Quantum and braided ZX calculus
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Publication:6362722
DOI10.1088/1751-8121/AC631FzbMath1507.81125arXiv2103.07264MaRDI QIDQ6362722
Publication date: 11 March 2021
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).
Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Quantum computation (81P68)
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