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.

Low check weight is a practically crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we explore the theory of weight-constrained stabilizer codes from various foundational perspectives including the complexity of computing code weight and the explicit boundary of feasible low-weight codes in both theoretical and practical settings. We first prove that calculating the optimal code weight is an $\mathsf{NP}$-hard problem, demonstrating the necessity of establishing bounds for weight that are analytical or efficiently computable. Then we systematically investigate the feasible code parameters with weight constraints. We provide various explicit analytical lower bounds and in particular completely characterize stabilizer codes with weight at most 3, showing that they have distance at most 2 and code rate at most 1/4. A $\sqrt{n}$-distance limit is suggested at weight 4, while good codes exist for constant weight 5. We also develop a powerful linear programming (LP) scheme for setting code parameter bounds with weight constraints, which yields exact optimal weight values for all code parameters with $n\leq 9$. We further refined this constraint from multiple perspectives by considering the generator weight distribution and overlap. In particular, we consider practical architectures and demonstrate how to apply our methods to e.g.~the IBM 127-qubit chip. To benchmark these bounds, we present several finite-size code constructions, including examples generated via reinforcement learning. Our study brings the weight as a crucial parameter into coding theory and provides guidance for code design and utility in practical scenarios.