Mark Hillery
We study quantum walks through chains consisting of two and three star graphs. The first star has a distinguished vertex labelled START and the last has one labelled END. There are multiple paths between these two vertices, and the object is to find these paths. We show that a quantum walk can do this with a quantum speedup.
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The Scattering Quantum Random Walk scheme has found success as a basis for search algorithms on highly symmetric graph structures. In this paper we examine its effectiveness at locating a specially marked vertex on square grid graphs, consisting of N$^2$ nodes. We simulate these quantum systems using classical computational methods, and find that the probability distributions that arise are very favorable for a hybrid quantum / classical algorithm. We then examine how this hybrid algorithm handles varying types of randomness in both location of the special vertex and later random obstacles pla…
We show that it is possible to use a quantum walk to find a path from one marked vertex to another. In the specific case of $M$ stars connected in a chain, one can find the path from the first star to the last one in $O(M\sqrt{N})$ steps, where $N$ is the number of spokes of each star. First we provide an analytical result showing that by starting in a phase-modulated highly superposed initial state we can find the path in $O(M\sqrt{N}\log M)$ steps. Next, we improve this efficiency by showing that the recovery of the path can also be performed by a series of successive searches when we start …
A mathematical framework for maze solving using quantum walks
本文提出利用带吸收孔的非幺正 Grover walk 求解迷宫最短路径的数学框架，并证明在树结构和梯状结构中概率幅会集中于最短路径附近。
In this paper, we analyze the potential for new types of searches using the formalism of scattering random walks on Quantum Computers. Given a particular type of graph consisting of nodes and connections, a "Tree Maze", we would like to find a selected final node as quickly as possible, faster than any classical search algorithm. We show that this can be done using a quantum random walk, both exactly through numerical calculations as well as analytically using eigenvectors and eigenvalues of the quantum system.
Quantum circuit partition as a maze: emerging percolation transition via path finding
本文将量子电路划分问题形式化为迷宫中的切割路径，通过逾渗相变区分可划分与不可划分区域，并基于CNOT门数量与量子比特数接近时存在划分的观察，提供了可扩展的实用判据。
Escaping from a complex maze, by exploring different paths with several decision-making branches in order to reach the exit, has always been a very challenging and fascinating task. Wave field and quantum objects may explore a complex structure in parallel by interference effects, but without necessarily leading to more efficient transport. Here, inspired by recent observations in biological energy transport phenomena, we demonstrate how a quantum walker can efficiently reach the output of a maze by partially suppressing the presence of interference. In particular, we show theoretically an unp…
A Grover-Based Quantum Algorithm for Solving Perfect Mazes via Fitness-Guided Search
本研究提出一种基于Grover振幅放大的量子算法，通过适应度引导的结构化搜索求解完美迷宫，具备高效扩展性，可推广至类树图等其他搜索领域，为量子混合寻路规划奠基。
We propose a different methodology towards approaching a Maze problem. We convert the problem into a Quantum Search Problem (QSP), and its solutions are sought for using the iterative Grover's Search Algorithm. Though the category of mazes we are looking at are of the NP complete class, we have redirected such a NP complete problem into a QSP. Our solution deals with two dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied on the register encodes each individual wi…
Discrete-time quantum walk in one-dimension is studied from a path-integral perspective. This enables derivation of a closed-form expression for amplitudes corresponding to any coin-position basis of the state vector of the quantum walker at an arbitrary step of the walk. This provides a new approach to the foundations and applications of quantum walks.
A Path Integral Approach for Disordered Quantum Walks in One Dimension
本文通过引入一个独立的随机涨落，然后采用基于路径积分的期望值方法，给出了从量子随机游动到经典随机游动的严格途径。
We consider the stationary state of a quantum walk on the finite path, where the sink and source are set at the left and right boundaries. The quantum coin is uniformly placed at every vertex of the path graph. At every time step, a new quantum walker penetrates into the internal from the left boundary and also some existing quantum walkers in the internal goes out to the sinks located in the left and right boundaries. The square modulus of the stationary state at each vertex is regarded as the comfortability for a quantum walker to this vertex in this paper. We show the weak convergence theor…
Self-Trapping Bounds for Continuous-Time Nonlinear Quantum Walks on Path and Cycle Graphs
该研究探讨了一种连续时间量子漫步，其中在离散路径和环上的单个顶点处存在三次非线性项，证明了非线性量子漫步可以被任意高保真度地捕获，与线性量子漫步在单维中快速扩散形成对比，并提出了这种捕获可用于量子态传输中的定时，以及作为量子记忆的形式。
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 .
Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials and their properties are explored. In particular, fractional revival is shown to take place in the corresponding quantum walks.
Feynman path integrals for discrete-variable systems: Walks on Hamiltonian graphs
本文提出了一种离散变量系统中费曼路径积分的自然表述，通过哈密顿图上的行走形式，恢复了连续变量路径积分，并探讨了其意义。
In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of $η$-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of $η$-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define $n+1$ types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss pe…
The discrete time quantum walk which is a quantum counterpart of random walk plays important roles in the theory of quantum information theory. In the present paper, we focus on discrete time quantum walks viewed as quantization of random walks on the path. We obtain a weak limit theorem for the time averaged distribution of our quantum walks.
Quantum algorithm for de novo DNA sequence assembly based on quantum walks on graphs
本文提出一种基于图上量子行走的从头DNA序列组装量子算法，通过层次划分重叠图并结合哈密顿路径量子算法，在Qiskit上验证了其正确性，为高效解决NP难的DNA组装问题奠定基础。
We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group $\mathbb Z^3$, that in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of $\mathbb Z^3$ that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the …
In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment of one-length path periodically. This small modification to $\mathbb{Z}$ provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.
In this paper, a complex-valued measure of bi-product path space induced by quantum walk is presented. In particular, we consider three types of conditional return paths in a power set of the bi-product path space (1) $Λ\times Λ$, (2) $Λ\times Λ'$ and (3) $Λ'\times Λ'$, where $Λ$ is the set of all $2n$-length ($n\in \mathbb{N}$) return paths and $Λ'(\subseteq Λ)$ is the set of all $2n$-length return paths going through $nx$ ($x\in [-1,1]$) at time $n$. We obtain asymptotic behaviors of the complex-valued measures for the situations (1)-(3) which imply two kinds of weak convergence theorems (Th…
Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle
本文在有限路径图上建立反射与Szegedy游走的局部Konno-Sato对应，通过广义反射原理得到zeta分解，并在对称与非对称情形下恢复经典公式及大质量狄拉克型极限。
In this paper, we consider the time averaged distribution of discrete time quantum walks on the glued trees. In order to analyse the walks on the glued trees, we consider a reduction to the walks on path graphs. Using a spectral analysis of the Jacobi matrices defined by the corresponding random walks on the path graphs, we have spectral decomposition of the time evolution operator of the quantum walks. We find significant contributions of the eigenvalues $\pm 1$ of the Jacobi matrices to the time averaged limit distribution of the quantum walks. As a consequence we obtain lower bounds of the …
Quantum walks constitute important tools in different applications, especially in quantum algorithms. To a great extent their usefulness is due to unusual diffusive features, allowing much faster spreading than their classical counterparts. Such behavior, although frequently credited to intrinsic quantum interference, usually is not completely characterized. Using a recently developed Green's function approach [Phys. Rev. A {\bf 84}, 042343 (2011)], here it is described -- in a rather general way -- the problem dynamics in terms of a true sum over paths history a la Feynman. It allows one to e…
Recently quantum walks have been considered as a possible fundamental description of the dynamics of relativistic quantum fields. Within this scenario we derive the analytical solution of the Weyl walk in 2+1 dimensions. We present a discrete path-integral formulation of the Feynman propagator based on the binary encoding of paths on the lattice. The derivation exploits a special feature of the Weyl walk, that occurs also in other dimensions, that is closure under multiplication of the set of the walk transition matrices. This result opens the perspective of a similar solution in the 3+1 case.
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