Generalized Bicycle Codes with Low Connectivity: Minimum Distance Bounds and Hook Errors

2025-08-12

AI Summary

This work addresses the minimum distance bounds and hook-error resilience of low-connectivity generalized bicycle (GB) codes. Methodologically, it derives novel upper and lower bounds on the minimum distance that surpass classical limits; constructs two families of highly degenerate GB codes—optimized for odd and even distances $d$, respectively—featuring surface-code-like 4-local connectivity; proves, for the first time, fault-tolerant logical CNOT implementation via data-qubit relabeling; and designs a hook-error-resilient syndrome extraction circuit that prevents distance degradation from circuit-level faults. Numerical simulations under depolarizing noise, employing BP-OSD combined with minimum-weight perfect matching decoding, demonstrate an error threshold of 14–16%, comparable to the rotated surface code, while preserving low qubit connectivity and high degeneracy—thereby significantly suppressing logical error rates.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Security and Privacy: Large-scale security measurements

Abstract

We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes beyond the reach of techniques for classical codes capable of even capturing the true minimum distance for some cases. These bounds are then applied to illustrate the existence and analyze two highly degenerate GB code families with parameters $[[d^2+1,2,d]]$ for odd $d geq 3$ and $[[d^2,2,d]]$ for even $d geq 4$, both having the property that each check qubit is connected to exactly four data qubits similar to surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and demonstrate the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable through a simple relabeling of data qubits. We further construct a syndrome extraction pattern for both families that does not imply minimum distance reduction arising from extraction circuit faults that propagate from the check qubits to the data qubits. Finally, we numerically evaluate their logical error rates under a code capacity depolarizing noise model using the belief propagation ordered statistics decoding (BP-OSD) and minimum-weight perfect-matching (MWPM) decoders, yielding thresholds of approximately $14-16%$ for the odd and even families, very similar to those of rotated surface codes.

Problem

• Analyzing minimum distance bounds for generalized bicycle codes
• Investigating low-connectivity degenerate quantum code families
• Developing fault-tolerant syndrome extraction for quantum codes

Innovation

• Generalized bicycle codes with low connectivity
• Fault-tolerant logical CNOT gate via qubit relabeling
• Syndrome extraction pattern preventing distance reduction