Title: Categorical Semantics of the ZX-calculus
Abstract: The ZX-calculus is a graphical language for qubit quantum circuits. In other words, it is a presentation for the full subcategory of complex matrices under the bilinear tensor product, where the objects are powers of 2. A consequence of Zanasi’s thesis is that the prop of linear spans over F_2 is equivalent to the phase free fragment of the ZX-calculus. We extend this correspondence to the affine and nonlinear cases. In the former case, we show that the fragment of the ZX-calculus with one pi-phase is a presentation for the full subcategory of spans of finite dimensional F_2-affine vector spaces, where the objects are non-empty affine vector spaces. In the latter case, we show that the fragment of the ZH-calculus with natural number H-boxes is a presentation for the full subcategory of spans of sets of finite functions where the objects are powers of 2 (the ZH -calculus is equivalent to the ZX-calculus). We must consider these full subcategories of spans because in these cases, because unlike in the linear case, the full categories of spans are not themselves props, having too many objects. These results are proven as modularly as possible, incrementally adding generators via pushout and distributive laws of props.