Ever wonder if there is a way to understand quantum computing using only a modest extension of high school mathematics (no complex numbers, no eigenvectors, no Fourier transform)? Look no further. Professor Chris Bernhardt introduces the main topics in a simple, clear, and logical way, with proofs and extremely easy-to-follow step-by-step examples.

This short textbook largely follows, in nine chapters, both the historical development and an increase in complexity of the quantum computing topics.

Chapter 1 starts with describing the measurements of the electron spin and photon polarization. Convincingly, they are fundamental in creating a proper mental representation of the quantum bits. Obtaining not a pseudo but a true random sequence of zeros and ones appears to be the first possible application mentioned here.

The next chapter sets, in Dirac’s notation, the basis of the linear algebra to be used from now on. It constitutes the fundamentals needed for understanding quantum computation. Included here are the definition and properties of “bra” and “ket” vectors: orthogonality, bases, matrix transformations. It finishes with a set of three algebraic tools that are frequently used in the following chapters: checking the orthonormality of a basis, expressing a ket in an orthonormal basis, and calculating the length of a ket.

Chapter 3 presents the electron spin and polarization as qubits in the simple terms of linear algebra. It explains, among other things, probability and interference. The chapter finishes with other simplified examples of quantum encryption communication applications using the BB84 protocol.

Chapter 4 characterizes the entanglement concept through a very, very gentle introduction to tensors. Entanglement implications related to superluminal communication and “spooky action at a distance” are debated through short historical and philosophical digressions. The first reference to a quantum gate (the controlled NOT gate, CNOT) appears here as an application.

Chapter 5 is dedicated to explaining the implications of Einstein’s failed proposals for local realism and hidden variables and the role that Bell inequality played. Bell’s inequality is clearly inferred from comparing the entangled qubits measured in three orthonormal bases to the classical probabilities where predetermined results are assumed. Bell’s inequality applied to the Ekert protocol for quantum key distribution is the exemplified application.

Chapter 6 brings us back to the classical computation gates, a universal set of gates and circuits with numerous examples. It touches on reversible computation and its link to information preservation and energy loss. It finishes unexpectedly with the Fredkin gate and the billiard ball computing models. In my opinion, this last section is not necessary. However, because of the analogy of billiard balls with atomic particles, it looked aesthetically pleasing to the author, as well as being justifiable from a historical perspective.

Chapter 7 switches back, in a logical and parallel manner, to the quantum counterpart of classical gates and circuits. This chapter is about the backbone of quantum computing that includes CNOT, Hadamard, X, Y, Z, and Toffoli quantum gates. It gives simple proof that a cloning quantum gate does not exist. Some basic applications follow: superdense coding, quantum teleportation, and quantum-based error corrections.

Chapter 8 describes more sophisticated and more powerful quantum algorithms, from Deutch to Deutch-Jozsa (deciding if a function is constant or balanced) and Simon’s algorithm (finding secret input binary strings that make a function have the same value). It introduces the Kronecker product, the dot product, and recursive Hadamard matrices. The author explains the P (polynomial), NP (nondeterministic-polynomial), BPP (bounded-error probabilistic polynomial), and quantic counterparts QP and BQP complexity classes, and the relevant gains obtained with quantum computing.

The final chapter discusses some of the profound impacts of quantum computing. For exemplification, it presents the fundamentals of Grover’s algorithm for searching data. Because it involves a higher level of math, the author sketches only the main ideas in Shor’s algorithm (factoring a product of large prime numbers, fundamental for RSA encryption). Synthesis in hardware, implementation difficulties, quantum supremacy, trends, and some predicted applications in chemistry and, indirectly, in biology, among others, finish the chapter and the book.

I think that the short index section and no list of references was an appropriate choice for this type of introductory textbook.

I highly recommend this introductory book if you have no prior knowledge of quantum computing but are curious about it, if you are an undergraduate student deciding on a field to follow, or maybe if you want to refresh in simple terms your forgotten knowledge of quantum-related mathematics. This book is actually for everyone, as the title says, and will trigger a strong interest to go deeper into the fascinating and exploding world of quantum computing.

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