Relation between coined quantum walks and quantum cellular automata
Masatoshi Hamada, Norio Konno, Etsuo Segawa
受Patel等人最近工作的启发，本文阐明了量子游动和量子元胞自动机之间的联系。因此，他们的结果自然是从联系中得出的。
Motivated by the recent work of Patel et al., this paper clarifies a connection between coined quantum walks and quantum cellular automata in a general setting. As a consequence, their result is naturally derived from the connection.
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Introduction to Quantum Cellular Automata
我们提供量子元胞自动机的介绍。
We present herein an introduction to implementing 2-color cellular automata on quantum annealing systems, such as the D-Wave quantum computer. We show that implementing nearest-neighbor cellular automata is possible. We present an implementation of Wolfram's cellular automata Rule 110, which has previously been shown to be a universal Turing machine, as a QUBO suitable for use on quantum annealing systems. We demonstrate back-propagation of cellular automata rule sets to determine initial cell states for a desired later system state. We show 2-D 2-color cellular automata, such as Conway's Game…
An algebraic study of unitary one dimensional quantum cellular automata
给出了线性量子元胞自动机，即一维量子元胞自动机的么正性的代数判据。我们得到这些都是通过直接的组合参数，并通过添加到模型中的限制，不改变量子元胞自动机的计算能力。我们所考虑的构形具有有限但无界的尺寸。
It is shown that irreversible classical cellular automata can be performed by quantum algorithm using additional ancilla registers. The algorithm for cellular automata states analysis has been proposed. This algorithm is based on the elements of Grover's algorithm - the inversion of amplitude of searched states and unitary transform of inversion about the average. The inversion of searched states amplitudes can be performed by quantum Toffoli gate.
Quantum cellular automata are important tools in understanding quantum dynamics, thanks to their simple and effective list of rules. Here we investigate explicitly how coherence is built and lost in the evolution of one-dimensional automata subject to noise. Our analysis illustrates the interplay between unitary and noisy dynamics, and draws considerations on their behaviour in the pseudo-frequency domain.
On the structure of Clifford quantum cellular automata
我们研究可逆量子元胞自动机的限制，这些也是克利福德操作。这意味着泡利算子（或离散Weyl算子）的张量积映射到泡利算子的张量积。因此，Clifford量子元胞自动机是由相空间中的辛元胞自动机诱导的。我们描述这些辛元胞自动机，并发现所有可能的本地规则必须是，一些全球性的转变，反射不变的起源。在一维情况下，我们还发现，每一个唯一确定的、在几何上不变的稳定子态都可以从一个乘积态通过一个Clifford元胞自动机时间步制备出来，从而刻画了这类稳定子态，并且我们证明了所有的一维Clifford量子元胞自动机都是由几个初等运算生成的.我们还表明，在周期性边界条件下，线性不变的稳定状态和线性不变的Clifford操作之间的对应关系成立。
Definition and evolution of quantum cellular automata with two qubits per cell
量子计算机实现的研究提出了细胞量子计算机架构。这些体系结构可以模拟量子元胞自动机的演化，从而可能模拟量子和经典物理系统和过程。然而，已知的是，除了平凡的情况下，每个细胞具有一个量子比特的一维均匀量子元胞自动机的幺正演化是不可能的。量子元胞自动机，每个细胞包括两个量子比特的定义和它们的演化研究使用量子计算机模拟器。演化是单一的，其线性表现为概率分布模式的周期性结构。
Reversible quantum cellular automata
我们将量子元胞自动机定义为具有离散时间动力学的无限量子晶格系统，使得时间步长与晶格平移互换并且具有严格有限的传播速度。与以前的定义相比，这使我们能够给出一个明确的特征的所有本地规则生成这样的自动机。同样的局部规则也产生周期性边界条件的自动机的全局时间步长。我们的主结构定理断言任何量子元胞自动机在结构上都是可逆的，即，它可以通过在广义Margolus分割方案中应用两个分块酉运算来获得。这意味着，与经典情况相反，最近邻量子元胞自动机的逆也是最近邻自动机。我们提出了几种量子元胞自动机的构造方法，基于幺正与它们的平移交换，基于（任意）可逆经典元胞自动机的量子化，基于量子线路，以及基于Clifford变换关于有限Weyl系统描述的单细胞。此外，我们指出如何量子随机游动可以被认为是特殊情况下的元胞自动机，即通过限制量子格子气自动机与本地粒子数守恒的单粒子部门。
Clifford quantum cellular automata (CQCAs) are a special kind of quantum cellular automata (QCAs) that incorporate Clifford group operations for the time evolution. Despite being classically simulable, they can be used as basic building blocks for universal quantum computation. This is due to the connection to translation-invariant stabilizer states and their entanglement properties. We will give a self-contained introduction to CQCAs and investigate the generation of entanglement under CQCA action. Furthermore, we will discuss finite configurations and applications of CQCAs.
