Searcharxiv⌕ Search

arXiv subjects

Naihong Hu

Publications and source records attributed to Naihong Hu.

At least 19 recordsLinked to original sources
arXiv2026

Quantum MDS codes from complements of unions of finite-field subsets

Let $q$ be an odd prime power. We use complements of unions of subsets of $\mathbb F_{q^2}$ as locator sets and establish a sufficient condition under which a generalized Reed--Solomon (GRS) code is Hermitian self-orthogonal. Using cosets of multiplicative subgroups and sets with prescribed trace or norm values, we construct five families of Hermitian self-orthogonal GRS codes over $\mathbb F_{q^2}$. The Hermitian construction then yields five corresponding families of $q$-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than $q/2+1$. By comparing codes of the same length, we give conditions under which our codes have strictly larger minimum distances than those obtainable from several previously known constructions based on trace maps, linear transformations, and cosets of multiplicative subgroups, either directly or via the propagation rule. We further show that such improvements occur for infinitely many values of $q$.

cs.IT↗
arXiv2026

Protected Logical Qudits in Kitaev Quantum Double Models via Stable Representations

Fault-tolerant quantum computation requires robust protection of encoded quantum information. In this work, we develop a representation-theoretic framework for constructing protected logical qudits in finite-group Kitaev quantum double models. By introducing $\varepsilon$-stable irreducible representations, we establish a necessary and sufficient existence criterion and derive ribbon--projector commutation relations that yield a $d$-dimensional protected logical subspace. We apply this construction to symmetric and alternating groups, obtaining logical qubits for $S_n$ ($n\ge 3$) and a logical qutrit for $A_4$. Moreover, the family $(\mathbb Z_2)^d\rtimes\mathbb Z_d$ realizes protected logical qudits of arbitrary dimension $d\ge 2$. Finally, for $D(A_4)$, we describe a scheme for universal logical qutrit computation using ribbon-based logical operations.

quant-ph↗
arXiv2026

Double-bosonization and Majid's conjecture (V): Grafting of Quantum Qroups

This paper aims to develop a grafting method to address Majid's conjecture. This method enables the construction of a larger target quantum group by grafting two given smaller quantum groups, and advances the study of the generation, classification, and construction of (quasi-)Hopf algebras. As a foundation for this construction, we establish a multi-tensor-product theory of generalized double-bosonization and use it to extract the crucial data for the associated braiding $R$-matrix. Beyond the braided monoidal category perspective arising from quantum subgroup representations, the grafting procedure needs to incorporate structural information from Lie-theoretic root systems. The resulting theoretic framework provides a one-stop strategy for resolving the generation problem concerning the quantum-groups tree in Majid's conjecture.

math.QA↗
arXiv2026

Quantification of the Resource Theory of Imaginarity Revisited

This work studies the decay of three imaginarity measures, namely the $l_1$-norm-based imaginarity, the robustness of imaginarity, and the relative-entropy-based imaginarity, under typical noise channels. For arbitrary single-qubit pure states subject to real quantum channels, we prove an exact attenuation theorem: the $l_1$-norm-based imaginarity and the robustness exhibit identical attenuation governed by a common channel-dependent factor, whereas the relative-entropy-based imaginarity additionally depends on the input-state orientation relative to the reference basis. We extend the investigation to two-qubit entangled and dual-rail states, establishing a quantitative link between photon-loss probability and residual imaginarity for photonic dual-rail encodings. For separable two-qubit states, we define a real-operation resource preorder and show that $Ω_{++}=|+\rangle\langle+|\otimes|+\rangle\langle+|$ is operationally maximal. We further distinguish operational de-imaginary power from reference-state imaginarity loss. For several real two-qubit channels we derive exact operational de-imaginary powers, while for other channels we provide analytic reference-state losses serving as benchmarks and rigorous lower bounds.

quant-ph↗
arXiv2026

Categorical Tensor-Graph Semantics for Quantum Algorithms

This paper investigates foundational quantum computing protocols from the intuitive perspective of categorical tensor-graph semantics within the category \textbf{FHilb}. While conventional Hilbert-space formalisms often conceal the structural nature of quantum algorithms behind high-dimensional matrix operations, the topological framework directly encodes algorithmic functionalities into their graphical skeletons. We provide a comprehensive topological reinterpretation of the Bernstein--Vazirani and Simon algorithms, demonstrating how topological transformations distill their core mathematical essence and clarify the operational mechanisms of oracles. Going beyond the standard qubit model, we construct explicit representations for the qutrit-adapted topological Deutsch--Jozsa and single-shot Grover algorithms. In particular, we establish a necessary and sufficient condition for the single-shot Grover search. We further implement CNOT gates via complementary Frobenius structures and investigate a diagrammatic decomposition scheme for the W-state preparation protocol. By bridging tensor category theory with practical quantum algorithmic design, this work furnishes a composable, scalable diagrammatic toolkit essential for automated circuit optimization across the evolving quantum hardware ecosystem.

