A simple example of these ideas is given by a single bit. In this case there are two possible classical states and corresponding to the values 0 and 1, respectively, for the bit. This defines a two dimensional vector space of superpositions for a quantum bit. There are a number of proposals for implementing quantum bits, i.e., devices whose quantum mechanical properties can be controlled to produce desired superpositions of two classical values. One example [14, 36] is an atom whose ground state corresponds to the value 0 and an excited state to the value 1. The use of lasers of appropriate frequencies can switch such an atom between the two states or create superpositions of the two classical states. This ability to manipulate quantum superpositions has been demonstrated in small cases [56]. Another possibility is through the use of atomically precise manipulations [14] using a scanning tunneling or atomic force microscope. This possibility of precise manipulation of chemical reactions has also been demonstrated [42]. There are also a number of other proposals under investigation [1, 49, 9], including the possibility of multiple simultaneous quantum operations [38].

A simple computation on a quantum bit is the logical NOT operation, i.e., and . This operator simply exchanges the state vector's components:

This operation can also be represented as multiplication by the permutation matrix . Another operator is given by the rotation matrix

This can be used to create superpositions from single classical states, e.g.,

This rotation matrix can also be used to illustrate interference,
an important way in which quantum computers differ from probabilistic
classical algorithms. First, consider a classical algorithm with two
methods for generating random bits, (producing a ``0'' with probability
) and (producing a ``0'' with probability
). Suppose a ``0'' represents a failure
(e.g., a probabilistic search that does not find a solution) while
``1'' represents a success. Finally, let the classical
algorithm consist of selecting one of these methods to use, with
probability *p* to pick
. Then the overall probability to obtain a
``0'' as the final result is just or

The best that can be done is to choose , giving a probability of for failure.

A quantum analog of this simple calculation can be obtained from a rotation with . Starting from the individual classical states this gives superpositions

which correspond to the generators and respectively, because of their respective probabilities of and to produce a ``0'' when measured. Starting instead from a superposition of the two classical states, , corresponds to the step of the classical algorithm where generator is selected with probability . The resulting state after applying the rotation, , has probability

to produce a ``0'' value. In this case the minimum value of the probability to obtain a ``0'' is not but in fact can be made to equal 0 with the choice . In this case the amplitudes from the two original states exactly cancel each other, an example of destructive interference.

As a final example, illustrating the limits of operations on superpositions, consider the simple classical program that sets a bit to the value one. That is, and . This operation is not reversible: knowing the result does not determine the original input. By linearity, , which in turn is . This state violates the normalization condition. Thus we see that this classical operation is not physically realizable for a quantum computer. Similarly, another common classical operation, making a copy of a bit, is also ruled out [51], forming the basis for quantum cryptography [3].

Wed Mar 20 16:29:17 PST 1996