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

Aleks Kissinger

TL;DR

This work establishes a precise correspondence between phase-free ZX diagrams and CSS stabiliser codes by showing that both are governed by the $\mathbb{F}_2$-linear subspaces, with phase-free ZX normal forms corresponding to code spaces generated by a subspace $S$ and its orthogonal $S^{\perp}$. It extends the translation to non-maximal CSS codes via encoder maps obtained by bending wires, enabling a diagrammatic encoding of logical qubits and their operators. The authors provide a fully graphical derivation of lattice surgery in the surface code, using the ZX embedding to relate physical operations to logical transformations. The results yield a complete ZX picture of CSS codes, including stabilisers, logical operators, and encoders, and point toward extensions to qudits and other stabiliser families, offering a practical, visual toolkit for fault-tolerant quantum computation.

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.

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

TL;DR

This work establishes a precise correspondence between phase-free ZX diagrams and CSS stabiliser codes by showing that both are governed by the -linear subspaces, with phase-free ZX normal forms corresponding to code spaces generated by a subspace and its orthogonal . It extends the translation to non-maximal CSS codes via encoder maps obtained by bending wires, enabling a diagrammatic encoding of logical qubits and their operators. The authors provide a fully graphical derivation of lattice surgery in the surface code, using the ZX embedding to relate physical operations to logical transformations. The results yield a complete ZX picture of CSS codes, including stabilisers, logical operators, and encoders, and point toward extensions to qudits and other stabiliser families, offering a practical, visual toolkit for fault-tolerant quantum computation.

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 -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 -linear subspace. Here, we will show that these two ways of describing a quantum state by an -linear subspace are in fact the same. Namely, the maximal CSS code generated by fixes the quantum state whose ZX normal form is also given by . 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.
Paper Structure (12 sections, 6 theorems, 29 equations, 2 figures)

This paper contains 12 sections, 6 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries
  4. ZX diagrams
  5. CSS codes
  6. Normal forms for phase-free ZX diagrams
  7. Stabilisers and Pauli ZX diagrams
  8. Maximal CSS codes as ZX diagrams
  9. Non-maximal CSS codes as ZX encoder maps
  10. Picturing the surface code
  11. Lattice surgery
  12. Conclusion and future work

Key Result

Theorem 2.2

If $\mathcal{S} \subseteq \mathcal{P}_n$ has $m$ generators, then $\textrm{Stab}(\mathcal{S})$ is a $2^{n-m}$ dimensional subspace of $(\mathbb C^2)^{\otimes n}$.

Figures (2)

  • Figure 1: The rules of the phase-free ZX calculus: the spider rules (sp) and strong complementarity (sc). Note the righthand-side of the (sc) rule is a complete bipartite graph of $m$ Z spiders and $n$ X spiders, with a normalisation factor $\nu := 2^{(m-1)(n-1)/2}$, which we typically drop when scalar factors are irrelevant.
  • Figure 2: The rules of the Pauli ZX calculus. Again, the righthand-side of the (sc) rule is a complete bipartite graph of $m$ Z spiders and $n$ X spiders, this time with a scalar factor $\nu' := (-1)^{jk} 2^{(m-1)(n-1)/2}$.

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 2.2: FTST
  • Definition 2.3
  • Example 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4: Completeness of phase-free ZX
  • ...and 5 more