arXiv subjects

Liang Kong

Publications and source records attributed to Liang Kong.

At least 19 recordsLinked to original sources

arXiv2026

An Efficient Fault-Tolerance Scheme for CKKS Computation on CPUs

Fully homomorphic encryption (FHE) enables computation on encrypted data, but its long ciphertext dataflow and high-dimensional modular arithmetic make it vulnerable to silent data corruption caused by transient hardware faults. Existing protection methods either target dedicated accelerators or impose substantial execution, modular-arithmetic, and memory-access overheads on CPUs. This work presents an efficient fault-tolerance scheme for CPU-based CKKS computation. It checks the input-output consistency of polynomial operators while reducing protection overhead at three levels. First, modulus-aware bucket checksum exploits wide CPU accumulators to reduce expensive modular reductions. Second, dataflow-fused in-operator checking embeds checksum accumulation into operator dataflows, avoiding separate scans of long ciphertext polynomials. Third, cross-operator check fusion eliminates redundant checksum computations between adjacent operators while preserving end-to-end checking invariants. We implement the scheme in OpenFHE and evaluate it on representative encrypted applications and ciphertext primitives under random single-bit transient faults. It achieves a 100 percent empirical detection rate across 150,000 non-crashing corrupted-result cases and maintains application accuracy close to the fault-free baseline over a wide range of fault rates. The scheme incurs only 6.0 percent to 8.4 percent runtime overhead, averaging 6.8 percent, and reduces average protection overhead by 4.9 times compared with direct checksum-based protection.

cs.AR↗

arXiv2026

On the Vulnerability of FHE Computation to Silent Data Corruption

Fully Homomorphic Encryption (FHE) is rapidly emerging as a promising foundation for privacy-preserving cloud services, enabling computation directly on encrypted data. As FHE implementations mature and begin moving toward practical deployment in domains such as secure finance, biomedical analytics, and privacy-preserving AI, a critical question remains insufficiently explored: how reliable is FHE computation on real hardware? This question is especially important because, compared with plaintext computation, FHE incurs much higher computational overhead, making it more susceptible to transient hardware faults. Moreover, data corruptions are likely to remain silent: the FHE service has no access to the underlying plaintext, causing unawareness even though the corresponding decrypted result has already been corrupted. To this end, we conduct a comprehensive evaluation of SDCs in FHE ciphertext computation. Through large-scale fault-injection experiments, we characterize the vulnerability of FHE to transient faults, and through a theoretical analysis of error-propagation behaviors, we gain deeper algorithmic insight into the mechanisms underlying this vulnerability. We further assess the effectiveness of different fault-tolerance mechanisms for mitigating these faults.

cs.CR↗

arXiv2025

The Future of Fully Homomorphic Encryption System: from a Storage I/O Perspective

Fully Homomorphic Encryption (FHE) allows computations to be performed on encrypted data, significantly enhancing user privacy. However, the I/O challenges associated with deploying FHE applications remains understudied. We analyze the impact of storage I/O on the performance of FHE applications and summarize key lessons from the status quo. Key results include that storage I/O can degrade the performance of ASICs by as much as 357$\times$ and reduce GPUs performance by up to 22$\times$.

cs.CR↗

arXiv2025

Generalized Witt and Morita equivalences

In this work, we introduce a family of new equivalence relations among fusion categories that are less refined than the usual Morita equivalence. We obtain abelian groups by quotienting these new equivalence relations from the commutative monoids of the equivalence classes of all fusion categories. Moreover, we upgrade them to equivalence relations among nondegenerate braided fusion categories that are more refined than the usual Witt equivalence. As a consequence, we obtain new abelian groups that are more refined than the usual Witt group. These new groups allow us to access the internal structures within Witt classes. We expect that they are useful in the classification program of (braided) fusion (higher) categories and in the study of gapless edges of 2+1D topological orders.

math.QA↗

arXiv2025

Phased Array Calibration based on Rotating-Element Harmonic Electric-Field Vector with Time Modulation

