Topological Quantum Error-Correcting Codes

• Topological quantum error-correcting codes (TQECC) are defined by encoding logical qubits in a system's global topological features using concepts like homology and non-contractible loops.
• They apply to various discretized manifolds—including torus, Klein bottle, and real projective plane—with each geometry offering unique code parameters and error correction properties.
• The practical design employs stabilizer Hamiltonians and decoding via minimum-weight perfect matching, achieving performance thresholds near a 10% error rate.

Topological quantum error-correcting codes (TQECC) are a class of quantum error-correcting codes in which logical qubits are encoded in the global topological features of a physical system, typically modeled as qubits or qudits arranged on a discretized manifold. These codes derive their error robustness, encoding rates, and logical operation structure from properties of the underlying topology, notably homology and cohomology groups. TQECC frameworks encompass but are not limited to the toric code, color codes, higher-dimensional generalizations, and codes realized on both orientable and non-orientable manifolds.

1. Algebraic Topological Construction of TQECC

A TQECC is defined using a cellulation $X_M$ of a manifold $M$ (for concreteness, a closed compact 2-manifold). Assign a physical qubit to each 1-cell (edge) $e\in E$, so the Hilbert space is $\mathcal H = \bigotimes_{e\in E} \mathbb{C}_e^2$. The stabilizer group $S$ is generated by:

• Star (vertex) operators: For each vertex $v\in V$,

$A_v = \bigotimes_{e\ni v} Z_e$

• Plaquette (face) operators: For each face $f\in F$,

$B_f = \bigotimes_{e\in\partial f} X_e$

All $A_v$ and $B_f$ are pairwise commuting and satisfy $\prod_v A_v = \prod_f B_f = I$. The code space is the simultaneous $+1$ eigenspace of all $A_v$, $B_f$, i.e., the ground space of the stabilizer Hamiltonian: $H = -J \sum_v A_v - J \sum_f B_f$

Logical operators are non-contractible loop operators: $$\bar X_\gamma = \bigotimes_{e\in\gamma} X_e,\quad \gamma \in C_1(M;\mathbb Z_2),\ \partial \gamma = 0,\ [\gamma]\neq 0$$

$$\bar Z_\beta = \bigotimes_{e\in\beta} Z_e,\quad \beta \in C^1(M;\mathbb Z_2),\ \delta \beta = 0,\ [\beta]\neq 0$$

where the intersection pairing modulo 2 ensures $\bar X_\gamma$ and $\bar Z_\beta$ anticommute if and only if $[\gamma]\cdot[\beta]=1\in\mathbb Z_2$ (Zou et al., 9 May 2025).

2. Topological Criteria for Quantum Memory and Logical Content

The fundamental requirement for a manifold $M$ to support a TQECC encoding nontrivial logical qubits is that the first homology group with $\mathbb{Z}_2$ coefficients is nontrivial: $$M\text{ admits a loop-based TQECC qubit code} \iff H_1(M;\mathbb Z_2) \ne 0$$

$k = \mathrm{rank}~H_1(M;\mathbb Z_2)$

Here, $k$ is the number of encoded qubits and equals the first $\mathbb{Z}_2$-Betti number $b_1$. For simply connected surfaces ($H_1 = 0$), TQECCs cannot encode logical qubits.

Poincaré duality over $\mathbb Z_2$ applies for compact manifolds (orientable or not), with

$H_k(M;\mathbb Z_2) \cong H^{n-k}(M;\mathbb Z_2)$

The ability to define TQECC thus extends to both orientable and non-orientable manifolds: (Zou et al., 9 May 2025).

3. Manifold-Dependent Code Families: Orientable and Non-Orientable Cases

TQECCs manifest distinct logical structures on different surface topologies:

Surface	$H_1(M;\mathbb{Z}_2)$	Encoded Qubits $k$	Logical Operator Structure
Torus $T^2$	$(\mathbb{Z}_2)^2$	$2$	Two inequiv. homology loops
Genus $g$ orientable	$(\mathbb{Z}_2)^{2g}$	$2g$	$2g$ non-contractible cycles
$\mathbb{R}P^2$	$\mathbb{Z}_2$	$1$	Loop traversing the Möbius edge
Klein bottle $K$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$2$	H/V non-contractible loops with flip
$g\mathbb{P}^2$	$(\mathbb{Z}_2)^{g}$	$g$	$g$ independent non-orientable cycles

Concrete construction for the Klein bottle: implement a square $L \times L$ lattice with twisted boundary conditions. Logical $\bar X$, $\bar Z$ are cycles corresponding to the two inequivalent non-contractible loops. For $L$ even, the distance for $Z$-loops on $K$ is $L+1$ (torus: $L$); for $L$ odd, distances coincide (Zou et al., 9 May 2025).

4. Generalization to Higher Dimensions

Given a closed $n$-manifold and cellulation, qubits can be placed on $i$-cells for $1 \leq i \leq n-1$, allowing code constructions sensitive to higher homology: $M \text{ supports %%%%54%%%%-dimensional TQECC} \iff H_i(M;\mathbb Z_2) \ne 0$ Logical content is given by the rank of $H_i(M;\mathbb Z_2)$. For example, on the $3$-torus $T^3$, both $H_1(T^3;\mathbb Z_2)$ and $H_2(T^3;\mathbb Z_2)$ are $(\mathbb{Z}_2)^3$, enabling independent 1-cell and 2-cell TQECC encodings. Stabilizers are determined via boundary and coboundary maps for the chosen cell dimension (Zou et al., 9 May 2025).

5. Novel Codes and Performance on Underexplored Topologies

The formalism of TQECC extends to surfaces beyond the torus and plane. New instances include:

• On $\mathbb{R}P^2$, a square-lattice cellulation with correct identification encodes 1 qubit; logic requires a physical realization of a Möbius traverse.
• On the Klein bottle, cellulation and identification yield two logical qubits with the minimum $Z$-distance improved by one over the torus in the even-$d$ case.

Simulations for the Klein bottle code validate the theoretical construction:

• For even code distances $d$, logical $Z$-errors have increased loop lengths $(L+1)$ versus $L$ on $T^2$, yielding a uniform improvement in $p_L$ (logical error rate), most notable at small $L$.
• $X$-error performance is identical on $K$ and $T^2$.

Decoding is performed with minimum-weight perfect matching (PyMatching). Threshold crossing of performance curves occurs at $p\approx 10\%$, identical within error to the toric code (Zou et al., 9 May 2025).

6. Fault-Tolerance and Theoretical Relevance

The expansion to arbitrary manifolds $M$ with $H_1(M;\mathbb Z_2)\ne 0$ substantially broadens the code family, with both practical and theoretical implications:

• Non-orientable surfaces such as $\mathbb{R}P^2$ and $K$ offer physically valid code platforms. While experimental realization on such exotic backgrounds is challenging, the theoretical construction is unimpeded.
• Higher-dimensional codes (e.g., $T^3$ and beyond) may be exploited for more sophisticated encoding and potentially increased fault tolerance.
• Variation in topology (especially non-orientable/boundary conditions) provides handles to optimize code distance vs. qubit overhead trade-offs.

Topological variability may thus serve as a resource for optimizing code families for engineered quantum memories, and the techniques of algebraic topology—Betti numbers, homology/cohomology, intersection pairings—fully characterize the logical structure and correctability landscape of TQECCs (Zou et al., 9 May 2025).