Quantum Tanner Color Codes on Qubits with Transversal Gates

TL;DR

The work develops Tanner color codes, a generalized framework unifying high-dimensional expander qLDPC codes with fault-tolerant transversal gates by encoding qubits on colorable simplicial complexes with sheaf-valued local codes. A central insight is an unfolding map that expresses X/logical operators of the Tanner code in terms of multiple copies of companion sheaf codes, enabling strictly transversal gates such as a $C^{D-1}Z$ across $D$ blocks and parallelizable Clifford gates. The authors instantiate the framework on expanding coset complexes, proving constant-rate behavior in a self-dual 2D family with transversal gates (and a Floquet variant that reduces check weight to 4), and provide a rigorous code-rate formula $\dim \mathcal{C}_{\mathcal{F}}(x,z)=\binom{D}{x+1}\dim H^{x+1}(\Delta,\mathcal{F})$ with concrete lower bounds like a rate of at least $7/64$ in 2D. This approach paves a path toward asymptotically good qLDPC codes that admit fault-tolerant, geometrically-local, and highly parallel gate implementations, potentially advancing scalable, fault-tolerant quantum computation. The combination of high-dimensional expansion, sheaf-theoretic code construction, and unfolded-transversal gates offers both theoretical and practical benefits for robust quantum information processing.

Abstract

This work focuses on growing our understanding of how high dimensional expanders (HDX) can be utilized to construct highly performant quantum codes. While asymptotically good qLDPC codes have been constructed on 2D HDX built from products of graphs, these constructions have a number of limitations, like lack of structure useful for fault-tolerant logic. We develop a framework for transversal logical gates that can naturally utilize symmetric non-product simplicial HDX, and we demonstrate a particular code in this framework that offers various advantages over prior constructions. Specifically, we study the generalization of color codes to \emph{Tanner color codes}, which encompass color, pin, and rainbow codes, and should enable constructions with better parameters. We prove an `unfolding' theorem that characterizes the logical operators of the Tanner color code in terms of logical operators from several colored copies of the companion sheaf code. We leverage this understanding of the logical operators to identify a local condition that ensures such a code on a $D$-dimensional complex has a strictly-transversal $\frac{2 π}{2^D}$-phase gate on a single block, $\frac{2 π}{2^\ell}$-phase gates on subsets of a single block for $\ell<D$, and $C^{D-1}Z$ across $D$ blocks that preserve the code space. We explicitly instantiate our paradigm in every dimension with codes on highly-symmetric expanding coset complexes. These are the first qubit codes explicitly defined on expanding (non-product) simplicial complexes. We investigate in detail the self-dual 2D family, which has large rate $\geq \frac{7}{64}$ and transversal $CZ$, $S$, and $H$ gates, among many other fault-tolerant (generalizations of) fold-transversal gates arising from the symmetry of the complex. We conjecture that it has constant relative distance. We conclude by describing a Floquet variant of this code with check weight 4.

Figures (3)

• Figure 1: Top line is a schematic depiction of each edge group for the 2D complex. Middle line is the vertex group for the color $0$ (teal); the other vertex groups follow similarly by permuting entries. Bottom line on the left depicts a generic triangle of the complex with each component simplex labeled by the appropriate coset. A gray circle suggests how one can think of constructing the vertex link of $gK_0$ on the right (with $q=3$) by covering the faces that include the vertex: the chosen triangle is represented as the bottom edge on this graph connecting vertices $gK_1$ and $gK_2$. Note that we could have equivalently labeled each vertex in the link by the corresponding edge it shares with $gK_0$, e.g. replace $gK_1 \to gK_{\{2\}^c}$, and similarly each edge would be labeled by a triangle, e.g. $gK_{\{0\}^c} \to g$. This latter notation is more consistent with thinking of the link itself as being the coset complex $\Delta\left(K_0;\left(K_{\{1\}^c},K_{\{2\}^c}\right)\right)$.
• Figure 2: Illustration of the chain map \ref{['BodyDiagram']} for two different color choices (purple and orange) for the standard color code on a triangulated 2D torus. On the left we have decorated the support of an $X$ logical with the physical $X$ operators acting on the qubits on the given edges. We then choose one of two color types $T$ that we restrict to in order to get the $T^c$-shrunk lattices with colored lines in the middle; note that the physical qubits remain on edges of either the original or shrunk lattice, but each shrunk lattice contains only a subset of the qubits that the logical $X$ was originally supported on. Finally we apply the inclusion and projection $\pi_\uparrow \circ \iota$ to get a logical $X$ operator of the corresponding color $T$ on the Tanner code on the right, which has qubits on triangles and the support of the logical shaded in the appropriate color.
• Figure 3: Pictorial representation of the periodic measurement sequence of the Floquet code. The overline on the $\overline{Z}$ operators is meant to evoke that these checks come from elements of the dual code $\overline{\mathcal{F}}_e$, which is what ensures they commute with the neighboring X vertex checks from ${\mathcal{F}}_v$. The $M$ subscript specifies which set of (edge) operators is being measured in the given round t. The gray operators with a slash denote previously measured check operators that are kicked out of the instantaneous stabilizer group that round. The remaining operators (together with the $M$ subscript operators) represent elements in the instantaneous stabilizer group. Note that a vertex operator can be reconstructed from the set of edge operators on any edge that includes it, and we only visually depict the non-redundant vertex operators that cannot be derived in this way. We see that any yellow (top-right) $X$ vertex operator is initialized at $\text{t}=0$ and preserved until $\text{t}=4$, at which point it is measured again; it is destroyed in the subsequent round $\text{t}=5$ before being reinitialized at $\text{t}=6$ (not shown) when the cycle repeats.

Theorems & Definitions (76)

• Remark
• Remark
• Claim 2.1
• proof
• Remark
• Definition 2.2
• Definition 2.3: Multi-even Space
• Definition 2.4: Multi-orthogonal Spaces
• Proposition 2.5
• Lemma 2.7