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Categorical quantum computation is way of using category theory to represent quantum computation. Samson Abramsky and Bob Coecke recast the Hilbert state formalism into the language of dagger compact categories. These categories allow the initial quantum structure to be represented using a graphical calculus that greatly simplifies categorical reasoning, in what resembles a two dimensional Dirac notation.

Categorical model
The category used to model quantum computation is a dagger compact category called FD-Hilb; the category of finite dimensional complex Hilbert spaces. Its objects are finite dimensional Hilbert spaces and its arrows are linear maps. Monoidal multiplication is represented by the Kronecker tensor product, while the monoidal unit object corresponds to the set of complex numbers $$I = \mathbb{C}$$. Associativity of the tensor and tensor identities are up to equality, so $$a_{A,B,C}$$, $$l_A$$ and $$r_A$$ are reduced to identity arrows.

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