  The real stabilizer fragment of quantum mechanics was shown to have a
complete axiomatization in terms of the angle-free fragment of the ZX-calculus.
This fragment of the ZX-calculus---although abstractly elegant---is stated in
terms of identities, such as spider fusion which generally do not have
interpretations as circuit transformations.
  We complete the category CNOT generated by the controlled not gate and the
computational ancillary bits, presented by circuit relations, to the real
stabilizer fragment of quantum mechanics. This is performed first, by adding
the Hadamard gate and the scalar sqrt 2 as generators. We then construct
translations to and from the angle-free fragment of the ZX-calculus, showing
that they are inverses.
  We then discuss how this could potentially lead to a complete axiomatization,
in terms of circuit relations, for the approximately universal fragment of
quantum mechanics generated by the Toffoli gate, Hadamard gate and
computational ancillary bits.
