Quantum machine learning -- lecture notes (2512.05151v1)

Abstract: Lecture notes on quantum machine learning for computer scientists.

• The paper introduces a detailed typology matrix categorizing quantum machine learning approaches based on data origin and processing method.
• The paper rigorously compares classical and quantum probability using geometric metrics and discusses universal quantum computation models with practical circuit examples.
• The paper examines hybrid quantum-classical algorithms, variational circuits, and challenges like barren plateaus while discussing quantum-inspired dequantization techniques.

Quantum Machine Learning—A Comprehensive Analysis

Introduction and Typology

The lecture notes "Quantum machine learning" (2512.05151) present an authoritative and technically detailed exposition of quantum machine learning (QML), addressing foundational theoretical constructs, algorithmic paradigms, and the implications for computational complexity, statistical learning, and practical implementation on current quantum hardware. The paper distinguishes four subareas in QML via a typology matrix based on the nature of both data origin (classical or quantum) and computational processing (classical or quantum), systematically mapping the principal research avenues and practical approaches.

Figure 1: Four approaches to quantum computing—typology based on data source and processing method.

Classical vs Quantum Probability—Geometry and Metrics

The notes provide a rigorous comparative framework for classical and quantum probability, emphasizing their geometric and operational distinctions. Classical probabilities are described in the Kolmogorov formalism, represented within an $N$-dimensional simplex. Key comparative tools—majorisation, Shannon entropy, relative entropy, and the Fisher-Rao metric—are introduced as quantitative measures for distinguishing probability distributions, supporting analysis of information-theoretic limits and statistical distances.

Figure 2: Probability simplex for $N=2$ and $N=3$, visualising the geometry of classical probability spaces.

The transition to quantum probability is articulated via extension to density matrices and the relaxation of diagonal restrictions, yielding density operators with richer geometric structure. The lecture explores the mapping between classical and quantum probabilities, the non-injectivity and non-surjectivity of the correspondence, and how quantum majorisation and entropic inequalities generalize the classical case.

Figure 3: The Bloch ball parameterising all single-qubit density matrices—pure states live on the surface (Bloch sphere), mixed states within.

Figure 4: Majorization and entropy relationships in qubit space—green states majorize red, with inequalities in von Neumann and quantum relative entropy.

Quantum Computation—Gate Model, Adiabatic, and Fault-Tolerant Algorithms

The core models of quantum computation—gate model, adiabatic quantum computation, measurement-based, and continuous variable models—are introduced as algorithmically universal up to polynomial overhead. The gate paradigm is analyzed through its composition of universal gate sets, the Solovay-Kitaev theorem, and the challenges of physical implementation, including noise and limited connectivity.

Quantum protocols, including Bell state preparation, teleportation, and foundational algorithms (Deutsch, Deutsch-Jozsa, Grover, and Quantum Fourier Transform), are presented via explicit circuit representations, emphasizing the principles of quantum parallelism and speedup classification. The phase estimation algorithm is highlighted for its centrality in exponential speedups for problems such as factoring (Shor) and order finding.

Figure 5: Example quantum circuit illustrating sequential gate application and measurement.

Figure 6: Circuit for Bell basis preparation—generation of maximally entangled two-qubit states.

Figure 7: Teleportation protocol—measurement and conditional operations transfer a qubit state across entanglement.

Figure 8: Schematic for Quantum Fourier Transform—efficient basis change critical for quantum algorithms.

Figure 9: Grover’s algorithm circuit—quadratic speedup for unstructured search.

Quantum Data Encodings and Hybrid Algorithms

Efficient data encoding strategies—basis, amplitude, qusample, and Hamiltonian encoding—are systematically compared for qubit budget, runtime, and suitability for various ML protocols.

Practical quantum machine learning chiefly relies on hybrid quantum-classical algorithms implemented on NISQ devices. The main paradigm is the variational quantum circuit framework, which leverages quantum circuits with tunable parameters, optimised by classical routines. Central algorithms include QAOA (quantum approximate optimization), variational eigensolvers, and variational classifiers. The optimization landscape is studied both in terms of convexity (kernel methods) and non-convexity (variational quantum circuits).

