17+2 pages, 5+1 figures, v3: revised section on dynamic tree codes; v4: clarified the explanation of syndrome extraction; added figs. 2 and 4; replaced the term "CSS Floquet code" with "CSS honeycomb code"
我们提出了一类新的二维和三维纠错动态码，它与任何父子系统码都没有明确的联系。我们将其中的二维码称为 CSS honeycomb code，它在几何上类似于 Hastings 和 Haah 提出的 honeycomb code，并且在动态上也嵌入了一个瞬态 toric code。然而，与 honeycomb code 不同的是，它具有显式的 CSS 结构，且其规范校验不构成子系统码。尽管如此，我们证明了我们的动态协议能够保持逻辑信息，并具备纠错阈值。我们将此构造推广到三维，得到了一种容错地交替实现两种 type-I fracton model（即 checkerboard model 和 X-cube model）的码。最后，我们通过证明可以在两个协议之间随机切换而不丢失信息，同时仍能测量错误伴随式，展示了我们的 CSS honeycomb code 协议与 honeycomb code 的兼容性。我们将这种更一般的非周期结构称为“dynamic tree codes”，并将其推广到了三维。我们构建了一种概率有限自动机方案，用于生成能够纠正任意单比特 Pauli 错误的 dynamic tree codes，这可以被视为迈向实用容错随机码开发的一步。
英文摘要：
We propose a new class of error-correcting dynamic codes in two and three dimensions that has no explicit connection to any parent subsystem code. The two-dimensional code, which we call the CSS honeycomb code, is geometrically similar to that of the honeycomb code by Hastings and Haah, and also dynamically embeds an instantaneous toric code. However, unlike the honeycomb code it possesses an explicit CSS structure and its gauge checks do not form a subsystem code. Nevertheless, we show that our dynamic protocol conserves logical information and possesses a threshold for error correction. We generalize this construction to three dimensions and obtain a code that fault-tolerantly alternates between realizing two type-I fracton models, the checkerboard and the X-cube model. Finally, we show the compatibility of our CSS honeycomb code protocol and the honeycomb code by showing the possibility of randomly switching between the two protocols without information loss while still measuring error syndromes. We call this more general aperiodic structure `dynamic tree codes', which we also generalize to three dimensions. We construct a probabilistic finite automaton prescription that generates dynamic tree codes correcting any single-qubit Pauli errors and can be viewed as a step towards the development of practical fault-tolerant random codes.
Dressed Floquet scars from protected zero modes in a Rydberg chain
Indian Association for the Cultivation of Science（印度科学文化促进协会）；University of Hyderabad（海得拉巴大学）；S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）
We analyze a simple model of quantum dynamics, which is a discrete-time deterministic version of the Frederickson-Andersen model. We argue that this model is integrable, with a quasiparticle description related to the classical hard-rod gas. Despite the integrability of the model, commutators of physical operators grow as in generic chaotic models, with a diffusively broadening front, and local operators obey the eigenstate thermalization hypothesis (ETH). However, large subsystems violate ETH; as a function of subsystem size, eigenstate entanglement first increases linearly and then saturates…
A protocol called the "honeycomb code", or generically a "Floquet code", was introduced by Hastings and Haah in \cite{hastings_dynamically_2021}. The honeycomb code is a subsystem code based on the honeycomb lattice with zero logical qubits but such that there exists a schedule for measuring two-body gauge checks leaving enough room at all times for two protected logical qubits. In this work we show a way to introduce boundaries to the system which curiously presents a rotating dynamics but has constant distance and is therefore not fault-tolerant.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) point-like topological excitations with restricted mobility, with non-trivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase t…
We consider a spatial analogue of the quantum error correction threshold. Given individual time-independent subsystems in which quantum information is coherent over sufficiently long lengths, we show how the information can be kept coherent for arbitrarily long lengths by forming time-independent composite systems. The subsystem coherence length exhibits threshold behavior. When it exceeds a length $ξ_{th}$, meaningful information can be extracted from the ground state of the composite system. Otherwise, the information is garbled. The threshold transition implies that the parent Hamiltonian o…
In this work, we introduce an approach to study quantum many-body dynamics, inspired by the Feynman-Vernon influence functional. Focusing on a family of interacting, Floquet spin chains, we consider a Keldysh path-integral description of the dynamics. The central object in our approach is the influence matrix (IM), which describes the effect of the system on the dynamics of a local subsystem. For translationally invariant models, we formulate a self-consistency equation for the influence matrix. For certain special values of the model parameters, we obtain an exact solution which represents a …
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 whether a generic multipartite pure state can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the simplest tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems' dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N-partite case, we provide conditions on the su…
Geometry of Knill-Laflamme Coefficients for Pauli Error Detection
The University of Texas at Dallas（德克萨斯大学达拉斯分校）；National Research Council Canada（加拿大国家研究委员会）；University of Waterloo（滑铁卢大学）；Iowa State University（爱荷华州立大学）
Homological Thresholds in Randomly Monitored Quantum Error-Correcting Codes
Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Department of Physics, Graduate School of Science, Tohoku University（东北大学理学研究科物理学系）
