Fuchuan Wei

Understanding nonstabilizerness (aka quantum magic) in many-body quantum systems, particularly its interplay with entanglement, represents an important quest in quantum computation and many-body physics. Drawing motivation from the study of quantum phases of matter and entanglement, we develop a systematic and rigorous theory of the notion of long-range magic (LRM)---nonstabilizerness that cannot be (approximately) erased by shallow local unitary circuits. By establishing connections to the theory of fault-tolerant logical gates, we show the emergence of LRM state families from quantum error-correcting codes. Then, denoting phases whose ground states all exhibit LRM as LRM phases, we prove concrete conditions under which a topological order constitutes an LRM phase, with prominent examples including certain non-Abelian topological orders. Finally, from the computational complexity perspective, we discuss the intrinsic quantumness of long-range magic from e.g. preparation and learning perspectives, and formulate a "no low-energy trivial magic" (NLTM) conjecture that has key motivation in the quantum PCP context for which our LRM results suggest a promising route. We also show how correlation functions can serve as diagnostics for LRM, demonstrating certain LRM state families by correlation properties. Our concepts and results admit nontrivial extensions to approximate (robust) versions and settings without geometric locality. This work leverages and sheds new light on the interplay between quantum resources, error correction and fault tolerance, many-body physics, and complexity theory.