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

• URL: http://arxiv.org/abs/2204.14038v1
• Date: Fri, 29 Apr 2022 12:17:51 GMT
• Title: Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)
• Authors: Aleks Kissinger
• Abstract summary: We show a direct correspondence between phase-free ZX diagrams and Calderbank-Shor-Steane codes. CSS codes are a family of quantum error correcting codes constructed from classical codes. We show that we can extend this translation to stabilisers and logical operators of any (possibly non-maximal) CSS code by "bending wires"
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we demonstrate a direct correspondence between phase-free ZX diagrams, a graphical notation for representing and manipulating a certain class of linear maps on qubits, and Calderbank-Shor-Steane (CSS) codes, a large family of quantum error correcting codes constructed from classical codes, including for example the Steane code, surface codes, and colour codes. The stabilisers of a CSS code have an especially nice structure arising from a pair of orthogonal $\mathbb F_2$-linear subspaces, or in the case of maximal CSS codes, a single subspace and its orthocomplement. On the other hand, phase-free ZX diagrams can always be efficiently reduced to a normal form given by the basis elements of an $\mathbb F_2$-linear subspace. Here, we will show that these two ways of describing a quantum state by an $\mathbb F_2$-linear subspace $S$ 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 maximal CSS code into a ZX diagram describing its associated state. We show that we can extend this translation to stabilisers and logical operators of any (possibly non-maximal) CSS code by "bending wires". To demonstrate the utility of this translation, we give a simple picture of the surface code and a fully graphical derivation of the action of physical lattice surgery operations on the space of logical qubits, completing the ZX presentation of lattice surgery initiated by de Beudrap and Horsman.

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