arXiv subjects

Zi-Wen Liu

Publications and source records attributed to Zi-Wen Liu.

At least 37 records · Page 2Linked to original sources

arXiv2026

Entirely nonlocal quantum magic without entanglement

Nonstabilizerness, or magic, is an archetypal \emph{quantum} resource that is necessary for quantum computational advantage. Here we uncover a phenomenon seemingly at odds with the quantum nature of magic: entirely nonlocal magic (ENM)---magic present only in correlations and absent from each party's marginal---can live without entanglement. We systematically study this separation and show it is universal and operationally reversible: every magical state or channel can be encoded into and recovered from a separable ENM realization using only local stabilizer processing and classical communication. We leverage this mechanism to devise an activation key protocol in which a classical key controls access to non-Clifford operations. We further formulate magic secret sharing, in which computational power inaccessible to any party alone becomes accessible through cooperation. On a superconducting quantum processor, we experimentally demonstrate activation key and network computing primitives, together with separable ENM state preparation and extraction protocols. Together, our results establish that magic can be classically activated, localized, and secret-shared without entanglement, providing new resource-control primitives for distributed quantum computation.

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arXiv2026

Intrinsic locality dimension of quantum codes

Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes, which is independent of the underlying geometry of quantum codes and naturally extends to non-integer values. Drawing on mathematical tools from fractal geometry and geometric measure theory, the intrinsic locality dimension accommodates flexible architectures and provides a quantitative measure of code connectivity, encompassing both topological codes and algebraic constructions such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we consider implications on thermal properties: toward fully characterizing the geometry requirement for self-correcting quantum memories (SCQMs), we present a conditional no-go result for SCQMs in dimension $3-ε$ and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.

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arXiv2026

Certifying localizable quantum properties with constant sample complexity

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 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 protocols to certify global properties -- including multipartite entanglement, circuit complexity, and quantum magic -- by witnessing them on a small, accessible subsystem. Remarkably, randomizing the local measurement bases extends this capability to certify state fidelity. Relying solely on local Pauli measurements, these protocols achieve constant sample complexity and robustness for almost all quantum states, including a wide range of physically relevant states. For certifying the fidelity of $n$-qubit states, this $O(1)$ scaling dramatically improves upon state-of-the-art protocols requiring $O(n^4)$ samples. Our unified framework provides both a practical toolkit for large-scale quantum certification and a novel lens into complex many-body systems.

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arXiv2026

Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check (qLDPC) codes with soundness $\tildeΘ(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates on qLDPC and quantum locally testable codes for the first time. Remarkably, our proofs proceed through highly general algebraic arguments. Building on insights from [Li et al.,~arXiv:2603.25831], we develop a general covering space framework for constructing and computing a rich family of cohomological invariant forms on sheaf codes that induce transversal logical multi-controlled-$Z$. To certify their nontriviality, we further demonstrate the existence of two-way product-expanding punctured Reed--Solomon codes, which is striking in light of the many negative examples for the product expansion behavior of ordinary Reed--Solomon codes. This approach directly overcomes the previous obstruction to realizing nontrivial logical operations while simultaneously preserving the code parameters. The claimed almost-good code results follow immediately as examples.

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arXiv2026

Witness expansion: A unified framework for analytical and measurable mixed-state resource detection

Quantum information science aims to harness different kinds of quantum resources to accomplish specific information-processing tasks. These resources also play an increasingly important role in addressing fundamental questions concerning quantum phases and dynamics. Therefore, developing powerful and practical methods for identifying and detecting quantum resources is of great significance, with applications ranging from benchmarking quantum devices to understanding the fundamental structure of quantum theory. In this work, we propose witness expansion, a unified framework for constructing nonlinear criteria for detecting quantum resources that are associated with a well-defined group of free unitaries. These criteria apply to both pure and mixed quantum states and are based on polynomial functions of the target state, which can be estimated experimentally using multiple copies of the state and evaluated analytically in certain physical models. We show how several well-known resource-detection quantities naturally emerge from our framework, including the $l_2$ norm of coherence, partial-transpose moments for entanglement, stabilizer entropy for nonstabilizerness (quantum magic), and fermionic antiflatness for fermionic non-Gaussianity. Beyond recovering these existing structures, our framework also yields new criteria for detecting qubit and qudit magic states, substantially enhancing witness-based detection capabilities. In addition, it gives, to the best of our knowledge, the first analytical criterion for detecting mixed-state fermionic non-Gaussianity with respect to the convex hull of pure fermionic Gaussian states that remains nontrivial for arbitrary numbers of qubits, demonstrating the broad applicability and conceptual unifying power of the framework.

