Quantum error-correcting codes are central to fault-tolerant quantum computation. Yet useful codes are often complex, making them difficult to design and analyze systematically. In this talk, we introduce a modular framework — the quantum lego method — based on tensor networks, which enables the construction and analysis of arbitrary quantum codes over qubits or qudits by assembling simpler, atomic building blocks. We begin with a pedagogical introduction to the quantum lego framework and its general utility in quantum code design and coding theory. We then show that transversal gates act as symmetries of the atomic lego blocks, and that tracking how these symmetries propagate through the tensor network composed of the quantum lego blocks yields codes with exotic transversal gates. This is achieved with relative ease compared to traditional construction approaches based on “quantizing” classical codes. As a proof of principle, we demonstrate that this technique produces new codes admitting transversal non-Clifford gates, and reveals significant overhead reductions for fault-tolerant logical gates in existing codes. Together, these results open the door to more efficient, automated code discovery.
