Terra incognita of the quantum dimension – Space of probabilities

The number of atoms which are involved in storing a “Black” or “White” bit state was 10^(17), in the year 1960, and has been exponentially reduced in the course of computer development. At present, we are at a point where only a few atoms are needed for this sort of data processing. However, with further miniaturization, we face quantum effects. Consequently, we can only operate with probabilities and interference, which means that we enter the quantum dimension.

Here, we discuss general conclusions resulting from the basic principles of quantum mechanics, viz. “probability” and “interference”, which applies to any quantum computer, independent of its explicit physical realization.

Each field on this chess board corresponds to one bit, that can assume either the value 1 or 0, or the colour “Black” or “White”. In contrast to the quasi-deterministic computer, where the value is either black or white, in case of a quantum computer, we operate with probabilities P(Black) and P(White) for black or white.

For one bit, due to conservation of probability, P■ + P □ =1, only one of the two probabilities can be freely selected. That means that one parameter is free. For the special case of P■ =0 and P □ =1 or P■ =1 and P □ =0, we obtain a deterministic computer. Thus, the classic computer can be considered a special case of the quantum computer.
For 2 bits, four combinations (□□, □■, ■□, ■■) arise, with four probabilities (P□□, P □■, P■□, P■■ ). Due to conservation of probability, three free parameters suffice to describe these probabilities.
For 3 bits, we obtain eight combinations. Due to conservation of probability, seven free parameters suffice to describe the probabilities.

The number of possible combinations doubles with each field, and is generally 2^N. The number of free parameters for the probabilities is 2^N -1 for N bits or N chessboard fields.