Calibration is crucial for ensuring the performance of phased array since amplitude-phase imbalance between elements results in significant performance degradation. While amplitude-only calibration methods offer advantages when phase measurements are impractical, conventional approaches face two key challenges: they typically require high-resolution phase shifters and remain susceptible to phase errors inherent in these components. To overcome these limitations, we propose a Rotating element Harmonic Electric-field Vector (RHEV) strategy, which enables precise calibration through time modulation principles. The proposed technique functions as follows. Two 1-bit phase shifters are periodically phase-switched at the same frequency, each generating corresponding harmonics. By adjusting the relative delay between their modulation timings, the phase difference between the $+1$st harmonics produced by the two elements can be precisely controlled, utilizing the time-shift property of the Fourier transform. Furthermore, the +1st harmonic generated by sequential modulation of individual elements exhibits a linear relationship with the amplitude of the modulated element, enabling amplitude ambiguity resolution. The proposed RHEV-based calibration method generates phase shifts through relative timing delays rather than physical phase shifter adjustments, rendering it less susceptible to phase shift errors. Additionally, since the calibration process exclusively utilizes the $+1$st harmonic, which is produced solely by the modulated unit, the method demonstrates consistent performance regardless of array size. Extensive numerical simulations, practical in-channel and over-the-air (OTA) calibration experiments demonstrate the effectiveness and distinct advantages of the proposed method.

eess.SP↗

arXiv2024

Categories of quantum liquids II

We continue to develop the theory of separable higher categories, including center functors, higher centralizers, modular extensions and group theoretical higher fusion categories. Moreover, we outline a theory of orthogonal higher categories to treat anti-unitary symmetries. Using these results we derive a systematic classification of gapped quantum liquids and predict many new SPT orders in spacetime dimension $\ge3$.

math.CT↗

arXiv2024

Enriched monoidal categories I: centers

This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.

math.CT↗

arXiv2024

Higher condensation theory

We develop a unified mathematical theory of defect condensations for topological orders in all dimensions based on higher categories, higher algebras and higher representations. A k-codimensional topological defect $A$ in an n+1D (potentially anomalous) topological order $C^{n+1}$ is condensable if it is equipped with the structure of a condensable $E_k$-algebra. Condensing such a defect $A$ amounts to a k-step process. In the first step, we condense the defect $A$ along one of its transversal directions, thus obtaining a (k-1)-codimensional defect $\Sigma A$, which is naturally equipped with the structure of a condensable $E_{k-1}$-algebra. In the second step, we condense the defect $\Sigma A$ in one of the remaining transversal directions, thus obtaining a (k-2)-codimensional defect $\Sigma^2 A$, so on and so forth. In the k-th step, we condense the 1-codimensional defect $\Sigma^{k-1}A$ along the only transversal direction, thus defining a phase transition from $C^{n+1}$ to a new n+1D topological order $D^{n+1}$. We give precise mathematical descriptions of each step in above process, including the precise mathematical characterization of the condensed phase $D^{n+1}$. When $C^{n+1}$ is anomaly-free, the same phase transition can be alternatively defined by replacing the last two steps by a single step of condensing the $E_2$-algebra $\Sigma^{k-2}A$ directly along the remaining two transversal directions. When n=2, this modified last step is precisely a usual anyon condensation in a 2+1D topological order. We derive many new mathematical results physically along the way. We also establish the connections among various notions of "gauging" symmetries. We also briefly discuss questions, generalizations and applications that naturally arise from our theory, including higher Morita theory, a theory of integrals and the condensations of liquid-like gapless defects in topological orders.

cond-mat.str-el↗

arXiv2023

Categories of quantum liquids I

We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion. Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their higher centers. We also outline a theory of unitary higher categories based on a $*$-version of condensation completion. After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all $n$D quantum liquids form a $*$-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.

hep-th↗

arXiv2023

Categorical computation

In quantum computing, the computation is achieved by linear operators in or between Hilbert spaces. In this work, we explore a new computation scheme, in which the linear operators in quantum computing are replaced by (higher) functors between two (higher) categories. If from Turing computing to quantum computing is the first quantization of computation, then this new scheme can be viewed as the second quantization of computation. The fundamental problem in realizing this idea is how to realize a (higher) functor physically. We provide a theoretical idea of realizing (higher) functors physically based on the physics of topological orders.

quant-ph↗

arXiv2023

String Condensations in 3+1D and Lagrangian Algebras

We present three Lagrangian algebras in the modular 2-category associated to the 3+1D $\mathbb{Z}_2$ topological order and discuss their physical interpretations, connecting algebras with gapped boundary conditions, and interestingly, maps (braided autoequivalences) exchanging algebras with bulk domain walls. A Lagrangian algebra, together with its modules and local modules, encapsulates detailed physical data of strings condensing at a gapped boundary. In particular, the condensed strings can terminate at boundaries in non-trivial ways. This phenomenon has no lower dimensional analogue and corresponds to novel mathematical structures associated to higher algebras. We provide a layered construction and also explicit lattice realizations of these boundaries and illustrate the correspondence between physics and mathematics of these boundary conditions. This is a first detailed study of the mathematics of Lagrangian algebras in modular 2-categories and their corresponding physics, that brings together rich phenomena of string condensations, gapped boundaries and domain walls in 3+1D topological orders.

