Quantum and braided ZX calculus

TL;DR

This paper extends ZX-calculus by incorporating noncommutative and noncocommutative Frobenius Hopf algebras, introduces *-structures, and explores braided tensor categories, with applications to quantum groups like u_q(sl_2).

Contribution

It generalizes ZX-calculus to braided tensor categories and quantum groups, providing new algebraic structures and a unified framework for quantum computing representations.

Findings

Extended ZX-calculus to noncommutative Hopf algebras.

Introduced *-structures and braided self-duality in ZX-calculus.

Provided examples based on quantum groups u_q(sl_2) and b_q(sl_2).

Abstract

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).