Some Good Error-Correcting Codes from Algebraic-Geometric Codes

量子码是量子化的码：基于有限域上量子化的稳定子码

Quantum codes are quantized codes: Stabilizer codes from quantization over finite fields

本文提出有限域上的离散几何量子化方法，将经典纠错码转化为量子纠错码，为编码理论中的量子-经典对应提供物理解释。

Quantum Error Correcting Codes From The Compression Formalism

We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not obtained by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the ``higher-rank numerical range''. We describe its basic properties and discuss possible further applications.

博弈论通过纳什均衡发现量子纠错码

Game-Theoretic Discovery of Quantum Error-Correcting Codes Through Nash Equilibria

提出博弈论框架，将码优化转化为竞争目标间的策略交互，通过纳什均衡系统生成具有期望性质的量子纠错码，并验证其能重新发现最优的[[15,7,3]]量子汉明码，且可扩展至100量子比特。

Local non-CSS quantum error correcting code on a three-dimensional lattice

We present a family of non-CSS quantum error-correcting code consisting of geometrically local stabilizer generators on a 3D lattice. We study the Hamiltonian constructed from ferromagnetic interaction of overcomplete set of local stabilizer generators. The degenerate ground state of the system is characterized by a quantum error-correcting code whose number of encoded qubits are equal to the second Betti number of the manifold. These models (i) have solely local interactions; (ii) admit a strong-weak duality relation with an Ising model on a dual lattice; (iii) have topological order in the g…

通过准几何方法变换量子比特

Transforming qubits via quasi-geometric approaches

本研究基于准几何方法，结合2D-QOCCCSs、AQECCs及Pauli X门、Hadamard H门实现少数量子比特到多纠错量子比特的变换，实验验证了其高纠错性能，为量子计算编码优化提供基础。

Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects

We discuss some aspects and examples of applications of dual algebraic pairs $({\cal G}_1,{\cal G}_2)$ in quantum many-body physics. They arise in models whose Hamiltonians $H$ have invariance groups $G_i$. Then one can take ${\cal G}_1 = G_i$ whereas another dual partner ${\cal G}_2= g^D$ is generated by $G_i$ invariants, possesses a Lie-algebraic structure and describes dynamic symmetry of models; herewith polynomial Lie algebras $\hat g = g^D$ appear in models with essentially nonlinear Hamiltonians. Such an approach leads to a geometrization of model kinematics and dynamics.

Algebraic Quantum Codes: Linking Quantum Mechanics and Discrete Mathematics

We present a general framework of quantum error-correcting codes (QECCs) as a subspace of a complex Hilbert space and the corresponding error models. Then we illustrate how QECCs can be constructed using techniques from algebraic coding theory. Additionally, we discuss secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs.

Entanglement-assisted Quantum Codes from Algebraic Geometry Codes

Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via the Euclidean and the Hermitian construction. Two of the families crea…

Fault-tolerant logical gates in quantum error-correcting codes

Recently, Bravyi and König have shown that there is a tradeoff between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant depth geometrically local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this paper, we elaborate this idea and provide several extensions and appli…

CSS量子纠错码的内射范数

The injective norm of CSS quantum error-correcting codes

本文将内射范数（几何纠缠度）的计算从Kitaev码推广至所有CSS码，得到无限量子态族的内射范数，并揭示了与拟阵理论和Edmonds交集定理的联系。

从不动点路径积分构造的拓扑纠错过程

Topological error correcting processes from fixed-point path integrals

提出将拓扑量子纠错码视为动态电路并关联不动点路径积分的统一范式，证明了多种现有码的等价性，并推导出两个新纠错码。

Majorana dimers and holographic quantum error-correcting codes

Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the AdS/CFT correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC). Relating its logical code basis to Majorana dimers, we ef…

基于AI的量子纠错码量子比特丢失解码

AI-Enabled Decoding of Qubit Loss for Quantum Error-Correcting Codes

针对量子比特丢失破坏稳定子代数结构的问题，提出基于时空图神经网络（STGNN）的解码器，同时纠正标准泡利错误和定位丢失位置，逻辑精度优于传统算法，并能在十轮后识别90%以上丢失位置。

Multipartite entangled states, symmetric matrices and error-correcting codes

A pure quantum state is called $k$-uniform if all its reductions to $k$-qudit are maximally mixed. We investigate the general constructions of $k$-uniform pure quantum states of $n$ subsystems with $d$ levels. We provide one construction via symmetric matrices and the second one through classical error-correcting codes. There are three main results arising from our constructions. Firstly, we show that for any given even $n\ge 2$, there always exists an $n/2$-uniform $n$-qudit quantum state of level $p$ for sufficiently large prime $p$. Secondly, both constructions show that their exist $k$-uni…

Algebraic techniques in designing quantum synchronizable codes

Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the previously known general framework for designing quantum synchronizable codes through more extensive use of the theory of finite fields. This makes it possible to widen the range of tolerable magnitude of block synchronization errors while giving mathematical insight into the algebraic mechanism of synchronization recovery. Also given are families of quantum synchronizable codes based on punctured Reed-Muller codes and their ambie…

Topological approach toward quantum codes with realistic physical constraints

The following open problems, which concern a fundamental limit on coding properties of quantum codes with realistic physical constraints, are analyzed and partially answered here: (a) the upper bound on code distances of quantum error-correcting codes with geometrically local generators, (b) the feasibility of a self-correcting quantum memory. To investigate these problems, we study stabilizer codes supported by local interaction terms with translation and scale symmetries on a $D$-dimensional lattice. Our analysis uses the notion of topology emerging in geometric shapes of logical operators, …

Geometric and algebraic aspects of spectrality in order unit spaces: a comparison

Two approaches to spectral theory of order unit spaces are compared: the spectral duality of Alfsen and Shultz and the spectral compression bases due to Foulis. While the former approach uses the geometric properties of an order unit space in duality with a base norm space, the latter notion is purely algebraic. It is shown that the Foulis approach is strictly more general and contains the Alfsen-Shultz approach as a special case. This is demonstrated on two types of examples: the JB-algebras which are Foulis spectral if and only if they are Rickart, and the centrally symmetric state spaces, w…

Geometric and Algebraic Topological Methods in Quantum Mechanics

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The present book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geomet…

An Algebraic-Geometric Characterization of Tripartite Entanglement

To characterize entanglement of tripartite $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ systems, we employ algebraic-geometric tools that are invariants under Stochastic Local Operation and Classical Communication (SLOCC), namely $k$-secant varieties and one-multilinear ranks. Indeed, by means of them, we present a classification of tripartite pure states in terms of a finite number of families and subfamilies. At the core of it stands out a fine-structure grouping of three-qutrit entanglement.

Geometric and algebraic approaches to quantum theory

We show how to formulate physical theory taking as a starting point the set of states (geometric approach). We discuss the relation of this formulation to the conventional approach to classical and quantum mechanics and the theory of complex systems. The equations of motion and the formulas for probabilities of physical quantities are analyzed. A heuristic proof of decoherence in our setting is used to justify the formulas for probabilities. We show that any physical theory theory can be obtained from classical theory if we restrict the set of observables. This remark can be used to construct …