Finite-rate sparse quantum codes aplenty

We introduce a methodology for generating random multi-qubit stabilizer codes based on solving constraint satisfaction problem (CSP) bipartite graphs. This framework allows us to enforce commutation, $X/Z$ balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in CSP that we can then solve numerically. Using state-of-the-art solver, obtain convincing evidence the existence of satisfiability threshold. Furthermore, extent satisfiable phase increases with number qubits. In phase, finding sparse becomes an easy problem. Moreover, observe found practically achieve channel capacity erasure noise. Our results show intermediate-size finite-rate quantum are find, while also demonstrating flexible good custom properties. therefore establish complete customizable pipeline code discovery.