Long-range nonstabilizerness and quantum codes, phases, and complexity

Understand

As a necessary resource for quantum computational advantage, quantum magic (nonstabilizerness) is of fundamental importance in the study of quantum computation and physics.

• We develop a systematic theory of \emph{long-range magic (LRM)} -- nonstabilizerness that cannot be erased by shallow unitary circuits -- and demonstrate its broad relevance.
• By bridging LRM with fault-tolerant logical gate theory, we show the emergence of LRM families from quantum error-correcting codes and devise a simple yet powerful method for testing transversal logical gates.
• Further, we introduce and characterize \emph{LRM phases} in which all ground states exhibit LRM, and identify certain non-Abelian topological orders as representative examples.