The Phase of the “1”

Abstract

In this chapter simulated qubits take on another dimension, the phase of the “1.” For each qubit this phase is held by a separate recursive neuron dedicated to recording phase. As in quantum theory, it is possible to employ this extra phase variable to develop useful algorithms. For instance, a simulated qubit may be prepared to be either [1 1]′ or [1 −1]′ but generally these cannot be distinguished by direct sampling and readout since both would give a true result. However after application of the h-transform, defined below, [1 1]′ is easily distinguished from [1 −1]′.

Two simulated qubits [a ₁ a ₂]′, [b ₁ b ₂]′ may be prepared to be [1 1]′, [1 1]′ meaning that each has 50 % probability of being true and 50 % probability of being false. Overall there are probabilities for the following combinations a ₁ b ₁, a ₁ b ₂, a ₂ b ₁, a ₂ b ₂. A multivibrator function is defined to be one that places a negative sign on one more of these terms, which is assumed to be accomplished by phase transforms on the simulated qubit vectors. Generalizing, for any number of simulated qubits, the locations of the negative signs will be congruent to the ones of a particular Boolean function. Negative signs are invisible in the sampling method, which provides only positive probabilities. So it is difficult to identify a multivibrator function by direct observation, should it be unknown.

However, for certain families of Boolean multivibrator functions, ways have been discovered to classify them. A variation on the Deutsch’s quantum algorithm permits classification of a multivibrator function of a single variable, using only one application of the function. A generalization to symmetric and antisymmetric functions, defined below, permits classification of binary multivibrator functions of many variables using only one application of the function. A variation on Grover’s algorithm, defined below, permits the relatively efficient identification of a code that satisfies a known multivibrator decoding function (a function that is true for exactly one binary combination as input).

Finally it is noted in this chapter that phase could be used to increase the capacity of biological memory. A data-packing example is given using n simulated qubits, in which each simulated qubit may be [1 0]′, [0 1]′, or [1 1]′ resulting in far more than the humble 2ⁿ codes that n bits usually provide.