Asmae Benhemou, Noah Berthusen

Transversal CNOTs are ubiquitous for entangling logical qubits of identical CSS codes pairwise. For distinct codes, the options are much more limited, and are typically known only for structurally related code families. We introduce an automated framework for synthesising inter-code logical CNOT circuits between arbitrary CSS codes using chain maps. Given a prescribed bipartite logical CNOT network between these codes, our method constructs the affine space of chain maps realising the desired logical action, and then searches this space for shallow and sparse physical circuit candidates. We benchmark this method on a range of heterogeneous CSS code pairs, recovering known transversal constructions, and finding new low-depth solutions, including distance-preserving and partially distance-preserving examples, which we demonstrate can be promoted to the full code distance using additional flag measurements. We discuss applications to code switching, magic-state injection, Pauli product measurements, and operations on concatenated codes, where bespoke chain maps offer favourable spacetime tradeoffs for logical interfaces tailored to heterogeneous architectures. Finally, we show how our framework straightforwardly extends to targeted logical CZ gates.