It is well known that many real world networks have the power-law degree distribution (scale-free property). However there are no rigorous results for continuous-time quantum walks on such realistic graphs. In this paper, we analyze space-time behaviors of continuous-time quantum walks and random walks on the threshold network model which is a reasonable candidate model having scale-free property. We show that the quantum walker exhibits localization at the starting point, although the random walker tends to spread uniformly.
In this paper we show how using complex valued edge weights in a graph can completely suppress the flow of probability amplitude in a continuous time quantum walk to specific vertices of the graph when the edge weights, graph topology and initial state of the quantum walk satisfy certain conditions. The conditions presented in this paper are derived from the so-called chiral quantum walk, a variant of the continuous time quantum walk which incorporates directional bias with respect to site transfer probabilities between vertices of a graph by using complex edge weights. We examine the necessit…
In this work, we show that ma-QAOA is equivalent to a restriction of continuous-time quantum walks on dynamic graphs. We then show it is universal for computation by finding the appropriate $B$ and $C$ operators and angles that implement the universal gate set consisting of the Hadamard, $π/8$ and Controlled-Not gates in the ma-QAOA framework. This result begins to bridge the gap between the continuous-time quantum walk model and gate model of quantum computation.
In this paper, we consider a continuous-time quantum walk based search algorithm. We introduce equitable partition of the graph and perfect state transfer on it. By these two methods, we can calculate the success probability and the finding time of the search algorithm. In addition, we gave some examples of graphs that we can calculate the success probability and the finding time.
We address continuous-time quantum walks on graphs, and discuss whether and how quantum-limited measurements on the walker may extract information on the tunnelling amplitude between the nodes of the graphs. For a few remarkable families of graphs, we evaluate the ultimate quantum bound to precision, i.e. we compute the quantum Fisher information (QFI), and assess the performances of incomplete measurements, i.e. measurements performed on a subset of the graph's nodes. We also optimize the QFI over the initial preparation of the walker and find the optimal measurement achieving the ultimate pr…
Spatial search by continuous-time quantum walks on crystal lattices
我们考虑的问题，搜索一般的$d$维格子的$N$顶点为一个单一的标记项目使用连续时间量子行走。我们要求局部性，但允许步行在小范围内周期性地变化。通过构建晶格哈密顿展示狄拉克点在其色散关系和利用狄拉克点附近的线性行为，我们开发的算法，解决了问题的$O(\sqrt N)$的时间为$d>2$和$O(\sqrt N \log N)$在$d=2$。特别是，我们表明，这样的算法存在，甚至在任何维度的超立方格。与以前的连续时间量子行走算法在低维超立方晶格，我们的方法不使用外部存储器。
The formalism of continuous-time quantum walks on graphs has been widely used in the study of quantum transport of energy and information, as well as in the development of quantum algorithms. In experimental settings, however, there is limited control over the coupling coefficients between the different nodes of the graph (which are usually considered to be real-valued), thereby restricting the types of quantum walks that can be implemented. In this work, we apply the idea of Floquet engineering in the context of continuous-time quantum walks, i.e., we define periodically-driven Hamiltonians w…
Continuous-time quantum walks (CTQWs) on static graphs provide efficient methods for search and sampling as well as a model for universal quantum computation. We consider an extension of CTQWs to the case of dynamic graphs, in which an ordered sequence of graphs governs free evolution of the quantum walk. We then consider how perfect state transfer during the quantum walk can be used to design dynamic graphs that implement a universal set of quantum logic gates. We give explicit examples for a complete logical basis, and we validate implementations using numerical simulations for quantum telep…
We address continuous-time quantum walks on graphs in the presence of time- and space-dependent noise. Noise is modeled as generalized dynamical percolation, i.e. classical time-dependent fluctuations affecting the tunneling amplitudes of the walker. In order to illustrate the general features of the model, we review recent results on two paradigmatic examples: the dynamics of quantum walks on the line and the effects of noise on the performances of quantum spatial search on the complete and the star graph. We also discuss future perspectives, including extension to many-particle quantum walk,…
Mixing and Decoherence in Continuous-Time Quantum Walks on Cycles
我们证明分析结果表明，退相干可以是有用的混合时间在有限周期的连续时间量子行走。这补充了Kendon和Tregenna（Physical Review A 67（2003），042315）对离散时间量子漫步的类似现象的数值观测。我们的连续时间量子行走的分析处理包括一个连续的监测，诱导退相干过程的所有顶点。我们确定的概率分布的动力学和观察如何混合时间经历从量子到经典行为的过渡，我们的退相干参数从零增长到无穷大。我们的研究结果表明，对于小的退相干率，混合时间与退相干线性改善，而对于大的退相干率，混合时间线性恶化的经典极限。在退相干速率的中间区域，我们的数值数据证实了存在一个唯一的最佳速率，使混合时间最小化。
In this paper, we consider Szegedy's walk, a type of discrete time quantum walk, and corresponding continuous time quantum walk related to the birth and death chain. We show that the scaling limit of time averaged distribution for the continuous time quantum walk induces that of Szegedy's walk if there exists the spectral gap on so-called the corresponding Jacobi matrix .
The discrete time quantum walk (DTQW) is a universal quantum computational model. Significant relationships between discrete and corresponding continuous quantum systems have been studied since the work of Pauli and Feynman. This work continues the study of relationships between discrete quantum models and their ostensive continuum counterparts by developing a formal transition between discrete and continuous quantum systems through a formal framework for continuum limits of the DTQW. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coin…
We address decoherence and classicalization of continuous-time quantum walks (CTQWs) on graphs. In particular, we investigate three different models of decoherence, and employ the quantum-classical (QC) dynamical distance as a figure of merit to assess whether, and to which extent, decoherence classicalizes the CTQW, i.e. turns it into the analogue classical process. We show that the dynamics arising from intrinsic decoherence, i.e. dephasing in the energy basis, do not fully classicalize the walker and partially preserves quantum features. On the other hand, dephasing in the position basis, a…
We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$. Given that $X$ has diameter $d$ and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of $X$. We also obtain a similar factorization for any graph $X$ in a Bose Mesner algebra.
We characterize a close connection between the continuous-time quantum-walk model and a discrete-time quantum-walk version, based on the staggered model with Hamiltonians in a class of Cayley graphs, which can be considered as a discretization of continuous-time quantum walks. This connection provides examples of perfect state transfer and instantaneous uniform mixing in the staggered model. On the other hand, we provide some more examples of perfect state transfer and instantaneous uniform mixing in the staggered model that cannot be reproduced by the continuous-time model.
The atom-optics kicked rotor can be used to prepare specific momentum distributions on a discrete basis set. We implement a continuous-time quantum walk and a quantum search protocol in this momentum basis. In particular we propose ways to identify a specific marked state from the final momentum distribution after the walker's evolution. Our protocol is guided by current experimental possibilities making it accessible to experimentally implemented quantum walks with Bose-Einstein condensates.
One of the significant breakthroughs in quantum computation is Grover's algorithm for unsorted database search. Recently, the applications of Grover's algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed so that the search is accelerated with improved threshold values. By taking advantage of the quantum tunneling effect, bett…
