1-qbit gates

The Pauli gates

The X gate swaps                                   The Y gate also swaps
the canonical basis                                  the canonical basis
vectors |0> and |1>                                  vectors |0> and |1>
on the 1-qbit space                                 but with a change of


The Z gate changes
the phase of the
canonical vector

The Hadamard gate                                                                                                            General gate


The Hadamard gate,               A general
denoted by H, acts                     gate acts
on the 1-qbit space              as a unitary
as a 2x2 unitary matrix.                matrix
It also coincides with
the Fourier transform
on the group with two
elements (binary bits).

The S-  and   T- gates


The S-gate changes                       The T-gate changes
the phase of the second             the phase of the second
qubit by 90 degrees.                           qubit by 45 degrees.

2-qbit gates and circuits

The CNOT gate


The  controlled NOT gate                 It can be used to "copy"
           (or CNOT)                                  an unknown bit or qubit
acts on the 2-qbits space.
It transforms the state  |xy>
into the state |x(x+y)>,
where the addition of binary
bits x and y is taken mod.2