Kevin T. Tian, Zhenghan Wang
Entanglement renormalization group flow of the Haah cubic code produces another fracton model with 4 qubits per lattice site, dubbed as the Haah B-code. We provide a schema that generalizes both models to stabilizer codes on any finite group with 2q qubits per site and labeled by multi-subsets of the finite group and a symmetric binary matrix.
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Fracton phases feature elementary excitations with fractionalized mobility and are exciting interest from multiple areas of theoretical physics. However, the most exotic 'type-II' fracton phases, like the Haah codes, currently have no known experimental diagnostics. Here, we explain how type-II fracton phases with gapless gauge modes, such as the $\mathrm{U}(1)$ Haah code, may be identified experimentally. Our analysis makes use of the 'multipole gauge theory' description of type-II fracton phases, which exhibits ultraviolet-infrared (UV-IR) mixing. We show that neutron scattering experiments …
Haah codes represent a singularly interesting gapped Hamiltonian schema that has resisted a natural generalization, although recent work shows that the closely related type I fracton models are more commonplace. These type I siblings of Haah codes are better understood, and a generalized topological quantum field theory framework has been proposed. Following the same conceptual framework, we outline a program to generalize Haah codes to all 3-manifolds using Hastings' LR stabilizer codes for finite groups.
We investigate two dimensional compactifications of three dimensional fractonic stabilizer models. We find the two dimensional topological phases produced as a function of compactification radius for the X-cube model and Haah's cubic code. Furthermore, we uncover translation symmetry-enrichment in the compactified cubic code that leads to twisted boundary conditions.
Exotic Symmetry Breaking Properties of Self-Dual Fracton Spin Models
本文研究四面体伊辛模型和分形伊辛模型两种自对偶分形子自旋模型，通过构造序参量、关联函数和对称性生成元，揭示其子维度对称性破缺的奇异性质，并发现两者均经历具有反常L^{-(D-1)}标度的强一阶相变，为理解棋盘码和Haah码的量子纠错性质奠定基础。
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter, but the existence of a TQFT description for these phases remains an open question. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground state wave functions. We explicitly compute the en…
Haah's 3D cubic code thermalizes rapidly
该研究严格证明了Haah三维立方码在任意正温度下快速热化，否定了其作为自校正量子记忆的候选资格，并首次给出分形码快速混合的解析证明。
We discuss the procedure for gauging on-site $\mathbb{Z}_2$ global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah's cubic code, and the checkerboard model. Where the former two models possess an on-site $\mathbb{Z}_2$ layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these mo…
A topological defect network (TDN) is formed by a network of topological defects embedded within a topological quantum field theory (TQFT). TDNs were introduced recently for the purpose of describing fracton topological phases of matter using the framework of defect TQFT. Their effectiveness has been demonstrated through numerous examples, yet a systematic construction was lacking. Here we solve this problem by formulating a method to construct TDNs for a wide range of lattice Hamiltonians. Our method takes a lattice Hamiltonian as input, applies an ungauging procedure, then creates a refined …
Error Resilience of Fracton Codes and Near Saturation of Code-Capacity Threshold in Three Dimensions
研究分数子码，尤其是checkerboard码，通过统计力学映射和并行退火蒙特卡罗模拟，发现其码容量阈值约为0.107(3)，接近理论极限，验证了双互双模型的广义熵关系。
Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions, we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah. We prove that given a translation-invariant fermionic system with general $q$-body interactions, where $q$ is even, a local mapping preserving global fermion parity to a dual Pauli spin model exists and is unique up to a choice of basis. Furthermore, the dual spin model is constructive, and we present various examples of these dualities. As an application, we bosonize fermionic s…
