I'm trying to read Quantum Mechanical Computer and to implement the CCNOT logical gate with Mathematica.
Since i wish to use the SWITCH implementation of the CNOT [Fig.8] i've realized that i need to use 7 cubits (there's no difference with registry cubits and command line cubits).
Also, i know that every operation is described by unitary matrix (we can only rotate ours cubit) i was thinking to use Pauli matrices, composed by tensorial product. As Feynman said, I'll obtain as results a $M=2^n * 2^n$.
The vector will rapresent every possibile configuration of my system (it's composed by $2^n$)
Also, using the observation that the upsite in the command line are invariant for my system (the expected falue of the number of the spinup commute with the hamiltonian of the spin) i can readuce the dimension of my matrix to $s*2^n$, and so the vector will be a unitary vector with only one $1$ in the corrispondence of the atom with the spin up.
I have to create an algebra of operation that preserve the invariant of the clock, so not every possible operation is allowed on my chain (ex: NOT is not allowed because i will have 2 or 0 spin up that is an unacceptalbe substate).
An example of a matrix that change my vector state from state $|1>$ to $|2>$ is the matrix $|2><1|$