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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

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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.

Blog Posts

The Story of a Photo: Transistor

Topological quantum computing: A short account for a college-level student in physics. Part I

A Collection of Resources for Learning Topological Quantum Computation

A Collection of Resources