Long-range nonstabilizerness from quantum codes, orders, and correlations

Abstract: 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 natural motivations from the study of quantum phases of matter and entanglement, we systematically investigate the notion of long-range magic (LRM), defined as nonstabilizerness that cannot be (approximately) erased by shallow local unitary circuits. In doing so, we prove a robust generalization of the Bravyi--K\"onig theorem. By establishing connections to the theory of fault-tolerant logical gates on quantum error-correcting codes, we show that certain families of topological stabilizer code states exhibit LRM. Then, we show that all ground states of topological orders that cannot be realized by topological stabilizer codes, such as Fibonacci topological order, exhibit LRM, which yields a no lowest-energy trivial magic'' result. Building on our considerations of LRM, we discuss the classicality of short-range magic from e.g.~preparation and learning perspectives, and put forward ano low-energy trivial magic'' (NLTM) conjecture that has key motivation in the quantum PCP context. We also connect correlation functions with LRM, demonstrating certain LRM state families by correlation properties. Most of our concepts and techniques do not rely on geometric locality and can be extended to systems with general connectivity. Our study leverages and sheds new light on the interplay between quantum resources, error correction and fault tolerance, complexity theory, and many-body physics.