5.2 Single qubit gates

In quantum computing, a gate is a reversible transformation of a qubit state to another qubit state , represented by an operator . It is convenient to write the single-qubit gate operators as 2 × 2 matrices. Therefore, the action of the gate can be written as follows:

The word reversible is important because it is a reminder that a gate operation is of a different nature from a measurement. The operation of the gate can be reversed so that it is possible to get back to the state , whereas, in general, a measurement makes an irreversible change to the qubit state. is defined as the operator needed to reverse the gate action and transform back to :

therefore

which means that

where is the identity operator, represented by the 2 × 2 matrix appearing in Equation 13. An operator, that obeys Equation 13 is called a unitary operator, hence gates are represented by unitary operators. The identity operator is itself a gate, denoted , and its symbol is shown in Figure ‘6.

You will now look at some gates, starting with the quantum NOT gate. (From now on the prefix quantum will be omitted as long as it is obvious the gates are quantum gates and not classical gates from the context.)