We revisit the notion of interacting Frobenius Hopf algebras for ZX-calculus
in quantum computing, with focus on allowing the algebras to be noncommutative
and coalgebras to be noncocommutative. We introduce the notion of *-structures
in ZX-calculus at this algebraic level and construct examples based on the
quantum group u_q(sl_2) at a root of unity. We provide an abstract formulation
of the Hadamard gate at this level and clarify its relationship to Hopf algebra
self-duality. We then solve the problem of extending the notion of interacting
Hopf algebras and ZX-calculus to take place in a braided tensor category. In
the ribbon case, the Hadamard gate coming from braided self-duality obeys a
modular identity. We give the example of b_q(sl_2), the self-dual braided
version of u_q(sl_2).Comment: 29 pages AMSlatex, 7 figures; correction to prop 2.7 and minor
improvement
