Researchers have developed an automated framework for synthesizing logical CNOT circuits between arbitrary CSS (stabilizer codes), even when these codes are different. Traditionally, transversal CNOT operations, essential for entangling logical qubits, have been limited to identical codes or structurally related code families. This new methodology, based on the use of chain maps, allows overcoming this limitation, opening the door to greater flexibility in the design of heterogeneous quantum architectures.

The proposed method constructs the affine space of chain maps that perform the desired logical CNOT action between two distinct CSS codes. Subsequently, this space is searched to identify shallow and sparse physical circuit candidates, thereby optimizing the operation's efficiency. The system was validated using a range of heterogeneous CSS code pairs, reproducing known transversal constructions and discovering new low-depth solutions. Among these, examples were found that preserve the code distance, either fully or partially, and it was demonstrated that this preservation can be extended to the full code distance using additional flag measurements.

This ability to generate CNOT operations between different quantum codes has significant implications for various applications in quantum computing. Its potential uses in code switching, magic-state injection, Pauli product measurements, and operations on concatenated codes are discussed. Custom chain maps offer spacetime tradeoffs for logical interfaces tailored to heterogeneous architectures. Furthermore, the framework is straightforwardly extendable to targeted logical CZ gates, further expanding its utility.