Figure 10: Schematic of the hybrid quantum-classical optimization loop for variational quantum algorithms.

Figure 11: Convex (RKHS-based kernel optimization) vs. non-convex (variational parameter space) quantum ML optimization landscapes.

Kernel Methods, Fourier Expressivity, and Data Power

All variational quantum models are shown to be inherently linear functions in the operator embedding space, and hence reducible to kernel methods within the quantum reproducing kernel Hilbert space (RKHS). The representer theorem holds, ensuring optimal quantum models' measurement operators are kernel expansions over the dataset.

Quantum models are further expressed as partial Fourier series, with accessible frequency spectrum determined by data encoding gates. The degree of Fourier expressivity can be efficiently amplified via sequential and parallel repetition of encoding gates, as well as through exponential encoding strategies.

Contrasting function approximation and prediction, the notes rigorously demonstrate the power of data: once training data is provided, classical models can often approximate the outputs of intractable quantum circuits efficiently, unless quantum model complexity metrics (e.g., kernel rank or geometric quantities) indicate otherwise.

Barren Plateaus, Mitigation, and Quantum CNNs

The barren plateau phenomenon in random parameterized quantum circuits (RPQCs)—exponentially vanishing gradients—is given a mathematical treatment, establishing that Haar-random unitaries or even 2-designs lead to untrainable landscapes as qubit number increases. The contrast with classical neural networks is explicit: quantum gradient estimation costs scale exponentially, rendering generic deep PQCs intractable for large $n$.

Notably, architectures such as Quantum Convolutional Neural Networks (QCNNs) mitigate barren plateaus due to their hierarchical structure and polynomial scaling of gradient variance. However, these architectures may be classically simulable, limiting genuine quantum advantage.

Figure 12: Loss function standard deviation scaling for variational ground states—no exponential suppression in vicinity of local minima.

Quantum-Inspired ML: Tensor Networks and Dequantization

Quantum-inspired classical algorithms are addressed, including tensor networks and dequantization methods. Tensor networks, originally developed for efficient representation of low-entanglement quantum many-body states, are applied for compact ML architectures, efficient contraction methods, and anomaly detection.

The dequantization framework critically examines the quantum speedup claims for algorithms relying on efficient state preparation and measurement, revealing that sample-and-query access conditions enable polynomial-time classical algorithms for tasks previously believed to admit quantum exponential advantage.

Figure 13: Mapping graph coloring to tensor network contraction—demonstrating NP-hardness.

Figure 14: Tensor network representation of quantum teleportation protocol—illustrating classical simulation via contraction.

Practical and Theoretical Implications, Future Directions

• Quantum Advantage: The notes provide nuanced claims. While exponential speedups exist for gate-based algorithms given efficient state preparation and measurement, substantial portions of ML tasks can be 'dequantized', and quantum models conferring practical advantage require careful geometric and statistical analysis of data.
• NISQ Computation: Hybrid models—especially variational circuits—are promising, but non-convexity and barren plateaus remain critical obstacles. Adiabatic minimization and problem-inspired ansatzes can mitigate these issues.
• Expressivity: Quantum expressivity, governed by the frequency spectrum of encodings and the design of variational circuits, is limited by data encoding and kernel structure. Deep Fourier analysis provides a robust language for quantifying these limits.
• Tensor Network ML: Quantum-inspired classical tensor network methods offer scalable and effective ML algorithms for structured data and anomaly detection.

Conclusion

The lecture notes (2512.05151) provide an exhaustive and technical overview of quantum machine learning, integrating the mathematical foundations of probability and quantum mechanics with the leading models of quantum computation and their intersection with statistical learning theory. The analysis delineates both the capabilities and limitations of quantum ML in computational and statistical terms, unambiguously highlighting the critical importance of geometric, kernel-based, and Fourier-based frameworks in assessing quantum advantage. The paper’s systematic approach to data encoding, hybrid algorithms, kernel methods, barren plateau analysis, and quantum-inspired classical computation sets a rigorous agenda for advancing both the theory and practice of quantum machine learning. Future work should focus on the construction of data regimes and quantum architectures where quantum models offer provable predictive advantage, as well as continued exploration of tensor network and dequantized hybrids for scalable ML.