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

TL;DR

This paper establishes bounds on the minimum distance of generalized bicycle codes, analyzes their structure and fault-tolerance, and evaluates their error thresholds, showing they perform comparably to surface codes.

Contribution

It introduces new bounds for generalized bicycle codes, constructs fault-tolerant logical operations, and evaluates their error thresholds under noise models.

Findings

Bounds on minimum distance for specific GB code families.

Existence of fault-tolerant logical CNOT gates.

Logical error rate thresholds of 14-16% similar to surface codes.

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.