It is well-known that 3D Type-I fracton models can be obtained from the condensations of stacked layers of 2D anyons. It is less obvious if 3D Type-II fractons can be understood from a similar perspective. In this paper, we affirm that this is the case: we produce the paradigm Type-II fracton model, Haah's cubic code, from a 2D layer construction. However, this is not a condensation of 2D anyons, but rather we start with stacks of 2D subsystem-symmetry protected topological states (SSPT). As this parent model is not topologically ordered in the strict sense, whereas the final state is, we refe…
We investigate the entanglement-renormalization group flows of translation-invariant topological stabilizer models in three dimensions. Fracton models are observed to bifurcate under entanglement renormalization, generically returning at least one copy of the original model. Based on this behavior we formulate the notion of bifurcated equivalence for fracton phases, generalizing foliated fracton equivalence. The notion of quotient superselection sectors is also generalized accordingly. We calculate bifurcating entanglement-renormalization group flows for a wide range of examples and, based on …
Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-p…
A holographic view of topological stabilizer codes
本文建立 Pauli 拓扑稳定子码体-边对应的通用框架，用有限域多项式刻画边界局域实现的阻碍，并推广至三维分形子模型。
Translation-invariant quantum low-density parity-check codes from compactified fracton models
本文研究了具有平移对称性和局部检查的量子纠错码，提出了一种统一框架，涵盖了多种平移不变码，包括许多分形码和所有阿贝尔双块群代数（A2BGA）码，包括双变量自行车（BB）码。
Fractons are emergent particles which are immobile in isolation, but which can move together in dipolar pairs or other small clusters. These exotic excitations naturally occur in certain quantum phases of matter described by tensor gauge theories. Previous research has focused on the properties of small numbers of fractons and their interactions, effectively mapping out the "Standard Model" of fractons. In the present work, however, we consider systems with a finite density of either fractons or their dipolar bound states, with a focus on the $U(1)$ fracton models. We study some of the phases …
A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules
该研究提出一种正方形晶格边三量子位、格点非可逆算子的稳定器码模型，发现新型禁闭分形子激发及非交换非阿贝尔融合规则，揭示其拓扑简并、禁闭特性及分形子液相等现象。
Towards Non-Invertible Anomalies from Generalized Ising Models
该研究提出基于d维广义Ising模型构建非可逆拓扑(d+1)维可解体模型的通用方法，推导体边对应条件，发现分形子序中存在单反常理论对应两种体模型的新现象，拓展了非可逆拓扑相的研究范畴。
Subsystem Symmetries and Fracton Models in Quantum Error Correction
该研究探究经典伊辛模型与量子纠错的关联，确定棋盘格码最优阈值达0.107(3)，证明分形子码是高鲁棒性量子存储器的优良候选，凸显统计力学框架的应用价值。
Preparing topological states with finite depth simultaneous commuting gates
本文提出利用有限深度且由长程同时交换双比特门构成的酉电路，制备二维阿贝尔与非阿贝尔拓扑有序态，并将方案扩展至三维 CSS 码、Haah 码及 X-Cube 分形子模型，电路所需门与辅助比特规模达最优 $O(L^2)$。
Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
研究通过簇态测量揭示CSS码与分数子模型间的异常流入及对偶性，构建了自对偶的非可逆对称量子模型。
Multipole symmetries are of interest both as a window on fracton physics and as a crucial ingredient in realizing new universality classes for quantum dynamics. Here we address the question of whether and when multipole symmetries can be spontaneously broken, both in thermal equilibrium and at zero temperature. We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder. We present both general results and explicit examples. Our results should …
Fragmented Topological Excitations in Generalized Hypergraph Product Codes
本研究通过广义超图积码揭示了碎片化拓扑激发的特性，发现其在不同维度中的拓扑性质及非单调基态退化现象。
The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the …
Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate, and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and König's bound [Phys. Rev. Le…
Braiding for the win: Harnessing braiding statistics in topological states to win quantum games
本文证明从拓扑或fracton序量子物质相提取的资源态在非局域量子博弈中具有稳健优势，核心方法是通过设计博弈利用特征braiding过程作为contextuality来源，并统一了利用mutual statistics的策略。
We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah…
As a quantum-informative window into quantum many-body physics, the concept and application of entanglement renormalization group (ERG) have been playing a vital role in the study of novel quantum phases of matter, especially long-range entangled (LRE) states in topologically ordered systems. For instance, by recursively applying local unitaries as well as adding/removing qubits that form product states, the 2D toric code ground states, i.e., fixed point of Z_2 topological order, are efficiently coarse-grained with respect to the system size. As a further improvement, the addition/removal of 2…
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