本文作者证明了，对于离散时间量子游走X^{（d）} t，其演化用Hadamard变换描述，弱极限定理成立：X^{（d）} t /\to dx / π（1-x^2）\sqrt{1 - 2 x^2} as \to \infty .本文证明了对于{ \it连续时间}量子行走X^{（c）}_t满足类似的弱极限定理：X^{（c）}_t /tPDMATHABMM dx / π \sqrt{1 - x^2} as \to \infty .量子游动的这些结果与对称离散和连续时间经典随机游动的中心极限定理形成了鲜明的对比：Y_{t} / \sqrt{t} \to e^{-x^2/2} dx / \sqrt{2 π} as t PDMATHAMM \infty。工作还涉及离散和连续时间量子行走之间的关系的问题。这一主题，在以前的文献中的一个长期的辩论的主题，在矩阵表示的形式主义和极限分布进行了详尽的比较，在这两种情况下。
英文摘要：
Concerning a discrete-time quantum walk X^{(d)}_t with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: X^{(d)}_t /t \to dx / π(1-x^2) \sqrt{1 - 2 x^2} as t \to \infty. The present paper shows that a similar type of weak limit theorems is satisfied for a {\it continuous-time} quantum walk X^{(c)}_t on the line as follows: X^{(c)}_t /t \to dx / π\sqrt{1 - x^2} as t \to \infty. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Y_{t}/ \sqrt{t} \to e^{-x^2/2} dx / \sqrt{2 π} as t \to \infty. The work deals also with issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.
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.
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 .
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.
The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution $U$ by lifting the eigenvalues of an induced self-adjoint matrix $T$ onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulab…
In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on one dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position, for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.
We consider the limit distributions of open quantum random walks on one-dimensional lattice space. We introduce a dual process to the original quantum walk process, which is quite similar to the relation of Schrödinger-Heisenberg representation in quantum mechanics. By this, we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them. In particular, it is possible to get rid of the initial state when we consider the evolution of the walk, it appears only in the last step of the computation.
Limit Theorems for Decoherent Two Dimensional Quantum Walks
在本文中，我们考虑由[Brun，T.A，et.al，Phys.Rev.A 67（2003）032304]引入的具有退相干算子的模型，该模型最近由[Ampadu，C.，Brun-Type Formalism for Decoherence in Two Dimensional Quantum Walks，Communication in Theoretical Physics To Appear，arXiv：1104.2061（2011）]以获得退相干量子行走的极限。
