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In this paper, we demonstrate a direct correspondence between phase-free ZXdiagrams, a graphical notation for representing and manipulating a certainclass of linear maps on qubits, and Calderbank-Shor-Steane (CSS) codes, a largefamily of quantum error correcting codes constructed from classical codes,including for example the Steane code, surface codes, and colour codes. Thestabilisers of a CSS code have an especially nice structure arising from a pairof orthogonal $\mathbb F_2$-linear subspaces, or in the case of maximal CSScodes, a single subspace and its orthocomplement. On the other hand, phase-freeZX diagrams can always be efficiently reduced to a normal form given by thebasis elements of an $\mathbb F_2$-linear subspace. Here, we will show thatthese two ways of describing a quantum state by an $\mathbb F_2$-linearsubspace $S$ are in fact the same. Namely, the maximal CSS code generated by$S$ fixes the quantum state whose ZX normal form is also given by $S$.  This insight gives us an immediate translation from stabilisers of a maximalCSS code into a ZX diagram describing its associated state. We show that we canextend this translation to stabilisers and logical operators of any (possiblynon-maximal) CSS code by "bending wires". To demonstrate the utility of thistranslation, we give a simple picture of the surface code and a fully graphicalderivation of the action of physical lattice surgery operations on the space oflogical qubits, completing the ZX presentation of lattice surgery initiated byde Beudrap and Horsman.

Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)