quant-ph↗
arXiv2026

The Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra $U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$

From the theory of finite-dimensional weight modules, we get the basic braided $R$-matrix $\widehat R$ of $U_{r, s}(\mathfrak{so}_{2n+1})$. For its FRT presentation $U(\widehat R)$, we achieve two word-formation methods of quantum Lyndon bases (whose bracketing rules are regulated by the $RLL$-formalism) and elucidate their distribution rule within the triangular $L$-matrix. Consequently, we contribute an algebraic proof for establishing an isomorphism between the Drinfeld-Jimbo presentation and the FRT presentation. In the affine setting, we first derive two spectral parameter-dependent $R$-matrices through the Yang-Baxterization. Next, we select the only one that satisfies the intertwining property with respect to the minimal affinization. Accordingly, we obtain the $RLL$ realization of $U_{r, s}(\widehat{\mathfrak{so}_{2n+1}})$ through the Gauss decompositions of the generating matrices. Finally, we contribute an algebraic proof to the Ding-Frenkel Isomorphism Theorem between the Drinfeld realization and the $RLL$ realization.

math.QA↗
arXiv2026

Harish-Chandra Theorem for the Multi-Parameter Quantum Groups of Okado-Yamane Type

This paper is devoted to studying the centre of the multi-parameter quantum group $U_{q,G}(\mathfrak{g})$ introduced by Okado and Yamane, where $\mathfrak{g}$ is a complex simple Lie algebra, and all parameters lie in general position. We mainly establish the Harish-Chandra theorem, proving that the Harish-Chandra homomorphism is an isomorphism; in particular, we determine the centre $Z(U_{q,G})\cong (U^0_\flat)^W$ is isomorphic to a polynomial algebra or a quotient algebra of a polynomial algebra. The same result holds for the $(U^0_\flat)^W$ of the two-parameter quantum group $U_{r,s}(\mathfrak{g})$.

math.QA↗
arXiv2026

A quantum shuffle approach to quantum affine super algebra of type $C(2)^{(2)}$ and its equitable presentation

In this study, we focus on the positive part $U_q^{+}$ of the quantum affine superalgebra $U_q(C(2)^{(2)})$. This algebra admits a presentation with two two generators $e_α$ and $e_{δ-α}$, which satisfy the cubic $q$-Serre relations. According to the work of Khoroshkin-Lukierski-Tolstoy, the Damiani and the Beck $PBW$ bases exist for this superalgebra. In this paper, we utilize the $q$-shuffle superalgebra and Catalan words to present these two bases in a closed-form expression. Ultimately, we present the bosonization of $U_q(C(2)^{(2)})$.

math.QA↗
arXiv2026

Some classifications of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:x^2y}$ of Kashina

Let $H$ be the $16$-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:x^2y}$ classified by Kashina. We figure out all simple Yetter-Drinfeld $H$-modules, and then determine all finite-dimensional Nichols algebras satisfying the constraint condition $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a simple object in $_H^H\mathcal{YD}$. Finally, we describe some liftings of the corresponding Radford biproducts $\mathcal{B}(V)\sharp H$, which provide some classifications of finite dimensional Hopf algebras with $H$ as their coradical.

math.QA↗
arXiv2026

Drinfeld Isomorphism for Novel Quantum Affine Algebra of Type $A_{1}^{(1)}$

In this paper, we first review the definition of the novel quantum affine algebra \(U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) of type \(A_{1}^{(1)}\) given in \cite{FHZ, HZhuang}. Furthermore, by introducing \(Ω\)-invariant generating functions, we construct the Drinfeld realization \(U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) of this algebra, and prove that \(U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) and \(U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) are algebraically isomorphic, which is known as the Drinfeld Isomorphism.

math.QA↗
arXiv2025

Novel isoclasses of one-parameter exotic small quantum groups originating from a two-parameter framework

The classification of one-parameter small quantum groups remains a fascinating open problem. This paper uncovers a novel phenomenon: beyond the Lusztig small quantum groups-equipped with double group-like elements -there exists a plethora of exotic small quantum groups, approximately fivefold more numerous than their standard counterparts, which originate from a two-parameter framework.

