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

Aleks Kissinger|arXiv (Cornell University)|Apr 29, 2022

Quantum Computing Algorithms and Architecture12 citations

TL;DR

This paper establishes a precise, one-to-one correspondence between phase-free ZX diagrams—graphical representations of quantum states—and Calderbank-Shor-Steane (CSS) codes, a major class of quantum error-correcting codes. By showing that the F₂-linear structure underlying both phase-free ZX diagrams and CSS codes is identical, the authors enable a fully graphical derivation of stabiliser generators, logical operators, and lattice surgery operations, particularly for surface codes, using only diagrammatic rules.

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.

Motivation & Objective

• To establish a rigorous, direct correspondence between phase-free ZX diagrams and CSS codes.
• To preserve the locality and structure of stabiliser generators in the ZX calculus, avoiding the loss of physical intuition seen in prior translations.
• To provide a fully diagrammatic framework for reasoning about stabiliser codes, especially surface codes, without relying on algebraic or matrix representations.
• To extend the ZX calculus to support systematic derivation of lattice surgery operations on CSS codes using only graphical rewrite rules.
• To lay the foundation for generalising these results to qudit systems and other stabiliser codes beyond the CSS family.

Proposed method

• Leverages the F₂-linear structure of phase-free ZX diagrams, which can be reduced to a normal form based on a subspace S.
• Uses the stabiliser generators of a maximal CSS code to directly construct a corresponding phase-free ZX diagram with the same subspace S.
• Applies graphical rewrite rules—such as the π-copy, complementarity, and strong complementarity rules—to simulate physical operations like lattice surgery.
• Derives X-split and Z-merge operations on surface code patches by manipulating ZX diagrams in the X- or Z-representation, preserving logical qubit structure.
• Uses error correction (e.c.) to absorb Pauli errors introduced during measurement-based operations, treating them as negligible for logical computation.
• Extends the derivation to arbitrary-sized surface code patches and generalises to other operations via colour symmetry and rotation.

Experimental results

Research questions

• RQ1 How can phase-free ZX diagrams be systematically mapped to stabiliser codes, particularly CSS codes, while preserving their underlying F₂-linear structure?
• RQ2 Can lattice surgery operations on surface codes be derived purely through diagrammatic rewriting in the ZX calculus, without resorting to algebraic or circuit-based methods?
• RQ3 To what extent can the ZX calculus represent logical operators and stabiliser generators of non-maximal CSS codes through 'bent wires' and graphical transformations?
• RQ4 How does the correspondence between ZX diagrams and CSS codes enable a more intuitive and rigorous understanding of topological quantum codes?
• RQ5 Can this framework be generalised to qudit systems and non-CSS stabiliser codes, and what new technical challenges arise?

Key findings

• Phase-free ZX diagrams and maximal CSS codes are in one-to-one correspondence via their shared F₂-linear subspace S, with the normal form of the diagram matching the code's stabiliser generators.
• The stabiliser generators of a CSS code can be directly translated into a ZX diagram using the X- or Z-representation, preserving the code's physical locality.
• Lattice surgery operations—X-split, Z-split, X-merge, Z-merge—can be derived entirely graphically using standard ZX rules, including π-copy, complementarity, and strong complementarity.
• The derivation of these operations is scalable and applies uniformly to surface code patches of any size, not just 3×3 examples.
• The same method applies to the Z-representation by colour duality and rotation, enabling a complete graphical picture of surface code operations.
• The framework provides a foundation for extending diagrammatic reasoning to defect braiding, colour codes, and qudit systems, suggesting broader applicability beyond qubits.