vs gate-based, and where classical heuristics already win.
  4. 42
    Optimization in AI PipelinesFeature selection, clustering, hyperparameter search, and resource allocation via QUBO.

Part VIII · Reality Check: Noise, Hardware, and Limits

7 chapters

Density matrices, noise, error correction, the 2026 hardware landscape (six modalities including topological), and a rigorous framework for evaluating advantage.

  1. 43
    Density Matrices and Mixed States NewPure vs mixed states, partial trace, reduced states, and why noise turns pure states mixed.
  2. 44
    Simulators Versus Real Quantum HardwareIdeal vs noisy simulation, shots, connectivity, queues, and circuit depth limits.
  3. 45
    Noise and Error MitigationNoise channels, decoherence, mitigation techniques, and the mitigation-vs-correction distinction.
  4. 46
    Error Correction and the Road to Fault Tolerance NewLogical qubits, threshold theorem, Willow below-threshold, IBM BB qLDPC, Quantinuum Helios 48-logical, AlphaQubit 2.
  5. 47
    The 2026 Hardware Landscape NewSix modalities: superconducting, trapped-ion, neutral-atom, photonic, silicon-spin, and topological (Majorana 1).
  6. 48
    Scaling Problems and Simulation LimitsExponential state spaces, tensor-network simulation boundaries, and practical limits.
  7. 49
    What Would Quantum Advantage Mean? Complexity, Dequantization, and Honest BaselinesBQP, dequantization, classical surrogates, Bowles-Ahmed-Schuld benchmarking, entanglement-induced advantage (2025), robust dequantization (2024).

Part IX · Advanced Topics in Quantum AI

8 chapters

Quantum data and learning separations, QNLP, quantum transformers, graph learning, scientific ML, neural quantum states, AI for quantum, and domain applications.

  1. 50
    Quantum Data and Learning Quantum SystemsPower of data, learning from quantum experiments, classical shadows, and quantum dynamics learning.
  2. 51
    Quantum Natural Language ProcessingCompositionality, grammar-and-meaning, tensor structure, and sentence circuits.
  3. 52
    Quantum Transformers and Attention NewQuantum attention blocks, and the quantum-hardware vs quantum-inspired-classical distinction (applied skeptically).
  4. 53
    Quantum Graph LearningQuantum walks, graph kernels, molecular graphs, and graph classification.
  5. 54
    Quantum AI for Scientific Machine LearningChemistry, materials, and simulation: the most plausible near-term advantage domain.
  6. 55
    Neural Quantum States NewClassical neural networks representing quantum wavefunctions. VMC, transformer wavefunctions, foundation NQS (2025). The baseline for Ch 54.
  7. 56
    AI for Quantum ComputingAlphaQubit 2 (near-optimal, real-time decoding), calibration, circuit search, and LLM-assisted coding.
  8. 57
    Quantum AI Across Domains NewDrug discovery (fastest-growing), finance, high-energy physics, and industrial optimization, all read through the claim-vs-proven lens.

Part X · Quantum AI Software, Hardware Access, and Research Practice New

4 chapters

The engineering and research-practice spine: reproducible environments, AI-assisted coding, real-hardware access, GPU/JIT acceleration, benchmarking, and statistics.

  1. 58
    The Quantum AI Development Environment and AI-Assisted Coding NewPinned envs, SDK decision map, testing quantum code, CI for notebooks, and the Build-With-AI (vibe coding) discipline.
  2. 59
    Compiling, Transpiling, and Running on Real Hardware NewTranspilation, OpenQASM 3 / QIR / primitives, Qiskit 2.3 C++ transpiler, Mitiq, and access recipes for IBM/Braket/IonQ.
  3. 60
    Performance, Scale, and Differentiable Quantum Programming NewBackend selection, Catalyst/JAX/lightning.gpu/CUDA-Q, shot-budget optimization, and circuit cutting.
  4. 61
    Datasets, Benchmarks, and the Statistics of Quantum Experiments NewShared benchmark harness, shot-noise statistics, bootstrap CIs, tests for advantage, and experiment tracking.

Part XI · Capstone Projects

8 chapters

Design, implement, evaluate, and explain complete quantum AI experiments with honest baselines, reproducible code, and the research-practice tools of Part X.

  1. 62
    How to Build a Quantum ML ExperimentResearch question, dataset, baseline, model, metrics, ablation, noise analysis, and reproducibility. The shared harness.
  2. 63
    Capstone 1: Quantum Kernel ClassifierDataset, classical baselines, quantum feature map, kernel SVM, paper-style report.
  3. 64
    Capstone 2: Variational Quantum ClassifierAnsatz design, training curves, classical comparison. Reproducible notebook and report.
  4. 65
    Capstone 3: Quantum Generative ModelBorn machine, distribution learning, sample evaluation, and failure analysis.
  5. 66
    Capstone 4: Quantum Optimization for AIQUBO / QAOA / quantum-inspired solver for feature selection, clustering, or scheduling, with a classical comparison.
  6. 67
    Capstone 5: Noise-Aware Quantum ML Study NewIdeal vs noisy vs hardware runs, mitigation applied and measured, shot-count sensitivity analysis.
  7. 68
    Capstone 6: Reproduce a Published Result with an AI Copilot NewEnd-to-end reproduction with error bars, a tuned classical baseline, and an explicit dequantization assessment. The copilot's contributions and corrections both documented.
  8. 69
    Final Perspective: What Quantum AI Is TodayMature, experimental, overhyped, and worth learning: an honest map of the field and how to keep studying.

Appendices & Back Matter

6 appendices
  1. A
    Linear Algebra and Probability ReferenceFull treatment behind Ch 0: vector spaces, unitarity, spectral decomposition, tensor products, and the book's key proof sketches.
  2. B
    Notation, Symbols, and Dirac Notation ReferenceComplete symbol table, Dirac notation alongside vector/matrix forms.
  3. C
    Environment Setup and the Companion RepositoryPinned versions, test suite, CI setup, hardware access, and how to use the companion repo.
  4. D
    Solutions and Instructor ResourcesSolution sketches, grading rubrics, the dependency graph in full, slides, and problem sets.
  5. E
    The Quantum AI Cookbook NewAll Recipe Boxes consolidated and indexed by task: encode, estimate, diagnose, run, align, track.
  6. F
    Reproduce-the-Paper Index and Research-Practice Checklist NewCurated reproduction exercises and a one-page checklist: pinned env, tests, seeds, construct-matched metrics, error bars, honest baselines.