Quantum Computing for Everyone

Quantum Computing for Everyone is an introduction to quantum computing, particularly the quantum circuit model, including qubits, entanglement, quantum gates, and quantum algorithms.  

 

After a discussion of the concept of spin in quantum physics and a review of some concepts from linear algebra the book introduces the notion of a qubit.  A qubit exists in a superposition of 0 and 1 states 

 

\( a_{0} | 0 \rangle + a_{1} | 1 \rangle. \)

 

In general, the coefficients \( a_{0} \) and \( a_{1} \) might be complex, but the author sticks with real coefficients to simplify the discussion.  The coefficients are normalized so that \( | a_{0} |^{2} + | a_{1} |^{2}=1 \).  Normally the state of the qubit is hidden until the qubit is measured.  The measurement process returns a conventional 0 or 1 bit.  The result is 0 with probability \( | a_{0} |^{2} \) and 1 with probability \( | a_{1} |^{2} \).  A collection of \( n \) qubits can be entangled in a superposition of \( 2^{n} \) states

 

\( a_{0} | 0\cdots 0 \rangle + \cdots + a_{2^{n}-1} | 1 \ldots 1 \rangle . \)

 

When the entangled qubits are measured, there are \( 2^{n} \) possible results, with probabilities \( | a_{i} |^{2} \).  

 

Bernhardt continues with a discussion and comparison of classical logic gates and quantum gates.  Quantum gates operate on one or more qubits by orthogonal (or unitary in the complex case) transformations of the coefficients \( a_{0}, a_{1}, \ldots, a_{2^{n}-1} \).  

 

It is the fact that \( n \) entangled qubits can produce \( 2^{n} \) possible outputs with a controlled probability distribution that gives quantum circuits their computational power.  For example, Google researchers recently published a paper in which they describe using a quantum computer to evaluate quantum circuits with 53 qubits.  This could (in theory!) be simulated with a conventional supercomputer by evaluating a unitary transform on a vector of \( 2^{53} \) ( about \( 9 \times 10^{15} \)) elements! 

 

After developing the quantum circuit model, the author goes on to discuss how quantum circuits can be used in algorithms, including Deutsch’s, Simon’s, and Grover’s algorithms.  Since the measurements of the output qubits are probabilistic, the analysis of these algorithms requires a background in both algorithms and probability theory.  

 

The author rounds out the chapter on quantum algorithms with a brief discussion of some issues in computational complexity.  There are deep theoretical questions about whether quantum computers are fundamentally more powerful than classical computers.  Readers of this book are unlikely to have the background in graduate-level computational complexity necessary to understand recent results in this area, so the discussion here is general rather than technical.    The final chapter of the book is a discussion of the broader impact of quantum computing.

 

I have two minor concerns.  The book includes no exercises.  Since languages and systems for quantum computation such as IBM's Q Experience are becoming available, some hands-on exercises could easily have been added to the book to help make the discussion of quantum circuits less abstract.  Another concern is that the quantum annealing approach to quantum computing is only mentioned briefly.  It would have been interesting to see a comparison of the quantum circuit approach with the quantum annealing approach.    

 

Although the book is not really accessible to general readers, it does succeed as a very readable introduction to the quantum circuit model that is accessible to readers with an undergraduate-level background in linear algebra, probability, and algorithms.  I recommend it highly as an introduction to quantum computing for readers with this background.

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Brian Borchers is a professor of Mathematics at New Mexico Tech and the editor of MAA Reviews.