Finite-Size Decoding Guarantees for Stabilizer Quantum Error-Correcting Codes

Abstract We develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. Rather than relying on asymptotic threshold argu ments, the analysis isolates decoder locality and control over syndrome connectivity as sufficient conditions for exponential suppression of logical errors at finite size. Under a stochastic Pauli noise model, we show that component-local decoders admit an explicit finite-size bound determined by the probability of large connected syndrome structures. The framework is formulated for stabilizer codes defined on bounded-degree syndrome graphs and is independent of any specific decoding algorithm. We prove a general finite-size decoding theorem and illustrate its application to the two-dimensional surface code, where the bound scales exponentially with the code distance. We further discuss the extension of the framework to quantum low-density parity-check and hypergraph-product codes. A minimal numerical illustration confirms the qualitative finite-size scaling predicted by the theory. These results provide a complementary perspective to fault-tolerance threshold theorems by mak ing explicit the mechanisms governing logical error suppression at finite code distance, with direct relevance to near- and intermediate-scale quantum devices.