Contextuality of Quantum Error-Correcting Codes (2502.02553v2)

Abstract: Quantum error correction is vital for fault-tolerant quantum computation, with deep connections to entanglement, magic, and uncertainty relations. Entanglement, for instance, has driven key advances like surface codes and has deepened our understanding of quantum gravity through holographic quantum codes. While these connections are well-explored, the role of contextuality, a fundamental nonclassical feature of quantum theory, remains unexplored. Notably, Bell nonlocality is a special case of contextuality, and prior works have established contextuality as a key resource for quantum computational advantage. In this work, we establish the first direct link between contextuality and quantum error-correcting codes. Using a sheaf-theoretic framework, we define contextuality for such codes and prove key results on its manifestation. Specifically, we prove the equivalence of definitions of contextuality from Abramsky--Brandenburger's sheaf-theoretic framework and Kirby--Love's tree-based approach for the partial closure of Pauli measurement sets. We present several findings, including the proof of a recent conjecture posed by Kim and Abramsky. Working within the partial closure, we further show that subsystem stabilizer codes with two or more gauge qubits are strongly contextual, while others are noncontextual. As a consequence, we highlight that several protocols for code-switching between stabilizer codes, which admit a universal transversal gate set, are strongly contextual. Our findings reveal a direct connection between contextuality and quantum error correction, offering new insights into the nonclassical resources enabling fault-tolerant quantum computation.