Universal Bit-Flip Error Correction for Arbitrary-Size Data Registers Using a Single Ancilla Qubit via Similarity Propagation

Quantum computing systems remain fragile due to decoherence and operational noise. Although large-scale quantum error correction offers robustness against general error models, its qubit and circuit overhead is often prohibitive for near-term devices. In this work, we present a scalable bit-flip error correction methodology that protects an arbitrary number of data qubits using only a single ancilla qubit, independent of register size. The method extends a previously introduced 4-qubit correction circuit for a 3-qubit data block and introduces a novel similarity circuit that enforces intra-triplet uniformity (always producing outputs of the form 000 or 111). Dedicated tail subcircuits handle the cases N = 3n + 1 and N = 3n + 2.