A simple quantum system is the two-level
spin-
particle. Its basis states, spin-down
and spin-up 
, may
be relabelled to represent binary zero and one, i.e.,
and 
, respectively. The state of a single such
particle is described by the wavefunction
.
The squares of the complex coefficients 
and 
represent the probabilities for finding the particle in the
corresponding states. Generalizing this to a set of k
spin-
particles we find that there are now 
basis states (quantum mechanical vectors that span a Hilbert space)
corresponding say to the 
possible bit-strings of length k.
For example, 
is one such state for
k=5.
The dimensionality of the Hilbert space grows exponentially with k. In some very real sense quantum computations make use of this enormous size latent in even the smallest systems.