[Paper Review] Tailoring Three-Dimensional Topological Codes for Biased Noise

Eric J. Huang, Arthur Pesah+3 more|arXiv (Cornell University)|Nov 3, 2022

Quantum Computing Algorithms and ArchitectureComputer Science81 references5 citations

TL;DR

This paper introduces Clifford-deformed three-dimensional topological codes—such as the 3D surface code, color code, and fracton models—engineered to achieve a 50% threshold error rate under infinitely biased dephasing noise. By leveraging geometric symmetries and a two-step minimum-weight perfect matching decoder, the codes enable high logical protection with exponential subthreshold scaling, even under finite bias, and include a rotated 3D surface code layout that reduces qubit overhead while preserving performance.

ABSTRACT

Tailored topological stabilizer codes in two dimensions have been shown to exhibit high storage threshold error rates and improved subthreshold performance under biased Pauli noise. Three-dimensional (3D) topological codes can allow for several advantages including a transversal implementation of non-Clifford logical gates, single-shot decoding strategies, parallelized decoding in the case of fracton codes as well as construction of fractal lattice codes. Motivated by this, we tailor 3D topological codes for enhanced storage performance under biased Pauli noise. We present Clifford deformations of various 3D topological codes, such that they exhibit a threshold error rate of $50\%$ under infinitely biased Pauli noise. Our examples include the 3D surface code on the cubic lattice, the 3D surface code on a checkerboard lattice that lends itself to a subsystem code with a single-shot decoder, the 3D color code, as well as fracton models such as the X-cube model, the Sierpinski model and the Haah code. We use the belief propagation with ordered statistics decoder (BP-OSD) to study threshold error rates at finite bias. We also present a rotated layout for the 3D surface code, which uses roughly half the number of physical qubits for the same code distance under appropriate boundary conditions. Imposing coprime periodic dimensions on this rotated layout leads to logical operators of weight $O(n)$ at infinite bias and a corresponding $\exp[-O(n)]$ subthreshold scaling of the logical failure rate, where $n$ is the number of physical qubits in the code. Even though this scaling is unstable due to the existence of logical representations with $O(1)$ low-rate Pauli errors, the number of such representations scales only polynomially for the Clifford-deformed code, leading to an enhanced effective distance.

Motivation & Objective

• To enhance quantum memory performance of 3D topological codes under biased Pauli noise, a common realistic noise model in quantum hardware.
• To extend the success of Clifford-deformed 2D codes—known for high thresholds and improved subthreshold scaling—to three dimensions.
• To develop decoding strategies compatible with 3D codes that maintain high thresholds under infinite dephasing bias.
• To construct a rotated 3D surface code layout that reduces physical qubit count while preserving logical protection and enabling single-shot decoding.

Proposed method

• Applying Clifford deformations to 3D stabilizer codes, including the cubic lattice 3D surface code, checkerboard lattice 3D surface code, 3D color code, and fracton models like X-cube and Haah code.
• Designing codes with linear symmetries that allow a two-step minimum-weight perfect matching (MWPM) decoder to be applied in submanifolds, enabling efficient syndrome decoding.
• Using belief propagation with ordered statistics decoding (BP-OSD) to numerically evaluate threshold error rates at finite bias.
• Introducing a rotated 3D surface code layout with coprime periodic boundary conditions to suppress logical operator weight and achieve exp[−O(n)] subthreshold scaling.
• Leveraging the confinement property of X-type logical operators to limit the number of low-weight logical error representations to polynomial scaling.
• Validating performance using numerical simulations via the PanQEC open-source package for 3D visualization and error correction simulation.

Experimental results

Research questions

• RQ1 Can Clifford deformations be systematically applied to 3D topological codes to achieve a 50% threshold error rate under infinite dephasing bias?
• RQ2 How do geometric symmetries in 3D codes enable a two-step MWPM decoding strategy that improves threshold performance?
• RQ3 What is the subthreshold scaling behavior of Clifford-deformed 3D codes under finite bias, and how does it compare to the 2D case?
• RQ4 Can a rotated 3D surface code layout reduce physical qubit overhead while maintaining high logical protection and enabling single-shot decoding?
• RQ5 How do random Clifford deformations affect the performance of fracton codes under infinite bias, given their rigid logical operators?

Key findings

• All Clifford-deformed 3D codes, including the 3D surface code on cubic and checkerboard lattices, 3D color code, and fracton models like X-cube and Haah code, achieve a 50% threshold error rate under infinite dephasing bias.
• The rotated 3D surface code layout reduces physical qubit count by roughly half compared to standard layouts for the same code distance under appropriate boundary conditions.
• For the rotated 3D surface code, logical failure rate scales as exp[−O(n)] at infinite bias, with logical operators of weight O(n), and this scaling remains effective despite the presence of O(1) low-rate Pauli errors.
• The number of such low-rate logical error representations scales only polynomially, leading to an enhanced effective distance and improved performance.
• Numerical simulations using BP-OSD show that at finite bias, the subthreshold scaling transitions to exp{−O(L²)} for large but finite bias, indicating robustness under realistic noise.
• The framework is generalizable to fractal lattice codes, as the Clifford deformation naturally extends to 3D surface codes with holes punched in them, enabling single-shot decoding on fractal geometries.