Mobius: Compiling Polynomial QLDPC Specifications into Fast Parameter and Logical Extraction

Mobius: Compiling Polynomial QLDPC Specifications into Fast Parameter and Logical Extraction This work presents Mobius, an algebraic compilation framework for efficiently extracting parameters and logical operators of Bivariate Bicycle (BB) quantum LDPC codes from their compact polynomial specifications. Existing approaches rely on constructing large binary parity-check matrices followed by Gaussian elimination, incurring cubic complexity in the number of qubits. Mobius eliminates this bottleneck by lifting the problem into the frequency domain over extension fields and decomposing the stabilizer constraints into independent orbit-wise subproblems via group-algebra techniques. The framework introduces an orbit-weighted rank aggregation method to compute code dimension with near-linear complexity, together with a trace-based reconstruction procedure that synthesizes explicit binary logical operators from frequency-domain kernels. To address non-semisimple cases arising from power-of-two lattice factors, a hybrid augmentation strategy ensures completeness while retaining computational efficiency. Experimental results demonstrate more than three orders of magnitude speedup compared to optimized binary elimination for large code instances, enabling scalable exploration of quantum LDPC code families that were previously computationally prohibitive.