Calderbank-Shor-Steane codes on group-valued qudits

TL;DR

This work generalizes Calderbank-Shor-Steane (CSS) quantum codes to group-valued qudits by introducing group CSS codes that combine Z-type checks enforcing group-word constraints with X-type checks given by left/right multiplications. It proves a rigidity result: for non-Abelian simple groups and under mild weight/distance assumptions, every G-covariant group-CSS code is equivalent to a CW quantum double on a 2D CW complex, linking stabilizer structure to CW topology. The authors develop a comprehensive CW-double formalism, detailing code spaces, logical operators, and rate/distance bounds, and show that non-manifold CW complexes enable constant-rate codes with logarithmic Z-distance, surpassing 2D-manifold limits. They further connect these codes to phases of matter, showing that SPT/SSB and non-invertible symmetry-protected topological states can be understood as CW quantum doubles, with ghost vertices capturing boundaries and defects. Finally, the paper provides both upper bounds and constructive examples of intrinsically non-Abelian codes with favorable rate and distance properties, while highlighting the rigidity that restricts purely geometric stabilizers for non-Abelian codes. Practical implications include a unified framework for non-Abelian topological order, defect engineering, and symmetry-based fault-tolerant architectures within CW complexes.

Abstract

Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting $X$- and $Z$-type checks. We introduce CSS-like codes on $G$-valued qudits for any finite group $G$ that reduce to qubit CSS codes for $G = \mathbb{Z}_2$ yet generalize the Kitaev quantum double model for general groups. The $X$-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while $Z$-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when $G$ is non-Abelian and simple, every $G$-covariant group-CSS code with suitably upper-bounded $Z$-check weight and lower-bounded $Z$-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.

Figures (17)

• Figure 1: We illustrate relationships between classical and quantum block codes on $n$ subsystems defined using a group $G$ (i.e., group-valued coordinates for the classical case, and group-valued qudits for the quantum case). Here, $\leq$ denotes a subgroup and $\subseteq$ denotes a subset. Group CSS codes are specified by subgroups $K_i \leq G$ defining $Z$-type constraints along with a subgroup $H \leq G^{n} \times G^{n}$ defining (left- and right-) $X$-type operators. Ordianry (Abelian) CSS codes fall under the group CSS framework if we allow for local blocking of sites and expansion of local dimension. We study a generalization of quantum doubles on 2D manifolds $M$ to quantum doubles on oriented 2D CW complexes $\Sigma^2$. While quantum doubles for non-Abelian $G$ utilize only the $\leq$2D structures of such complexes, restricting to an Abelian group $G = A$ allows the quantum double to utilize the features of an arbitrary (not necessarily 2D) CW complex $\Sigma$. We identify certain $G$ for which, under mild additional assumptions, all $G$-CSS codes are quantum double codes on a 2D CW complex.
• Figure 2: Generalized $Z$-checks of group CSS codes define the admissible group elements $\mathbf{g}\in G^n$ whose computational basis states $|\mathbf{g}\rangle$ are used to construct the codewords. Here, the square region depicts all such labels, and the three circular regions of the Venn diagram denote the subsets satisfying the constraints $Z^{K_i} = 1$ for each "local" subgroup $K_{i} \leq G$. Their intersection forms the set $\mathcal{C}_Z$ of admissible basis labels for the code. All regions reduce to subgroups of $G^n$ for Abelian $G$, but neither the three circular regions nor their intersection are subgroups for non-Abelian codes.
• Figure 3: Figure illustrating quantum double codes on 2D CW complexes. Zero-cells are represented by dots, 1-cells by oriented edges, and 2-cells are shown shaded in gray. Left: The gluing map $\varphi^2:\partial e^2 \to \Sigma^1$ is defined by mapping the red dotted line continuously to the black dotted line. Group-valued degrees of freedom are attached to the edges of the graph, with the blue kets denoting the group words associated to the edges in the dotted loop. To associate a QD code to this complex, the two-cell is associated to a $Z$-check, which enforces the constraint $ab^{-1}c^{-1}d = 1$. An $X$-check is associated to each vertex acting on all adjacent qudits. Right: A 2D CW complex $\Sigma$ obtained by gluing 2-cells to the wedge sum of two circles labeled by $r, s$ as indicated by the orientations has the fundamental group $\pi_1(\Sigma) = \mathrm D_8 = \langle r,s \ | \ r^4 = 1, s^2=1, srs = r^{-1} \rangle$. This is called a presentation complex, where the circles in the wedge sum are generators and the 2-cells are relations. This illustrates a general procedure for engineering a CW complex $\Sigma$ with $\pi_1(\Sigma) = G$ for any group $G$, which controls the codespace, as implied by Eq. \ref{['eq:codespace_rough']}. These complexes are often not manifolds.
• Figure 4: Associating a rough double to a CW complex. The open circles show ghost checks. The $X$-checks associated to each vertex are shown by their action on the qudits $\ket{a}, \ket{b}, \ket{c}, \ket{d}$ in red. The gray shaded region shows a $Z$-check $\mathbb I(a^{-1}bcd^{-1} = 1)$ with a discontinuity illustrated by the dotted line. Once ghost checks are identified to a ghost vertex as shown by the double-headed arrow, the gluing rule for the $Z$-check becomes continuous.
• Figure 5: Example of codespace construction for a rough quantum double. Each box shaded gray hosts a $Z$-check, and all vertices except for the open circles host $X$-checks. The green edges correspond to $T_1$, a tree rooted at the first ghost check, and the blue edges correspond to $T_2$, a tree rooted at the second ghost check. We can see that $T = T_1 \sqcup T_2$ is a spanning tree for $\Sigma/Y$. The black edges are trivial once $T$ is fixed to the identity, and the red edges can be non-trivial. The loop corresponding to the edges labeled $a$ connects the two ghost vertices and is nullhomotopic in $\Sigma$ but non-trivial in $\Sigma/Y$. The edges labeled $b$ correspond to the loop wrapping around the puncture.

Theorems & Definitions (71)

• Definition 3.1: Classical group bit code
• Remark 3.2
• Definition 3.3
• Definition 3.4: Group GKP code
• Definition 3.5: Generalized $Z$-checks
• Definition 3.6: Generalized $X$-checks
• Definition 3.7: Group CSS code
• Definition 3.8: Doubled Hilbert space
• Definition 3.9: Orbit of $X$-checks
• Proposition 3.10