Quantum Lego: building quantum error correcting codes from tensor networks

Inspired by quantum code toy models in quantum gravity, we propose the quantum legos framework for crafting novel quantum error correcting codes. This universal framework generalizes code concatenation, and is able to create any quantum code by combining a few types of simple lego blocks. For codes with a suitable quantum lego construction, we provide the current best algorithm for computing quantum weight enumerators using tensor network methods. As a corollary, it offers a more efficient way for finding quantum code distances that is up to exponentially faster than existing ones. The enumerator also provides an analytical expression for error probabilities, thereby inducing the optimal decoder when the code is subject to any i.i.d. single qubit/qudit error.