quantum walk
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Author(s):  
François David ◽  
Thordur Jonsson

Abstract We study continuous time quantum random walk on a comb with infinite teeth and show that the return probability to the starting point decays with time t as t−1. We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spine with a finite probability. The walk along the spine and into the teeth behaves qualitatively as a quantum random walk on a line. This behaviour is quite different from that of classical random walk on the comb.


Author(s):  
Wen-Qiang Liu ◽  
Xin-Jie Zhou ◽  
Hai-Rui Wei

Abstract Unitary operation is an essential step for quantum information processing. We first propose an iterative procedure for decomposing a general unitary operation without resorting to controlled-NOT gate and single-qubit rotation library. Based on the results of decomposition, we design two compact architectures to deterministically implement arbitrary two-qubit polarization-spatial and spatial-polarization collective unitary operations, respectively. The involved linear optical elements are reduced from 25 to 20 and 21 to 20, respectively. Moreover, the parameterized quantum computation can be flexibly manipulated by wave plates and phase shifters. As an application, we construct the specific quantum circuits to realize two-dimensional quantum walk and quantum Fourier transformation. Our schemes are simple and feasible with the current technology.


2022 ◽  
Author(s):  
Jingbo Zhao ◽  
Tian Zhang ◽  
Jianwei Jiang ◽  
Tong Fang ◽  
Hongyang Ma

Abstract Aiming at solving the trouble that digital image information is easily intercepted and tampered during transmission, we proposed a color image encryption scheme based on alternate quantum random walk and controlled Rubik’s Cube transformation. At the first, the color image is separated into three channels: channel R, channel G and channel B. Besides, a random sequence is generated by alternate quantum walk. Then the six faces of the Rubik’s Cube are decomposed and arranged in a specific order on a two-dimensional plane, and each pixel of the image is randomly mapped to the Rubik’s Cube. The whirling of the Rubik’s Cube is controlled by a random sequence to realize image scrambling and encryption. The scrambled image acquired by Rubik’s Cube whirling and the random sequence received by alternate quantum walk are bitwise-XORed to obtain a single-channel encrypted image. Finally the three-channel image is merged to acquire the final encrypted image. The decryption procedure is the reverse procedure of the encryption procedure. The key space of this scheme is theoretically infinite. After simulation experiments, the information entropy after encryption reaches 7.999, the NPCR is 99.5978%, and the UACI is 33.4317%. The encryption scheme with high robustness and security has a excellent encryption effect which is effective to resist statistical attacks, force attacks, and other differential attacks.


2022 ◽  
Vol 22 (1&2) ◽  
pp. 53-85
Author(s):  
Thomas G. Wong

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.


2021 ◽  
Vol 104 (6) ◽  
Author(s):  
Jacob Rapoza ◽  
Thomas G. Wong

2021 ◽  
Vol 9 ◽  
Author(s):  
T. Bennett ◽  
E. Matwiejew ◽  
S. Marsh ◽  
J. B. Wang

This paper demonstrates the applicability of the Quantum Walk-based Optimisation Algorithm (QWOA) to the Capacitated Vehicle Routing Problem (CVRP). Efficient algorithms are developed for the indexing and unindexing of the solution space and for implementing the required alternating phase-walk unitaries, which are the core components of QWOA. Results of numerical simulation demonstrate that the QWOA is capable of producing convergence to near-optimal solutions for a randomly generated eight location CVRP. Preparation of the amplified quantum state in this example problem is demonstrated to produce higher-quality solutions than expected from classical random sampling of equivalent computational effort.


Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2263
Author(s):  
Leo Matsuoka ◽  
Kenta Yuki ◽  
Hynek Lavička ◽  
Etsuo Segawa

Maze-solving by natural phenomena is a symbolic result of the autonomous optimization induced by a natural system. We present a method for finding the shortest path on a maze consisting of a bipartite graph using a discrete-time quantum walk, which is a toy model of many kinds of quantum systems. By evolving the amplitude distribution according to the quantum walk on a kind of network with sinks, which is the exit of the amplitude, the amplitude distribution remains eternally on the paths between two self-loops indicating the start and the goal of the maze. We performed a numerical analysis of some simple cases and found that the shortest paths were detected by the chain of the maximum trapped densities in most cases of bipartite graphs. The counterintuitive dependence of the convergence steps on the size of the structure of the network was observed in some cases, implying that the asymmetry of the network accelerates or decelerates the convergence process. The relation between the amplitude remaining and distance of the path is also discussed briefly.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mrinal Kanti Giri ◽  
Suman Mondal ◽  
Bhanu Pratap Das ◽  
Tapan Mishra

AbstractWe investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the slow and fast particles perform independent particle quantum walks when the interaction between them is weak. However, stronger inter-particle interactions result in quantum walks by the repulsively bound pair formed between the two particles. For different initial states when the walkers are on different sites initially, the quantum walk performed by the slow particle is almost independent of that of the fast particle, which exhibits reflected and transmitted components across the particle with large hopping strength for weak interactions. Beyond a critical value of the interaction strength, the wave function of the fast particle ceases to penetrate through the slow particle signalling a spatial phase separation. However, when the two particles are initially at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both of them from each other. We analyze the above mentioned features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.


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