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 . 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: