---
title: Hyperbolic Color Codes
url: https://www.emergentmind.com/topics/hyperbolic-color-codes
type: topic
---

# Hyperbolic Color Codes

Hyperbolic color codes are quantum error-correcting codes constructed from colorable cellulations of compact hyperbolic manifolds. In the two-dimensional setting, they place qubits on vertices of trivalent, three-face-colorable tilings and associate both $X$- and $Z$-type stabilizers with faces. Their defining geometric feature is negative curvature: the number of encoded qubits can grow linearly with the blocklength while the distance grows logarithmically. Higher-dimensional generalizations, defined on colorable triangulations of arithmetic hyperbolic manifolds, can instead achieve constant rate and polynomial distance. Hyperbolic color-code geometries also underlie Floquet constructions with weight-two measurements, finite encoding rate, and efficient graph-based decoding.

## 1. Definition and stabilizer framework

A CSS code is specified by binary parity-check matrices $H_X$ and $H_Z$ satisfying

$$
H_XH_Z^T=0.
$$

For $n$ physical qubits, the parameters are

$$
[[n,k,d]],\qquad
k=n-\operatorname{rank}(H_X)-\operatorname{rank}(H_Z),
$$

where $d$ is the minimum weight of a nontrivial logical $X$- or $Z$-operator.

A two-dimensional color code is defined on a trivalent tiling of a closed surface whose faces admit a proper three-coloring. Qubits are placed on vertices. If $A$ is the face-vertex incidence matrix, then

$$
H_X=H_Z=A,
\qquad
A_{f,v}=1\Longleftrightarrow v\in f.
$$

Thus each face $f$ supports the stabilizers

$$
B_f^X=\prod_{v\in f}X_v,
\qquad
B_f^Z=\prod_{v\in f}Z_v.
$$

For a tiling with vertex set $V$, edge set $E$, and face set $F$, the blocklength is $n=|V|$. The color-code dimension is

$$
k=4-2\chi,
\qquad
\chi=|V|-|E|+|F|.
$$

For an orientable surface of genus $g$,

$$
\chi=2-2g,
\qquad
k=4g.
$$

The factor of two relative to the $k=2g$ logical qubits of an ordinary surface code reflects the doubled homological structure of color codes. Logical operators can be described using colored strings on shrunk lattices. A string has a color, a Pauli type, and a nontrivial homology class; only two of the three color descriptions are independent because their product is stabilizer-equivalent to the identity.

The trivalence and bounded face sizes imply bounded stabilizer weights and bounded qubit participation degrees. Consequently, these families have quantum low-density parity-check (qLDPC) structure whenever the local combinatorics remain uniformly bounded.

## 2. Hyperbolic constructions and their two-dimensional parameters

The first systematic family of hyperbolic color codes was constructed from finite quotients of regular hyperbolic tilings generated by triangle-group Cayley graphs [1301.6588]. Integers $\ell,m$ are chosen so that

$$
\frac1\ell+\frac1m<\frac12.
$$

The corresponding triangle group is

$$
T(\ell,m)=
\left\langle a,b\ \middle|\ a^2=b^\ell=(ab)^m=1\right\rangle.
$$

Its Cayley graph contains $\ell$-gonal faces generated by powers of $b$ and $2m$-gonal faces generated by $(ab)^m$. Finite groups $T_r(\ell,m)$ reproduce the infinite group relations up to length $2r+1$, producing finite tilings $\tau_r(\ell,m)$ satisfying

$$
|T_r(\ell,m)|\le C^r.
$$

Every radius-$r$ neighborhood in the quotient is isomorphic to a radius-$r$ neighborhood in the infinite hyperbolic tiling. This local-planarity property is central to the distance proof: a sufficiently small operator is supported inside a planar region and is therefore stabilizer-trivial.

The finite quotient need not itself have three-colorable faces. If $H_r$ is the graph whose vertices correspond to the $2m$-gonal faces and whose edges represent shared boundaries, then

$$
\tau_r\text{ has 3-colorable faces}
\Longleftrightarrow
H_r\text{ is bipartite}.
$$

When this condition fails, a double cover $\widetilde{\tau}_r$ is formed with vertex set

$$
\widetilde V=V(\tau_r)\times\mathbb F_2,
$$

and edges

$$
\{(x,i),(y,i+1)\}.
$$

The cover is three-face-colorable and preserves the radius-$r$ local geometry.

For the double cover, $n=2|V|$, while trivalence gives $|E|=\frac32|V|$. The Euler characteristic is

$$
\chi(\tau_r)
=
\left(\frac1\ell+\frac1m-\frac12\right)|V|,
$$

and the resulting color code has

$$
k=
2\left(\frac12-\frac1\ell-\frac1m\right)n+4.
$$

Hence the asymptotic rate is

$$
\lim_{r\to\infty}\frac{k}{n}
=
1-\frac2\ell-\frac2m,
$$

which is strictly positive in the hyperbolic regime.

If

$$
M=\max\{\ell,2m\},
$$

the construction satisfies

$$
d\ge \frac{r}{3M}
$$

up to the strict-inequality convention in the proof. Since $\log n=O(r)$, the distance obeys

$$
d=\Omega(\log n).
$$

Thus the family has

$$
k=\Theta(n),
\qquad
d=\Omega(\log n),
$$

with bounded stabilizer weights. The same work proves the tradeoff

$$
kd^2\le C(\log k)^2n
$$

for bounded-degree, bounded-face-size surface and color-code tilings [1301.6588]. Consequently, any such family with $k=\Theta(n)$ satisfies

$$
d=O(\log n).
$$

The logarithmic distance is therefore asymptotically optimal within this two-dimensional bounded-geometry class, although it is not a universal limitation on qLDPC codes.

## 3. Hyperbolic polygons, examples, and geometric caveats

A separate construction uses compact orientable surfaces of genus $g\ge2$ represented as quotients of the hyperbolic plane by Fuchsian groups [1804.06382]. A regular fundamental polygon, often of type $\{4g,4g\}$, is formed by pairing opposite sides. Hyperbolic tessellations of type $\{p,q\}$ satisfy

$$
\frac1p+\frac1q<\frac12.
$$

For two-dimensional color codes, the relevant tessellations are trivalent, so $q=3$, and their faces must be three-colorable. The construction reports examples including

$$
[[16,8,4]],\qquad
[[20,12,4]],\qquad
[[24,16,4]],\qquad
[[28,20,4]].
$$

It also reports the family

$$
[[8+4g,\,4g,\,4]],
$$

whose rate is

$$
\frac{k}{n}
=
\frac{g}{g+2}
\longrightarrow1.
$$

This family has fixed distance $d=4$, so its high rate does not constitute asymptotic goodness: the distance remains bounded as $n$ grows.

The paper estimates distance using the hyperbolic side length $l_{p,q}$ and the diameter $D_{p,q}$ of the circumscribed polygon,

$$
l_{p,q}
=
\operatorname{arccosh}
\left[
\frac{\cos^2(\pi/q)+\cos(2\pi/p)}
{\sin^2(\pi/q)}
\right],
$$

$$
D_{p,q}
=
2\operatorname{arccosh}
\left[
\frac{\cos(\pi/p)\cos(\pi/q)}
{\sin(\pi/p)\sin(\pi/q)}
\right].
$$

Defining

$$
AR_{p,q}=l_{p,q}+D_{p,q},
$$

the reported estimate is

$$
d=
2\left\lceil\frac{d_h}{AR_{p,q}}\right\rceil.
$$

For $\{8,3\}$, this gives the reported code $[[16,8,4]]$; for $\{10,3\}$, the paper reports $[[10,8,2]]$.

The construction has an important internal consistency issue. It states that qubits are placed on vertices but later uses $n=2E$. In the standard trivalent vertex-qubit convention,

$$
3V=2E,
$$

so $n=V=2E/3$, not $2E$. The reported face counts, blocklengths, and parameters therefore do not all follow unambiguously from the standard cellulation identities as printed. The parameter families should consequently be understood as claims within the paper’s bookkeeping convention rather than as consequences that are fully consistent with the standard two-dimensional color-code counting formulas.

## 4. Floquet realizations on hyperbolic color-code tilings

Hyperbolic color-code tilings can be converted into Floquet codes by replacing high-weight face stabilizers with periodic measurements of weight-two edge operators. A tiling suitable for this purpose is trivalent and three-face-colorable. For a uniform $r.g.b$ tiling,

$$
|E|=\frac{3|V|}{2},
\qquad
|F|=|V|
\left(\frac1r+\frac1g+\frac1b\right).
$$

The associated Floquet code places physical qubits on vertices and assigns each edge a Pauli type determined by its color:

$$
P^R=Z,\qquad
P^G=Y,\qquad
P^B=X.
$$

The edge checks are therefore $ZZ$, $YY$, and $XX$. Products of edge checks around a face reproduce face-type stabilizers. The number of logical qubits equals the first homology dimension,

$$
k=\dim H_1
=
|V|
\left(
\frac12-\frac1r-\frac1g-\frac1b
\right)+2.
$$

For hyperbolic tilings, this gives finite asymptotic rate. The $8.8.8$ tiling on the genus-two Bolza surface yields a small code with four logical qubits and sixteen data qubits,

$$
[[16,4,d]].
$$

The hyperbolic Floquet construction studied in [2308.03750] uses three measurement sub-rounds,

$$
R\text{-checks}\longrightarrow G\text{-checks}\longrightarrow B\text{-checks},
$$

and has weight-two checks, finite encoding rate, and logarithmic distance. For ordinary hyperbolic families,

$$
k=\Theta(n),
\qquad
d=O(\log n).
$$

Semi-hyperbolic refinement subdivides the dual tiling with parameter $l$. It preserves the topology and logical-qubit count while increasing the blocklength and distance:

$$
n_l=l^2n,
\qquad
k_l=k,
\qquad
d_l\ge C_1ld.
$$

For a fixed seed surface,

$$
d_l=O(\sqrt{n_l}).
$$

The principal resource is the use of low-weight measurements. Under direct two-qubit measurement noise, semi-hyperbolic Floquet codes were numerically compared with planar honeycomb codes, with reported resource reductions of up to $48\times$ in the tested regime. Under standard circuit-level depolarizing noise, the paper reports approximately $30\times$ improvement over planar honeycomb codes and a $5.6\times$ improvement over surface-code circuits for a large instance. These are finite-size simulation results rather than asymptotic or rigorous threshold statements.

A later construction, the hyperbolic color Floquet (HCF) code, uses only $XX$ and $ZZ$ measurements on a six-step schedule,

$$
\mathrm{rXX}
\rightarrow
\mathrm{gZZ}
\rightarrow
\mathrm{bXX}
\rightarrow
\mathrm{rZZ}
\rightarrow
\mathrm{gXX}
\rightarrow
\mathrm{bZZ}.
$$

For the regular $\{8,3\}$ family,

$$
n=|V|,
\qquad
k=2+\frac n8,
\qquad
\lim_{n\to\infty}\frac{k}{n}=\frac18,
$$

and

$$
d=\Theta(\log n).
$$

Because only $XX$ and $ZZ$ checks are measured, each single-fault event generally affects at most two detectors. The resulting syndrome is graph-valued rather than hypergraph-valued, allowing minimum-weight perfect matching (MWPM) decoding. The reported circuit-level threshold is approximately $1.5\%$ for the studied noise model [2509.24110].

## 5. Logical gates, decoding, and architectural constraints

Color-code geometry supports colored string operators, shrunk lattices, holes, twists, and nontrivial genus encodings. Joint logical-Pauli measurements can implement the Clifford group using an appropriately encoded logical ancilla. The method includes logical Pauli operations, $S$, $SHS$, Hadamard generation, and CNOT through measurements such as

$$
\bar Z_c\bar Z_a,
\qquad
\bar X_a\bar X_t,
\qquad
\bar Z\bar Y_a.
$$

The joint measurement protocol temporarily modifies stabilizers along a logical string, measures the string qubits individually, reconstructs the syndrome, restores the original checks, and performs $O(d)$ rounds of syndrome extraction. A twist-defect ancilla can represent logical $\bar Y$ as a string and avoids the unfavorable CAT-state scaling associated with graph-like logical operators. The framework is explicitly demonstrated for hyperbolic surface codes and ordinary color codes in [1804.04144]; it does not itself construct or resource-count a hyperbolic color code.

Hyperbolic color-code decoders are naturally related to homological decoding. Nontrivial logical operators correspond to noncontractible cycles, while sufficiently small error patterns lie in locally planar regions and are topologically trivial. Floquet variants use detector graphs and MWPM when individual faults produce only pairs of detection events. By contrast, schedules involving $XX$, $YY$, and $ZZ$ can produce hypergraph syndromes in which a single fault flips three or more detectors.

The principal architectural limitation is that hyperbolic adjacency is not generally realizable through strictly local nearest-neighbor interactions in a planar Euclidean layout. Implementations may require long-range couplings, nonplanar connectivity, modular architectures, or semi-hyperbolic decompositions. Distance preservation during logical measurements also requires careful routing of strings, placement of defects, and control of temporary boundaries.

The Clifford constructions do not provide a non-Clifford $T$ gate by themselves. Universality therefore still requires magic-state injection and distillation unless a separate transversal or hardware-native non-Clifford mechanism is available.

## 6. Higher-dimensional polynomial-distance constructions

The logarithmic-distance limitation applies to the two-dimensional bounded-geometry families described above. A 2026 construction extends hyperbolic color codes to arbitrary dimensions $D\ge4$ using arithmetic hyperbolic manifolds and colorable triangulations [2609.16125].

A type-$i$ $D$-dimensional color code is defined on a $D$-dimensional simplicial complex $X$ whose vertices admit a proper $(D+1)$-coloring. Physical qubits occupy $D$-simplices, while $X$- and $Z$-stabilizers are indexed by $(i-1)$- and $(D-i-1)$-simplices. The logical-qubit count is

$$
k_{CC_i(X)}=\binom{D}{i}b_i(X),
$$

where $b_i(X)$ is the $i$th Betti number. The barycentric subdivision of any triangulation of a closed $D$-manifold supplies the required colorable complex.

The manifolds are arithmetic hyperbolic covers

$$
M_N=\Gamma_N\backslash\mathbb H^D,
$$

with blocklength proportional to the manifold volume:

$$
n=\Theta(\operatorname{vol}(M_N)).
$$

The distance is controlled by higher-dimensional homological systoles. For $2\le i\le D-2$,

$$
d_{CC_i(X_N)}
=
\Omega\left(n^{\gamma(i,D)}\right),
$$

where

$$
\gamma(i,D)
=
\min\left(
\frac{2(i-1)}{D(D+1)},
\frac{2(D-i-1)}{D(D+1)}
\right).
$$

The logical-qubit scaling is governed by

$$
\delta(i,D)=\frac{2i(D-i)}{D(D+1)}.
$$

For even dimension $D=2m$ and middle type $i=m$, the family has

$$
k=\Theta(n),
$$

and the distance satisfies

$$
d=
\Omega\left(
n^{\frac{m-1}{m(2m+1)}}
\right).
$$

In particular, for $D=4$ and $i=2$,

$$
d=\Omega(n^{0.1}).
$$

These are qLDPC families because the construction uses finite covers of a fixed base complex, preserving bounded stabilizer weights and bounded qubit degrees. They therefore establish constant rate together with polynomial distance in higher-dimensional hyperbolic color codes, unlike the two-dimensional families whose distance is logarithmic.

The construction does not establish an efficient decoder with polynomial decoding radius and a finite threshold, a complete circuit-level fault-tolerance theorem, self-correcting computation, or optimality of the rate and distance exponents. Its significance is instead the separation of two-dimensional and higher-dimensional behavior: two-dimensional hyperbolic color codes provide constant rate with logarithmic distance, whereas even-dimensional middle-type codes on arithmetic hyperbolic manifolds can provide constant rate with polynomial distance.

## 7. Extensions and related structures

Hyperbolic color-code ideas also appear in rainbow codes, which generalize color codes and pin codes to colored simplicial complexes that need not be manifold-like [2408.13130]. In these constructions, hypergraph-product complexes are interpreted as collections of color codes joined along domain walls. Selected maximal and rainbow subgraphs become stabilizers, allowing the removal of constant-weight logical operators that occur in some hypergraph-product pin codes.

Combining these joins with quasi-hyperbolic color codes yields qubit LDPC families with

$$
[[n,\Theta(n),\Theta(\log n)]],
$$

and transversal single-qubit $T/T^\dagger$ operations implementing logical $CCZ$ gates. The family is not asymptotically good because $d/n\to0$, but its constant rate and unbounded distance make the formal magic-state-yield parameter

$$
\gamma=\log_d(n/k)
$$

tend to zero.

The geometric coloring requirements of hyperbolic lattices should not be confused with continuous distance-graph coloring. A result on a five-chromatic same-distance graph in the hyperbolic plane proves that, for a particular distance $d\approx1.375033509$, four colors cannot avoid monochromatic pairs at that exact distance [2303.06801]. This is not a quantum-code theorem: hyperbolic color codes require a combinatorial coloring of cells or faces satisfying stabilizer and commutation conditions, not a coloring of every pair of points at a fixed metric separation.

Taken together, the constructions define several distinct regimes:

- **Two-dimensional static hyperbolic color codes**: $k=\Theta(n)$ and $d=\Omega(\log n)$, with bounded stabilizer weights.
- **Two-dimensional hyperbolic Floquet codes**: finite rate, weight-two measurements, logarithmic distance, and homological decoding.
- **Semi-hyperbolic Floquet codes**: refined distance scaling approaching $O(\sqrt n)$ for fixed-topology seeds, at the cost of reduced rate and increased resources.
- **Rainbow and quasi-hyperbolic constructions**: constant rate, logarithmic distance, and transversal single-qubit non-Clifford gates.
- **Higher-dimensional arithmetic hyperbolic color codes**: constant rate and polynomial distance for even-dimensional middle types.

The central tradeoff is therefore dimension- and architecture-dependent. Hyperbolic geometry supplies extensive logical degeneracy and qLDPC locality in the intrinsic geometry, while curvature limits two-dimensional systolic distance to logarithmic growth. Higher-dimensional topology, arithmetic covers, domain-wall joins, and dynamical measurement schedules provide distinct mechanisms for modifying this rate-distance-decoding tradeoff.

Source: https://www.emergentmind.com/topics/hyperbolic-color-codes