For quite some time it has often been difficult for students and young researchers interested in quantum computing to bridge the gap between computer science and quantum physics. What remains are the parallel evolutions of quantum computing technologies and the educational preparation of future programmers, circuit designers, and users.

This compact textbook bases its pedagogical approach on algebra and some key paradigms specific to quantum physics, and brilliantly takes readers by the hand to lead them from the basics to dense formulations of some specific algorithmic implementations, for example, quantum Fourier transforms and quantum search. This is achieved visibly by accumulated teaching experience, but also by a comprehensive account of relevant research results over the past many years. The volume also includes examples or proofs of progressing complexity, nicely nested together, and thus pedagogically helps the reader eventually evolve toward quantum circuit design. Other strong points are 87 exercises (with solutions), an extensive index of subjects, a name index of key researchers with lifetime data, and relevant reference lists attached to each chapter. The volume also summarizes formulae in handy tabular forms.

The scientific exposé is rigorous. That said, it insufficiently highlights the differences between results on approximations and correct functional results with error bounds. While it’s fine to have segregated the methodological links between quantum physics and quantum computing in Appendix A, the exposé is a bit lightweight and definitions as well as key results are missing about decoherence, non-uniqueness of results, quantum memory aspects, and the currently preferred experimental architectures for data entry and readout with error correction. There are literature results on semi-group, Lie, and color algebras for addressing decoherence.

In a future edition, or a companion textbook, the authors would be well inspired to apply a similar pedagogy to the evolving set of quantum computing programming environments and languages, which exploit the algebraic results summarized here. This volume, due to its pedagogy and clarity, is strongly recommended to all computer scientists and physicists wanting to embark upon the voyage that quantum computing is bound to have in specific domains despite the many technological and empirical hurdles. This can only help in the analysis, diffusion, and creation of new algorithms in those fields where quantum computing may offer significant advantages.

It is unusual to run across a concise guide of definitions and established basic results, especially in the rapidly evolving field of quantum computing. Thus the high value of this volume lies precisely in the method and pedagogy involved in bringing together key results in a practical way; it is also enhanced by a comprehensive set of 87 short exercises with answers/solutions that will vastly expand the reader’s horizon. I strongly recommend *Concise guide to quantum computing* to all scientists with a mathematical or computer science background, who want to understand from a computational point of view the fundamental notions and results in the field; such readers may wish to start studying the field, or to further explore research results, or even to develop new applications of quantum computing.