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arXiv2026

No Universal Purification in Quantum Mechanics

Many central tasks in fundamental physics and quantum information processing are possible only insofar as mixed quantum states can be made purer. In this work, we prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental principle of quantum information processing. We first establish that no quantum operation can transform a finite number of copies of an unknown quantum state or channel into an exactly pure output that depends non-trivially on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Building on this, we show that, upon relaxing the requirement of exact purity, one can establish quantitative sample-complexity lower bounds for approximate purification that hold for arbitrary physically allowed strategies, whose scaling matches the performance of purification-related tasks across several different areas of quantum information processing. Moreover, this lower bound leads to a generalized standard quantum limit for learning arbitrary functions of a quantum state, greatly extending earlier results based on quantum Fisher information and revealing a deep connection between purification and quantum learning. Extending this principle to other important settings, we establish, for the first time, an exponential sample-complexity lower bound for approximate pure dilation state preparation and a no-go theorem for approximate bosonic Gaussian state purification with passive Gaussian operations, establishing much more stringent limitations under practical operational constraints.

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arXiv2026

Approximate quantum error correction theory of non-isometric codes

Non-isometric encoding arises in various important contexts in quantum error correction, most notably in the finite-energy, non-ideal codewords inevitable in experimental realizations of continuous-variable codes, and holographic quantum gravity. In this work, we present a general and systematic theory of non-isometric quantum error-correcting codes. In particular, we employ the approximate quantum error correction framework to quantitatively study the fundamental limitations imposed by non-isometric encodings on the accuracy of quantum error correction and implementation of logical operations. We apply our theory to analyze GKP and tiger codes under energy constraints, and discuss the implications to holography.

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arXiv2026

Can scrambling protect quantum state distinguishability under noise?

Quantum state distinguishability is a fundamental concept in quantum information science that underpins a wide range of important practical tasks. Traditionally formulated for pairs of states, quantum state distinguishability is here extended to quantum state ensembles, which we characterize through the average pairwise trace distance. Motivated by both theoretical and practical interest in noisy quantum information processing, we ask whether ``minimally'' scrambled ensembles modeled by 2-designs protect distinguishability under noise, which sheds light on the fundamental competition between noise and information scrambling. Using a rigorous decoupling approach, we establish tight bounds on noisy ensemble distinguishability. We show that the distinguishability of noisy 2-design ensembles exhibits a sharp threshold and phase-transition behavior governed by channel conditional entropy: below the threshold, the states remain mutually distinguishable with high probability, while above it, distinguishability undergoes a sudden power-law decay and then collapses exponentially. On the other hand, under local purity-shrinking noise, post-measured noisy 2-design ensembles become exponentially indistinguishable for any measurement, precluding a noise threshold for learning tasks such as shadow tomography. These results reveal a sharp difference between unmeasured and post-measured scrambled ensembles: the former can retain high distinguishability for sufficiently small noise, whereas the latter exhibits no such protected regime. We discuss the implications of these results for crucial tasks ranging from quantum communication and cryptography to learning.

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arXiv2026

On the KAK Decomposition and Equivalence Classes

The KAK decomposition is a fundamental tool in Lie theory and quantum computing. Despite its widespread use, the mathematical foundations remain incomplete, particularly regarding the precise conditions for the decomposition and the characterization of equivalence classes under multiplication by elements of $K$. Here, we present a mathematical theory of the KAK decomposition for connected compact semisimple Lie groups and derive the decomposition for $\mathrm{SU}(4)$. In particular, we clarify the relationship between various definitions of a Cartan decomposition in the literature and give a complete proof of a general KAK decomposition theorem. We then distinguish two distinct notions of KAK equivalence classes, double coset equivalence and projective equivalence, thereby addressing mathematical inconsistencies regarding KAK classification in the literature. Specifically, for $\mathrm{SU}(4)$, we show that local equivalence classes under multiplication by $\mathrm{SU}(2)\otimes \mathrm{SU}(2)$ are geometrically represented not by the usual "Weyl chamber" as claimed in the existing literature. Instead, the "Weyl chamber" is only recovered by the projective-local equivalence which disregards global phases. We develop a systematic theory for determining equivalence and uniqueness for both notions of equivalence. Our work establishes a rigorous Lie-theoretic foundation for the theory of quantum gates and circuits.

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arXiv2026

Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection

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. 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.

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arXiv2026

Approximate Quantum Error Correction with 1D Log-Depth Circuits

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 decoupling theorems tailored 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.

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arXiv2026

Long-range nonstabilizerness and quantum codes, phases, and complexity

As a necessary resource for quantum computational advantage, quantum magic (nonstabilizerness) is of fundamental importance in the study of quantum computation and physics. We develop a systematic theory of \emph{long-range magic (LRM)} -- nonstabilizerness that cannot be erased by shallow unitary circuits -- and demonstrate its broad relevance. By bridging LRM with fault-tolerant logical gate theory, we show the emergence of LRM families from quantum error-correcting codes and devise a simple yet powerful method for testing transversal logical gates. Further, we introduce and characterize \emph{LRM phases} in which all ground states exhibit LRM, and identify certain non-Abelian topological orders as representative examples. Then, adopting a complexity theory perspective, we demonstrate the classicality of non-LRM systems in e.g.~preparation and learning settings, and present a ``no low-energy trivial magic'' (NLTM) conjecture with key motivation in the quantum PCP context, for which our LRM results suggest a promising route. Additionally, we demonstrate how to diagnose LRM with correlation functions. The concepts and results admit substantive extensions to approximate (robust) and nongeometric scenarios. Our LRM theory illuminates profound new connections among quantum resources, computational advantage, error correction and fault tolerance, many-body physics, and complexity theory.

