Parameters of the generalised bicycle (GB) quantum code family

Establish the parameters (n, k, d) of the generalised bicycle quantum LDPC codes defined by lift l = 2^m − 1 and sets A and B chosen so that the polynomials A(x) and B(x) generate the parity-check matrices of the classical simplex codes; specifically, prove that for each integer m > 3 the resulting GB code has parameters [[2(2^m − 1), 2m, m + (m − 4)^2]].

Background

The paper instantiates the Pinnacle Architecture using a specific family of generalised bicycle (GB) quantum LDPC codes whose parity checks are derived from polynomials generating classical simplex codes. These GB codes are described by a lift l = 2^m − 1 and sets A and B, and their structure enables low-weight parity checks and practical gadget-based logical measurements for fault-tolerant computation.

For resource estimation and architecture design, knowing the exact parameters (n, k, d) of this GB code family is crucial, as these determine the number of physical and logical qubits and the code distance needed to achieve target logical error rates. The authors explicitly conjecture closed-form parameters for this family, which, if proven, would solidify the scaling behavior used throughout the architecture’s benchmarks and support confident extrapolation to larger code instances.