Cylindrical and Möbius Quantum Codes for Asymmetric Pauli Errors

TL;DR

The paper tackles the prevalence of asymmetrical Pauli errors in quantum devices by introducing two CSS topological code families, cylindrical and Möbius, derived as fiber-bundle specializations of topological codes via chain-complex techniques. Using tensor-product (and twisted) constructions, the authors derive code parameters (e.g., $[[L^2+L(L-1),1,L]]$ for cylindrical with distance $d=L$) and provide both analytical and numerical assessments of performance under the minimum weight perfect matching (MWPM) decoder. Central to the analysis are weight-enumerator polynomials $L(z)$ obtained from quantum MacWilliams identities, enabling upper bounds on the logical error rate $p_L$ and precise beta-coefficients $\beta_j$ for small codes. Numerical results show that cylindrical and Möbius codes outperform standard surface codes across depolarizing and phase-flip channels, with Möbius codes offering the strongest gains under high asymmetry, highlighting their potential for fault-tolerant quantum hardware with biased noise and limited connectivity.

Abstract

In the implementation of quantum information systems, one type of Pauli error, such as phase-flip errors, may occur more frequently than others, like bit-flip errors. For this reason, quantum error-correcting codes that handle asymmetric errors are critical to mitigating the impact of such impairments. To this aim, several asymmetric quantum codes have been proposed. These include variants of surface codes like the XZZX and ZZZY surface codes, tailored to preserve quantum information in the presence of error asymmetries. In this work, we propose two classes of Calderbank, Shor and Steane (CSS) topological codes, referred to as cylindrical and Möbius codes, particular cases of the fiber bundle family. Cylindrical codes maintain a fully planar structure, while Möbius codes are quasi-planar, with minimal non-local qubit interactions. We construct these codes employing the algebraic chain complexes formalism, providing theoretical upper bounds for the logical error rate. Our results demonstrate that cylindrical and Möbius codes outperform standard surface codes when using the minimum weight perfect matching (MWPM) decoder.

Figures (8)

• Figure 1: Pictorial representation of a $[[13, 1, 3]]$ surface code. (a) Actual physical representation of the qubits of the code. Data qubits in circles and ancilla qubits on squares. (b) Simplified representation of the same lattice.
• Figure 2: (a) $[[ 13,1,3 ]]$ surface code. Data qubits are depicted as circles, blue ancillas represent $\bm{Z}$ stabilizers while red ancillas stand for $\bm{X}$ stabilizers. Examples of $\mathbb{ \bm{X} _L}$ and $\mathbb{ \bm{Z} _L}$ logical operators are drawn on the lattice. (b) $[[ 15,1,3 ]]$ cylindrical code. (c) $[[ 15,1,3 ]]$ Möbius code.
• Figure 3: Some comparison between logical operators on the surface and the cylindrical codes. The pattern highlighted in (b) is not a logical operator. In (a) and (b) error patterns are composed by $\bm{Z}$ operators, while in (c) and (d) are composed by $\bm{X}$ operators.
• Figure 4: Logical error rate over physical error rate, $[[13,1,3]]$ surface code and $[[15,1,3]]$ cylindrical code with MWPM decoder. Depolarizing and phase flip channels channel. The curves refer to the asymptotic approximations \ref{['eq:error_probWithBetaApprox']}. Squares and circles correspond to MWPM decoding.
• Figure 5: Asymptotic behaviour of the logical error rate over channel asymmetry $A$ for surface and cylindrical, and Möbius codes. The logical error probabilities are computed at physical error probability $p = 0.01$. Solid line: exact \ref{['eq:error_probWithBetaApprox']}. Dashed line: upper bound \ref{['eq:boundCor']}.

Theorems & Definitions (6)

• Example 1: Topological interpretation of classical linear codes
• Lemma 1: CSS binary construction SarKlaRot:09
• Example 2: Topological interpretation of quantum CSS codes
• Theorem 1
• proof
• Corollary 1