Zi-Wen Liu

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.

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.

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.

Characterizing increasingly complex quantum systems is a central task in quantum information science, yet experimental costs often scale prohibitively with system size. Certifying key properties---such as entanglement, circuit complexity, and quantum magic---using simple local measurements is highly desirable but challenging. In this work, we introduce a highly general certification framework based on a physical phenomenon that we call localizable quantumness: for generic many-body states, essential quantum properties are robustly preserved within the projected ensembles on small subsystems after performing local projective measurements on the rest of the system. Leveraging this insight, we develop certification protocols that certify global properties by witnessing them on a small, accessible subsystem. Our method dramatically reduces experimental cost by relying solely on local Pauli measurements, while achieving constant sample complexity, constant-level robustness, and soundness for mixed states---exponentially improving the sample complexity and overcoming major limitations of previous methods. We further present a random-basis variant to certify state fidelity, with numerical evidence strongly suggesting it maintains constant sample complexity and robustness for generic states, representing a substantial improvement over existing methods. Our results provide a practical, scalable toolkit for certifying large-scale quantum processors and offer a novel lens for understanding complex many-body quantum systems.

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.

We prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental limitation of quantum information processing. In particular, no quantum operation can transform a finite number of copies of an unknown quantum state or channel into a pure state or channel that depends on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Relaxing the requirement of exact pure output, we further extend our result to establish quantitative sample complexity bounds for approximate purification, independent of any task details or operational constraints. To illustrate the practical consequences of this principle, we examine the task of approximately preparing pure dilation and, for the first time, prove an exponential lower bound on the required sample complexity.

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products, in sharp contrast to the PPT criterion. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.