Quantum parallelism arises from the ability of a quantum memory
register to exist in a superposition of base states. Each component
of this superposition may be thought of as a single argument to a
function. A function performed once on the register in a
superposition of states is performed on each of the components of the
superposition. Since the number of possible states is 2^{n} where
*n* is the number of qubits in the quantum register, you can perform
in one operation on a quantum computer what would take an exponential
number of operations on a classical computer. This is fantastic, but
the more superposed states that exist in you register, the smaller the
probability that you will measure any particular one will become.

As an example suppose that you are using a quantum computer to
calculate the function
(*x*) = 2**x* mod 7, where *x* is
the superposition of integers between 0 and 7 inclusive. You could
prepare a quantum register that was in a equally weighted
superposition of the states 0-7. Then you could perform the
2**x* mod 7 operation once, and the register would contain the equally
weighted superposition of 1,2,4,6,1,3,5,0 states, these being the
outputs of the function
2**x* mod 7 for inputs 0 - 7. When
measuring the quantum register you would have a 2/8 chance of
measuring 1, and a 1/8 chance of measuring any of the other
outputs. It would seem that this sort of parallelism is not useful,
as the more we benefit from parallelism the less likely we are to
measure a value of a function for a particular input. Some clever
algorithms have been devised, most notably by Peter Shor which succeed
in using quantum parallelism on a function where there is interest in
some property of all the inputs, not just a particular one.

This kind of parallelism is very appealing for simulation on a
parallel computer. A *n* bit quantum register contains a
superposition of each of its 2^{n} possible base states, and we
represent this by an array of 2^{n} complex numbers which are
probabilities of measuring the quantum register to be the
corresponding base state. To perform an operation on the quantum
register, we simply modify each of the 2^{n} array locations. By
splitting the calculations of how to change the probability values of
the array locations into even ranges via process elements, we get
nearly linear speedup.