Quantum Walks and Reversible Cellular Automata
本文研究了一维离散时间量子行走的可逆元胞自动机的分布性质和守恒量之间的联系。作为推论，我们给出了量子行走的详细信息。
Quantum Cellular Automata
量子元胞自动机（QCA）的评论，包括早期和最近的建议。QCA是（经典）元胞自动机（CA），特别是可逆CA的推广。后者将在短期内进行审查。概述了早期尝试由不同的作者定义一维QCA。这些原来有严重的缺点，讨论以及。随后提出的各种建议，一些作者的一般定义的一维和更高的维QCA的审查和他们的属性，如普遍性和可逆性进行了讨论。
Quantum Circuits for Elementary Cellular Automata
该研究确定了22条可通过量子电路模拟的初等元胞自动机规则，且为每条规则设计了门数为O(N)的量子实现电路。
Quantum Cellular Automata from Lattice Field Theories
我们将格场论的方法应用于元胞自动机的量子化。本文利用路径积分和格点场论的算符形式，讨论了五类主要的元胞自动机：玻色子、费米子、超对称、自旋和量子点的量子化。我们表明，量子化的超对称元胞自动机是有关最近讨论的字符串位模型的索恩和伯格曼，并代表了一个链接的元胞自动机理论的基础物理。我们讨论了自旋和量子点元胞自动机在实验实现和量子计算中的重要性。以往的量子元胞自动机研究都是以波函数值作为元胞内容，离散化的线性狄拉克方程作为更新方程。我们表明，我们的方法量子化的费米子元胞自动机包括这种利用作为一个字段方程，并允许通过晶格场相互作用的非线性。
Gauge-invariance is a fundamental concept in Physics -- known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of Cellular Automata. More precisely, the notions of gauge-invariance and gauge-equivalence in Cellular Automata are formalized. A step-by-step gauging procedure to enforce this symmetry upon a given Cellular Automaton is developed, and three examples of gauge-invariant Cellular Automata are examined.
Very much as its classical counterpart, quantum cellular automata are expected to be a great tool for simulating complex quantum systems. Here we introduce a partitioned model of quantum cellular automata and show how it can simulate, with the same amount of resources (in terms of effective Hilbert space dimension), various models of quantum walks. All the algorithms developed within quantum walk models are thus directly inherited by the quantum cellular automata. The latter, however, has its structure based on local interactions between qubits, and as such it can be more suitable for present …
Unitarity in one dimensional nonlinear quantum cellular automata
整体演化的么正性是离散时空中有限态模型的一个极其严格的条件。量子元胞自动机尤其受到严格的限制。在以前的工作中，我们证明了一个简单的No-go定理，排除了线性量子元胞自动机的非平凡的齐次演化。在这里，我们仔细定义了一般的量子元胞自动机，以调查的可能性，有非平凡的均匀幺正演化时的局部规则是非线性的。由于幺正整体跃迁振幅是由局部跃迁振幅的乘积构成的，因此无限晶格需要与周期晶格不同的处理方法。我们证明了这两种情况下的酉定理，表示等价的$1+1$尺寸的全球统一性和某些限制的本地规则，然后表明，这些限制可以解决各种多参数家庭的非线性量子元胞自动机。幺正性定理和无限情形下的满射性定理也暗示了一维元胞自动机的幺正性是可判定的。
Quantum cellular automata consist in arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and univ…
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