Machine learning has proved to be very successful in computer science, with applications to many areas in human life. Quantum computing is a marvelous new computation model with applications to some very hard problems. This book is at the crossroads of both scientific areas.

This book is not written for the general public, but instead for specialists in machine learning who are interested in quantum computing and how it is used to improve machine learning techniques. The rigor, the mathematical detail, and the inclusion of proofs are very important contributions, although some parts are difficult to follow. And the fact that all the algorithms are written in either Cirq from Google Research or Qiskit from IBM is a remarkable contribution, allowing for very concise and exact definitions for those algorithms and ideas.

Chapter 1 provides a very simple yet complete description of the aspects of quantum physics related to quantum computing. The different quantum circuits illustrating ideas such as quantum teleportation are very insightful. Chapter 2 is more into the mathematical foundations of linear algebra and quantum computing, necessary for understanding the rest of the book, and illustrated with many simple examples and very gentle explanations. Chapter 3 is the inflection point where quantum algorithms are introduced. These are the basic building blocks for far more complex algorithms later in the book, where it is important to understand how quantum entanglement and quantum teleportation, for example, are used in practice for computing. Chapter 4 also provides some basics for more complex algorithms, but focuses only on Fourier transformations since they are the cornerstone of quantum computing. I particularly liked the part on hidden subgroup problems, as I didn’t know that quantum computing is applicable to this very general class of problems related to modular exponentiation.

The previous chapters are all preliminaries for quantum machine learning. Chapter 5 presents all the main machine learning routines in terms of quantum computing, including Euclidean distance, *k*-means clustering, and support vector machines. It is very interesting to realize how the reversibility property of quantum computing affects machine learning. Chapter 6 presents quantum deep learning and how layers are defined and combined. The possibility of fully quantum neural networks is very interesting. The final chapter (7) is devoted to optimizations techniques, since many machine learning techniques rely on learning using a cost function.

In summary: the book is well written and easy to read. Concepts, ideas, and algorithms are very well illustrated with simple examples but then also explained in exquisite mathematical detail, followed by concise yet nicely explained codification in Cirq or Qiskit.

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