The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.
Szegedy's quantum walk is a quantization of a classical random walk or Markov chain, where the walk occurs on the edges of the bipartite double cover of the original graph. To search, one can simply quantize a Markov chain with absorbing vertices. Recently, Santos proposed two alternative search algorithms that instead utilize the sign-flip oracle in Grover's algorithm rather than absorbing vertices. In this paper, we show that these two algorithms are exactly equivalent to two algorithms involving coined quantum walks, which are walks on the vertices of the original graph with an internal deg…
We analyze the equivalence between discrete-time coined quantum walks and Szegedy's quantum walks. We characterize a class of flip-flop coined models with generalized Grover coin on a graph $Γ$ that can be directly converted into Szegedy's model on the subdivision graph of $Γ$ and we describe a method to convert one model into the other. This method improves previous results in literature that need to use the staggered model and the concept of line graph, which are avoided here.
We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy's quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has $l$ integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight $l$. We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we genera…
We present and study a two-particle quantum walk on the line in which the two particles interact via a long-range Coulombian-like interaction. We obtain the spectrum of the system as well as study the type of molecules that form, attending to the bosonic or fermionic nature of the walkers. The usual loss of distinction between attractive and repulsive forces does not entirely apply in our model because of the long-range of the interaction.
N-dimensional alternate coined quantum walks from a dispersion relation perspective
我们通过推广最近的提议[Di Franco et al.，物理修订版信函106，080502（2011）]。与N维量子行走的标准定义QW_N相比，这种N维交替量子行走AQW_N只需要一个硬币量子位。通过分析AQW_2和AQW_3的色散关系，我们讨论了它们的量子扩散性质，特别揭示了魔鬼点的存在。这使我们能够突出与其他众所周知的物理现象的有趣相似之处。我们还证明了AQW_3产生了真正的多体纠缠。最后讨论了AQW_N的可实现性。
Quantum walks with memory(QWM) are a type of modified quantum walks that record the walker's latest path. The general model of coined QWM is presented in Phys. Rev. A 93, 042323 (2016). In this paper, we present general model of Szegedy QWM. Importantly, the relation of coined QWM and Szegedy QWM is revealed. By transforming coined QWM to Szegedy QWM, some amazing results about QWM are founded.
The goal of teleportation is to transfer the state of one particle to another particle. In coined quantum walks, conditional shift operators can introduce entanglement between position space and coin space. This entanglement resource can be used as a quantum channel for teleportation, as proposed by Wang, Shang and Xue [Quantum Inf. Process. 16, 221 (2017)]. Here, we demonstrate the implementation of quantum teleportation using quantum walks on a five-qubit quantum computer and a 32-qubit simulator provided by IBM quantum experience beta platform. We show the teleportation of single-qubit, two…
In recent years, quantum walks have been widely researched and have shown exciting properties. One such is a quadratic speed-up in hitting time compared to its classical counterpart. In this paper, we design a quantum circuit for the MNRS algorithm, which finds a marked node in a graph with a quantum walk, and use it to find a hitting time for the marked nodes in the walk. We do this by implementing the circuit on IBM quantum simulators and show that the execution on a noise-free simulator results in hitting times that agree with the theoretical expectations. We also run the algorithm on a moc…
Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices
波利亚数刻画了随机游动的递归性。我们把这个概念推广到量子行走[M.什特费滕茨阿克岛Jex和T.《吻，物理学》，Rev. Lett. \textbf{100}，020501（2008）]，其基于特定的测量方案。量子游走的波利亚数一般取决于硬币的选择和初始硬币状态，这与经典随机游走不同，经典随机游走的晶格维度唯一地决定了它。我们分析了几个例子来描述各种可能的递归性质。首先，我们证明，对于所有空间维度上由独立硬币驱动的量子行走类，波利亚数与初始条件和实际硬币算子无关，因此类似于经典行走的性质。我们提供了这类量子行走的Pólya数的分析估计。其次，我们研究了2-D格罗弗步行，表现出本地化，因此是经常性的，除了一个特定的初始状态的步行是短暂的。我们推广的Grover步行表明，可以构建在任意尺寸的量子步行是经常性的。这与只对$d=1,2$维循环的经典游动形成了很大的对比。最后，我们分析了2-D傅里叶游走的递归性。除了初始态的二维子空间，这种量子行走是经常性的。我们给出了Pólya数依赖于初始状态的解析公式。
在本文中，我们完成了由两位作者在先前的工作中开始的关于线上离散量子行走的分析[J. Phys. A 36：8775-8795（2003）quant-ph/0303105 ]。我们得到一致收敛渐近的“指数衰减”区域的主要峰值的薛定谔（或波动力学）图片的前沿。这个计算需要我们推广的方法的固定阶段，我们描述了这种扩展的一些细节，包括自包含的证明所需的所有技术引理。我们还严格地建立了这个系统的路径积分和波动力学表示之间的精确费曼等价性，使用特殊函数理论的一些技术。结合前面的工作，我们现在可以通过两种途径证明每个定理。
We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep…
