Current quantum computing uses 2x2 (Pauli) operators
and (4x4) Dirac operators expressed as matrices.
But Hestenes
https://en.wikipedia.org/wiki/David_Hestenes
showed that these are simple subdomains of the Clifford
(aka Geometric) Algebra. Axiom implements the Clifford
algebra so it should be possible to re-formulate the
quantum algorithms using operations expressed in
this more general form.
It is possible that these more general Clifford forms would
improve clarity and expressiveness for quantum algorithms.
In particular, applying Clifford notation to the Bloch sphere
rotations seems like an interesting idea.
Also of interest is that the Hadamard gate, expressed as a
matrix [[1,1],[1,-1]], is fundamental quantum operator. But it
is also the method of constructing communication codes so
that multiple endpoints can fully share the same channel using
the full channel bandwidth. This leads to the thought that
quantum communication is intimately linked with these higher
codes. See
http://www.uow.edu.au/~jennie/WEBPDF/2005_12.pdf
A Geometric Algebra formulation of these matrix operations
provides a more general, coordinate-free language.
This leads to the need for more work on the Clifford domain.
Tim
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