Zi-Wen Liu

Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose new decoupling theorems for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.

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.

Abstract The manipulation of quantum resources such as entanglement and coherence lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to purify the quantum resources, since they are inevitably contaminated by noises and thus often lost their power or become unreliable for direct usage. In these two works, we establish a theory of the universal limitations on the accuracy and efficiency of resource purification tasks which apply to any well-behaved resource theory, for both state (static) and channel (dynamical) resources. The general results bring new insights and imply various forms of fundamental limits to a broad range of problems of great theoretical and practical importance, including magic state distillation and fault tolerant quantum computing, quantum error correction, quantum Shannon theory, and quantum circuit synthesis.