math.QA↗
arXiv2025

Automorphism group of Suzuki's Hopf algebra

In this paper, we calculate explicitly automorphism group of the Suzuki's Hopf algebra $A_{Nn}^{μλ}$ by viewing Yetter-Drinfeld modules as invariants of Hopf algebra automorphisms.

math.QA↗
arXiv2025

Harish-Chandra Theorem for Two-parameter Quantum Groups

This paper is devoted to investigating the centre of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ via establishing the Harish-Chandra homomorphism. Based on the Rosso form and the representation theory of weight modules, we prove that when rank $\mathfrak{g}$ is even, the Harish-Chandra homomorphism is an isomorphism, and in particular, the centre of the quantum group $\breve{U}_{r,s}(\mathfrak{g})$ of the weight lattice type is a polynomial algebra $\mathbb{K}[z_{\varpi_1},\cdots,z_{\varpi_n}]$, where canonical central elements $z_λ\; (λ\in Λ^+)$ are turned out to be uniformly expressed. For rank $\mathfrak{g}$ to be odd, we figure out a new invertible extra central generator $z_*$, which doesn't survive in $U_q(\mathfrak g)$, then the centre of $\breve{U}_{r,s}(\mathfrak{g})$ contains $\mathbb{K}[z_{\varpi_1},\cdots,z_{\varpi_n}]\otimes_\mathbb K\mathbb K[z_*^{\frac{1}{\ell}}, z_*^{-\frac{1}{\ell}}]$, where $\ell=2$, except $\ell=4$ for $D_{2k+1}$.

math.QA↗
arXiv2024

Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology over Quantum Grassmann Superalgebra

We explore the indecomposable submodule structure of quantum Grassmann super-algebra $Ω_q(m|n)$ and its truncated objects $Ω_q(m|n,\textbf{r})$ in the case when $q=\varepsilon$ is an $\ell$-th root of unity. A net-like weave-lifting method is developed to show the indecomposability of all the homogeneous super subspaces $Ω_q^{(s)}(m|n,\textbf{r})$ and $Ω_q^{(s)}(m|n)$ as $\mathcal U_q(\mathfrak{gl}(m|n))$-modules by defining "energy grade" to depict their "$\ell$-adic" phenomenon. Their Loewy filtrations are described, the Loewy layers and dimensions are determined by combinatorial identities. The quantum super de Rham cochain short complex $(\mathcal D_q(m|n)^{(\bullet)},d^\bullet)$ is constructed and proved to be acyclic (Poincaré Lemma), where $\mathcal D_q(m|n)=Ω_q(m|n)\otimes \sqcap_q(m|n)$ and $\sqcap_q(m|n)$ is the quantum exterior super-algebra, over which we define the $q$-differentials. %such that the product structure of $\sqcap_q(m|n)$, the quantum exterior super-algebra, is well-matched everywhere. However, the truncated quantum de Rham cochain subcomplexes $(\mathcal D_q(m|n,\textbf{r})^{(\bullet)},d^\bullet)$ we mainly consider are no longer acyclic and the resulting quantum super de Rham cohomologies $H^s_{DR}(\mathcal D_q(m|n, \mathbf r)^{(\bullet)})$ are highly nontrivial.

math.QA↗
arXiv2024

$RLL$-realization of two-parameter quantum affine algebra in type $D_n^{(1)}$

We obtain the basic $R$-matrix of the two-parameter Quantum group $U=U_{r,s}\mathcal(\mathfrak{so}_{2n})$ via its weight representation theory and determine its $R$-matrix with spectral parameters for the two-parameter quantum affine algebra $U=U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n}})$. Using the Gauss decomposition of the $R$-matrix realization of $U=U_{r,s}\mathcal(\mathfrak{so}_{2n})$, we study the commutation relations of the Gaussian generators and finally arrive at its $RLL$-formalism of the Drinfeld realization of two-parameter quantum affine algebra $U=U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n}})$.

math.QA↗
arXiv2023

Enhanced quantum channel uncertainty relations by skew information

By revisiting the mathematical foundation of the uncertainty relation, skew information-based uncertainty sequences are developed for any two quantum channels. A reinforced version of the Cauchy-Schwarz inequality is adopted to improve the uncertainty relation, and a sampling technique of observables' coordinates is used to offset randomness in the inequality. It is shown that the lower bounds of the uncertainty relations are tighter than some previous studies.

quant-ph↗