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.