The staggered quantum walk model allows to establish an unprecedented connection between discrete-time quantum walks and graph theory. We call attention to the fact that a large subclass of the coined model is included in Szegedy's model, which in its turn is entirely included in the staggered model. In order to compare those three quantum walk models, we put them in the staggered formalism and we show that the Szegedy and coined models are defined on a special subclass of graphs. This inclusion scheme is also true when the searching framework is added. We use graph theory to characterize whic…
IRT Saint Exupery（圣埃克苏佩里技术研究所）；Université Paul Sabatier（保罗·萨巴捷大学）；IRIT（图卢兹信息学研究所）
Quantum walks on binary trees are used in many quantum algorithms to achieve important speedup over classical algorithms. The formulation of this kind of algorithms as quantum circuit presents the advantage of being easily readable, executable on circuit based quantum computers and simulators and optimal on the usage of resources. We propose a strategy to compose quantum circuit that performs quantum walk on binary trees following universal gate model quantum computation principles. We give a particular attention to NAND formula evaluation algorithm as it could have many applications in game t…
A quantum walk on a toral phase space involving translations in position and its conjugate momentum is studied in the simple context of a coined walker in discrete time. The resultant walk, with a family of coins parametrized by an angle is such that its spectrum is exactly solvable with eigenangles for odd parity lattices being equally spaced, a feature that is remarkably independent of the coin. The eigenvectors are naturally specified in terms the $q-$Pochhammer symbol, but can also be written in terms of elementary functions, and their entanglement can be analytically found. While the phas…
In this work we explain the necessity for consistently labeled rotation maps for efficiently computing coined discrete time quantum walks on regular graphs.
We introduce a new framework to analyze quantum algorithms with the renormalization group (RG). To this end, we present a detailed analysis of the real-space RG for discrete-time quantum walks on fractal networks and show how deep insights into the analytic structure as well as generic results about the long-time behavior can be extracted. The RG-flow for such a walk on a dual Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained explicitly from elementary algebraic manipulations, after transforming the unitary evolution equation into Laplace space. Unlike for classical rand…
We consider the two-state space-inhomogeneous coined quantum walk (QW) in one dimension. For a general setting, we obtain the stationary measure of the QW by solving the eigenvalue problem. As a corollary, stationary measures of the multi-defect model and space-homogeneous QW are derived. The former is a generalization of the previous works on one-defect model and the latter is a generalization of the result given by Konno and Takei (2015).
In this paper, we consider an extended coined Szegedy model and discuss the existence of the point spectrum of induced quantum walks in terms of recurrence properties of the underlying birth and death process. We obtain that if the underlying random walk is not null recurrent, then the point spectrum exists in the induced quantum walks. As an application, we provide a simple computational way of the dispersion relation of the edge state part for the topological phase model driven by quantum walk using the recurrence properties of underlying birth and death process.
Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that for the coined quantum walks on star graph and complete graph with self-loops perfect state transfer between the sender and receiver vertex is achieved for arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on complete graph allows for state transfer with unit fidelity in the limit o…
Decoherence phenomenon has been widely studied in different types of quantum walks. In this work we show how to model decoherence inspired by percolation on staggered quantum walks. Two models of unitary noise are described: breaking polygons and breaking vertices. The evolution operators subject to these noises are obtained and the equivalence to the coined quantum walk model is presented. Further, we numerically analyze the effect of these decoherence models on the two-dimensional grid of $4$-cliques. We examine how these perturbations affect the quantum walk based search algorithm in this g…