cond-mat.str-el↗

arXiv2022

An invitation to topological orders and category theory

Although it has been a well-known fact, for more than two decades, that category theory is needed for the study of topological orders, it is still a non-trivial challenge for students and working physicists to master the abstract language of category theory. In this work, for those who have no background in category theory, we explain in great details how the structure of a (braided) fusion category naturally emerges from lattice models and physical intuitions. Moreover, we show that nearly all mathematical notions and constructions in fusion categories and its representation theory, such as (monoidal) functors, Drinfeld center, module categories, Morita equivalence, condensation completion and fusion 2-categories, naturally emerge from lattice models and physical intuitions. In this process, we also introduce some basic notions and important results of topological orders.

cond-mat.str-el↗

arXiv2022

One dimensional gapped quantum phases and enriched fusion categories

In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these models. In the end, we provide a classification of and the categorical descriptions of all 1-dimensional (the spatial dimension) gapped quantum phases with a finite onsite symmetry.

cond-mat.str-el↗

arXiv2022

Categories of quantum liquids III

We continue our study of the categories of quantum liquids started in a previous work. We combine local quantum symmetries with topological skeletons into a single mathematical theory of topological nets and defect nets. In particular, we introduce the notion of a topological net, which is motivated from and generalizes that of a conformal net, and the notion of a defect net which generalizes that of a defect between conformal nets. We give explicit examples of them. Moreover, we construct the category of topological $n$-nets with $k$-morphisms defined by defect $n$-nets of codimension $k$, and show that the category of $n$D quantum liquids can be extracted from it and computed explicitly via the condensation theory of topological nets.

hep-th↗

arXiv2021

Anyon condensation and tensor categories

Instead of studying anyon condensation in concrete models, we take an abstract approach. Assume that a system of anyons, which form a modular tensor category D, is obtained via an anyon condensation from another system of anyons (i.e. another modular tensor category C). By a bootstrap analysis, we derive the relation between C and D from natural physical requirements. It turns out that the tensor unit of D can be identified with a connected commutative separable algebra A in C. The modular tensor category D consists of all deconfined particles and can be identified with the category of local $A$-modules in C. If this condensation occurs in a 2d region in the C-phase, then it also produces a 1d gapped domain wall between the C-phase and the D-phase. The confined and deconfined particles accumulate on the wall and form a fusion category that is precisely the category of right A-modules in C. We also consider condensations that are confined to a 1d line. We show how to determine the algebra A from physical macroscopic data. We provide examples of anyon condensation in the toric code model, Kitaev quantum double models and Levin-Wen types of lattice models and in some chiral topological phases. In the end, we briefly discuss Witt equivalence between 2d topological phases. We also attach to this paper an Erratum and Addendum to the original version of "Anyon condensation and tensor categories" published in [Nucl. Phys. B 886 (2014) 436-482].

cond-mat.str-el↗

arXiv2020

Measuring the Unique Identifiers of Topological Order Based on Boundary-Bulk Duality and Anyon Condensation

A topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that $R$- and $F$-matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the $R$- and $F$-matrices be experimentally measured? By using quantum simulations based on a toric code model with boundaries and state-of-the-art technology, we show that the braidings, i.e. the $R$-matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The $F$-matrices can also be measured in a scattering quantum circuit involving the fusion of three anyons in two different orders. Thus we provide an experimental protocol for measuring the unique identifiers of topological order.

quant-ph↗

arXiv2020

Defects in the 3-dimensional toric code model form a braided fusion 2-category

It was well known that there are $e$-particles and $m$-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional $\mathbb{Z}_2$ topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional $\mathbb{Z}_2$ topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.

cond-mat.str-el↗

arXiv2020

Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also refer to non-invertible gravitational anomaly as categorical symmetry to stress its connection to symmetry. This provides a holographic and entanglement view of symmetries. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low energy excitations, as long as they have the same categorical symmetry; (4) the classification of gapped liquid phases for bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry). This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.

cond-mat.str-el↗