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:

- 4.5.1 Forming the Initial Superposition
- 4.5.2 Adjusting Phases
- 4.5.3 The Mixing Matrix
- 4.5.4 Required Search Time

Feb. 1999