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Elementary quantum notation:

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.

Samuel L.~Braunstein
Wed Aug 23 11:54:31 IDT 1995