16 - Quantum bits and quantum gates

Summary

This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit, which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices, which are also called operators. I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere. The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible, because they are characterized by unitary matrices; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I, X, Y, Z, H, CNOT, CCNOT, CROSSOVER or SWAP, controlled-U, and controlled-controlled-U. These gates can be used to form quantum circuits, involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product, as progressively introduced through the above description, is eventually formalized.