Quantum computing is a complex and highly interdisciplinary field that assumes deep knowledge of particle physics, quantum mechanics theory, mathematics, and information theory, to name the most important disciplines outside the also essential fields of technology. This thorough and compact volume focuses almost exclusively on quantum mechanics assumptions, theorems with their proofs, and a range of specific codes in relation to a range of generic quantum mechanics algorithms.

Chapter 1 reviews a set of postulates and basic formalisms, such as state vectors, Schmidt decomposition, mixed states, and formal entanglement, with some brief explanations on the links between the principles behind the measurement of the state vector and their probabilistic interpretation. On that basis, chapter 2 gives some basic specifications for several types of quantum gates, including NOT, AND, and OR, in Dirac bra-ket or operator notations. Logically, chapter 3 looks at quantum bit specifications, CNOT, and toy models with measurement schemes, but does not address (as its title claims) any realization or technology needed in quantum computing.

Some established quantum algorithms are formalized in chapter 4: teleportation, parallelism, quantum Fourier transform, quantum phase estimation, and factorization. Going into further depth on those, chapter 6 focuses on quantum error correction and stabilizers via different approaches. This follows chapter 5’s coverage of quantum decoherence and Kraus operations and the distance between quantum states, which would have benefitted from being placed first in the chapter sequence. Finally, the interesting chapter 7 is a good attempt at bridging information theory with quantum mechanics in that it discusses the essential concept of using quantum entanglement entropy as a physical resource. Subsequently, three appendices summarize key notions in mathematics and one appendix gives solutions to selected problems formulated at the end of each chapter.

Essential companion contributions to this volume are a Mathematica software application named Q3, which allows users to conduct symbolic calculations and some simulations, and a “quantum workbook” that is a set of Wolfram notebook files providing full proofs of the quantum mechanics results in the book itself (readers can also amend these for new results). The book provides an extensive bibliography and index. The style of the exposé is to mix some explanations with detailed formal Dirac specifications or gate structures; as this approach is used from the very beginning, the book does not offer a better pedagogical progression than [1] and thus has more the character of a dense compilation. When combined with its two software companions, this volume is well suited for a advanced graduate or first-year PhD course in quantum mechanics, with ample time available for self-study. It would benefit from some clearly explained elements in particle physics and technology to illustrate the difficulties in quantum computing.