Graphical Algebraic Geometry creates universal diagrammatic languages for commutative algebras and affine varieties that also characterize the qudit ZH calculus for quantum computation.
Interacting Hopf Algebras
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abstract
We introduce the theory IH of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IH are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid-comonoid pairs. This construction is instrumental in showing that IH is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R.
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The qufinite ZXW calculus is complete for the category FHilb of finite-dimensional Hilbert spaces, as any diagram rewrites to a unique normal form.
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Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi
Graphical Algebraic Geometry creates universal diagrammatic languages for commutative algebras and affine varieties that also characterize the qudit ZH calculus for quantum computation.
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Completeness of qufinite ZXW calculus, a graphical language for finite-dimensional quantum theory
The qufinite ZXW calculus is complete for the category FHilb of finite-dimensional Hilbert spaces, as any diagram rewrites to a unique normal form.