The Category TOF

We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points, is an equivalence into the full subcategory of sets and partial isomorphisms with objects finite powers of the two element set. The structure of the proof builds – and follows the proof of Cockett et al. – which provided a full set of identities for the cnot gate with computational ancillary bits. Thus, first it is shown that TOF is a discrete inverse category in which all of the identities for the cnot gate hold; and then a normal form for the restriction idempotents is constructed which corresponds precisely to subobjects of the total points of TOF. This is then used to show that TOF is equivalent to FPinj2, the full subcategory of sets and partial isomorphisms in which objects have cardinality a power of 2.