Quantum Lego and XP Stabilizer Codes

We apply the recent graphical framework of "Quantum Lego" to XP stabilizer codes where the stabilizer group is generally non-Abelian. We show that the idea of operator matching continues to hold for such codes and is sufficient for generating all their XP symmetries provided the resulting code is XP. We provide an efficient classical algorithm for tracking these symmetries under tensor contraction or conjoining. This constitutes a partial extension of the algorithm implied by the Gottesman-Knill theorem beyond Pauli stabilizer states and Clifford operations. Because conjoining transformations generate quantum operations that are universal, the XP symmetries obtained from these algorithms do not uniquely identify the resulting tensors in general. Using this extended framework, we provide examples of novel XP stabilizer codes with a higher distance than existing non-trivial XP regular codes and a $[[8,1,2]]$ Pauli stabilizer code with a fault-tolerant $T$ gate. For XP regular codes, we also construct a tensor-network-based maximum likelihood decoder for any independently and identically distributed single qubit error channel using weight enumerators.