Foundations of Quantum Programming

2.2.3 Controlled Gates

One-qubit gates are not enough for any useful quantum computation. In this subsection, we introduce an important class of multiple-qubit gates, namely the controlled gates.

The most frequently used among them is the CNOT operator C defined in Example 2.1.7. Here, we look at it in a different way. Let q₁,q₂ be qubit variables. Then C[q₁,q₂] is a two-qubit gate with q₁ as the control qubit and q₂ as the target qubit. It acts as follows:

$\begin{array}{l}\hfill C[q_1,q_2]|i_1,i_2\rangle =|i_1,i_1\oplus i_2\rangle \end{array}$

for i₁,i₂ ∈{0,1}, where ⊕ is addition modulo 2; that is, if q₁ is set to |1〉, then q₂ is flipped, otherwise q₂ is left unchanged. As a simple generalization of ...