Conceptually, the operation of Eq. (12) can be performed classically
by matrix multiplication. However, since the matrices have 
rows
and columns, this is not be a practical algorithm. As described in
§2, quantum computers can rapidly perform many matrix
operations of this size. Here we show how this is possible for the
operations used by this algorithm.
For describing the implementation, it is useful to denote the
individual components in a superposition explicitly. Traditionally,
this is done using the ket notation introduced by
Dirac [18]. For instance, the superposition described by the
state vector of Eq. (1) is equivalently
written as 
where 
just represents
a unit basis vector corresponding to the assignment s. An example of
these alternate, and equivalent, notations is:
