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Quantum machine learning -- lecture notes

Bojan Žunkovič

TL;DR

These lecture notes synthesize quantum machine learning by contrasting classical and quantum probability, introducing density-matrix formalisms, and detailing multiple computation models. They map data into quantum embeddings, explore kernel formalism, and present hybrid quantum-classical algorithms (e.g., QAOA, VQE, and variational classifiers) with data-encoding strategies (basis, amplitude, Q-sample, Hamiltonian). The text highlights practical ML paths on NISQ devices, including block encoding, linear-combination-of-unitaries, and gradient strategies like the stochastic parameter-shift rule, while acknowledging fundamental hurdles such as barren plateaus. Collectively, the notes lay a framework for understanding where quantum advantages can emerge in ML, through Fourier-analytic perspectives, kernel methods, and quantum probabilistic modeling, and they outline both the opportunities and the current technological limitations for real-world quantum ML deployment.

Abstract

Lecture notes on quantum machine learning for computer scientists.

Quantum machine learning -- lecture notes

TL;DR

These lecture notes synthesize quantum machine learning by contrasting classical and quantum probability, introducing density-matrix formalisms, and detailing multiple computation models. They map data into quantum embeddings, explore kernel formalism, and present hybrid quantum-classical algorithms (e.g., QAOA, VQE, and variational classifiers) with data-encoding strategies (basis, amplitude, Q-sample, Hamiltonian). The text highlights practical ML paths on NISQ devices, including block encoding, linear-combination-of-unitaries, and gradient strategies like the stochastic parameter-shift rule, while acknowledging fundamental hurdles such as barren plateaus. Collectively, the notes lay a framework for understanding where quantum advantages can emerge in ML, through Fourier-analytic perspectives, kernel methods, and quantum probabilistic modeling, and they outline both the opportunities and the current technological limitations for real-world quantum ML deployment.

Abstract

Lecture notes on quantum machine learning for computer scientists.

Paper Structure

This paper contains 109 sections, 1 theorem, 345 equations, 27 figures, 5 tables.

Table of Contents

  1. Approaches to QML
  2. Classical and quantum probability
  3. Classical probability
  4. Probability simplex
  5. Majorisation
  6. Shanon entropy
  7. Relative entropy
  8. Fisher-Rao metric
  9. Quantum probability
  10. Is the mapping to quantum probability unique?
  11. Determines the quantum probability as a unique classical probability?
  12. Pure and mixed states
  13. Superposition and $\left| Schroedinger \right\rangle$
  14. Collapse of the probability
  15. Majorisation of quantum probabilities
  16. ...and 94 more sections

Key Result

Proposition 1

If a classical algorithm without training data can efficiently compute $f(x)$ for arbitrary $U_{\text{QNN}}$ and $O$, then $\mathrm{BPP} = \mathrm{BQP}$.

Figures (27)

  • Figure 1.1: Four approaches to quantum computing based on the data source and the algorithm used to process the data.
  • Figure 2.1: The space of all probabilities. In the case of $N=2$ (left), it is a line; in the case of $n=3$, it is a triangle.
  • Figure 2.2: Majorisation relation in the $N=3$ probability simplex. The vectors in the red region majorise the vector $\vec{x}$, and the vectors in the green region are majorised by the vector $\vec{x}$.
  • Figure 2.3: Binary code trees for four characters. The right tree has a shorter expected length if $p_1>p_2+p_3$.
  • Figure 2.4: The left plot represents the standard probability simplex for $N=2$ and the $L_1$ distance between the probabilities $P$ and $Q$. The right plot shows the deformed simplex in new coordinates with a flat Fisher-Rao metric and a geometric distance between the probabilities given by the angle between the square root probability vectors.
  • ...and 22 more figures

Theorems & Definitions (38)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 28 more