Typing Tensor Calculus in 2-Categories

We define semi-additive 2-categories; 2-categories enriched over the category of semiadditive categories with 2-biproducts between objects. We provide both limit-form and algebraic definitions, and demonstrate both definitions are equivalent. We further propose 2-morphisms of these categories are tensor with 4 indices and demonstrate the details of horizontal and vertical compositions.

Monoidal 2-Categories: A Review

We recover Kapranov and Voevodsky's definition of monoidal 2-categories from the algebraic definition of weak 3-category (or tricategory) by Gurski. Baez and Neuchl reviewed the semi-strict definition of monoidal 2-category. Stay, one the other hand, spelled out the definition but without tensorators. Schommer Pries cited Stay. The combination of both, constructs the full description of a monoidal 2-category. We show two unit polytopes spelled out by Stay are excessive. Stay’s diagrams also need to be revised, as in the presence of tensorators, filling 2-morphisms will be modified based on the modified tensor product


Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics Unitary fusion categories formalise the algebraic theory of topological quantum computation. We rectify confusion around a category describing an anyonic theory and a category describing topological quantum computation. We show that the latter is a subcategory of Hilb. We represent elements of the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang-Baxter equation is directly connected to an instance of the P-rule of the ZX-calculus. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as π/2 phase rotations, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons.

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A Collection of Resources for Learning Topological Quantum Computation

A Collection of Resources