Petri-Toffoli-Fredkin gates

The logic gates for reversible logic are these $8\times 8$matrices. They are almost trivial and they are all clearly equivalent up to row operations! They simply swap two rows of the input vectors. Here is the Toffoli gate matrix, or CCN (controlled controlled NOT) matrix, acting on the combined vectors:

$$\left( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\\ ... ...1 & 1\\ 1 & 0 & 0\\ 1 & 0 & 1\\ 1 & 1 & 0\\ 1 & 1 & 1\\ \end{array}\right)$$ (25)

This is a collection of matrices I and $\sigma_1$, which are the only possibilities for performing row operations on real, positive vectors. This form must therefore be universal, up to further row operations.

Notice what has been achieved by making this construction Look at the left hand side (the result) and try to identify the $~{\rm\bf AND}~$ operation.

$$\left( \begin{array}{cccc} & A & B & C\\ \hline X & 0 & 0 & 0\\ ... ... 0 & 0\\ ~ & 1 & 0 & 1\\ X & 1 & 1 & 1\\ ~ & 1 & 1 & 0\\ \end{array}\right)$$ (26)

Take the first row as giving the result of the operation and look at the coding of the other bits:

$$\begin{array}{cc\vert cc} & A & B & C\\ \hline X & 0 & 0 & 0\\ X & 0 & 0 & 1\\ X & 0 & 1 & 0\\ X & 1 & 1 & 1\\ \end{array}$$ (27)

This is simply the $~{\rm\bf AND}~$ gate, with binary numbered zeroes, i.e. to make a unique 1:1 mapping, instead of three equivalent or symmetrical zeroes, we have effectively labelled the zeroes one, two and three, in binary. The same applies to the other operations, and indeed this follows from the group properties of the matrices I and $\sigma_1$.

To summarize, by looking for a combined linear mapping from vectors of sufficient size, we find a representation which is invertible and thus reversible. The consequence of this is that these operations have a change of being represented directly as physical systems, using fundamental properties of physical law. This assumes that one can construct a system with mappings which correspond to the same representations. Since we see that only the Pauli matrices were needed to do this, we expect the be able to use reducible representations of spin systems to achieve binary computation. This clearly applies to quantum mechanics, and thus one can probably build quantum computers.