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arXiv2025

No-go theorems for logical gates on product quantum codes

Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.

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arXiv2025

SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning

Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. SU(d) is a continuous group of particular interest since it represents a fundamental type of non-Abelian symmetry and also plays a vital role in quantum computation. Here, we explicate three particularly interesting applications of symmetric random unitaries in diverse contexts ranging from physics to quantum computing: information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning. First, we show that, in the presence of SU(d) symmetry, the local conserved quantities would exhibit residual values even at $t \rightarrow \infty$ which decays as $Ω(1/n^{3/2})$ under local Pauli basis for qubits and $Ω(1/n^{(d+2)^2/2})$ under symmetric basis for general qudits with respect to the system size, in contrast to O(1/n) decay for U(1) case and the exponential decay for no-symmetry case in the sense of out-of-time ordered correlator. Second, we show that SU(d)-symmetric unitaries can be used to construct asymptotically optimal (in the sense of saturating the fundamental limits on the code error, or the approximate Eastin--Knill theorems) SU(d)-covariant codes that encode any constant number of logical qudits, extending [Kong & Liu; PRXQ 3, 020314 (2022)]. Finally, we derive an overpartameterization threshold via the quantum neural tangent kernel required for exponential convergence guarantee of generic ansatz for geometric quantum machine learning, which reveals that the number of parameters required scales only with the dimension of desired subspaces rather than the entire Hilbert space. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.

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arXiv2025

Tessellation codes: encoded quantum gates by geometric rotation

We utilize the symmetry groups of regular tessellations on two-dimensional surfaces of different constant curvatures, including spheres, Euclidean planes and hyperbolic planes, to encode a qubit or qudit into the physical degrees of freedom on these surfaces, which we call tessellation codes. We show that tessellation codes exhibit decent error correction properties by analysis via geometric considerations and the representation theory of the isometry groups on the corresponding surfaces. Interestingly, we demonstrate how this formalism enables the implementation of certain logical operations through geometric rotations of surfaces in real space, opening a new approach to logical quantum computation. We provide a variety of concrete constructions of such codes associated with different tessellations, which give rise to different logical groups. This formalism sheds a new light on quantum code and logical operation construction.

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arXiv2025

Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage

We show that a subset of the basis for the irreducible representations of a tensor-product SU(2) rotation forms a covariant approximate quantum error-correcting code with transversal U(1) logical gates. Generalizing previous work on ``thermodynamic codes" to general local spin and different irreducible representations using only properties of the angular momentum algebra, we obtain bounds on the code inaccuracy under generic noise on any known $d$ sites, under independent and identically distributed noise, and under heralded $d$-local erasures. We demonstrate that this family of codes protects a probe state with quantum Fisher information surpassing the standard quantum limit when the sensing parameter couples to the generator of the U(1) logical gate.

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arXiv2025

Spacetime Quantum Circuit Complexity via Measurements

Quantum circuit complexity is a fundamental concept whose importance permeates quantum information, computation, many-body physics and high-energy physics. While extensively studied in closed systems, its characterization and behaviors in the widely important setting where the system is embedded within a larger one -- encompassing measurement-assisted state preparation -- lack systematic understanding. We introduce the notion of embedded complexity that characterizes the complexity of projected states and measurement operators in this general setting incorporating auxiliary systems and measurements. For random circuits and certain strongly interacting time-independent Hamiltonian dynamics, we show that the embedded complexity is lower-bounded by the circuit volume -- the total number of gates acting on both the subsystem and its complement. This strengthens the complexity linear growth theorems, enriches the understanding of deep thermalization, and indicates that measurement-assisted methods generically cannot yield significant advantages in state preparation cost, contrary to expectations. We further demonstrate a spacetime conversion of certain circuit models that concentrates circuit volume onto a subsystem, and showcase applications for random circuit sampling and shadow tomography. Our theory establishes a unified framework for space and time aspects of quantum circuit complexity, yielding profound new insights and applications across quantum information and physics.

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arXiv2025

Noise-strength-adapted approximate quantum codes inspired by machine learning

We demonstrate that machine learning provides a powerful tool for discovering new approximate quantum error-correcting (AQEC) codes beyond conventional algebraic frameworks. Building upon direct observations through hybrid quantum-classical learning, we discover two new 4-qubit amplitude damping codes with an innovative noise-strength-adaptive (NSA) feature where the codeword varies with noise strength. They are NSA self-complementary and NSA pair-complementary codes. We show that they can both outperform conventional codes for amplitude damping (AD) noise. The 4-qubit self-complementary NSA code outperforms the standard LNCY AD code in fidelity and Knill-Laflamme condition violation. The pair-complementary code, which has no known non-NSA analog, achieves even better performance with higher-order loss suppression and better fidelity. We further generalize both approaches to families of NSA AD codes for arbitrary system size, as well as an NSA variant of the 0-2-4 binomial code for single-photon loss. Our results demonstrate that adaptation to noise strength can systematically lead to significant improvements in error correction capability, and also showcase how machine learning can help discover new valuable code formalisms that may not emerge from traditional design approaches.

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