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We mainly study exceptional configuration for coined quantum walk search. For searching on a two-dimensional grid by AKR algorithm, we find some new classes of exceptional configurations that cannot be found by the AKR algorithm effectively and the known diagonal configuration can be regarded as its special case. Meanwhile, we give two modified quantum walk models that can improve the success probability in the exceptional configurations by numerical simulation. Furthermore, we introduce the concept of generalized exceptional configuration and consider search by quantum walk on a cycle with Grover coin. We find that the most common coin combination model (G, −), where G is a Grover diffusion transformation, is a generalized exceptional configuration when just searching one marked vertex on the cycle. In the end, we find generalized exceptional configuration has a different evolution of quantum coherence from exceptional configuration. These extend largely the range of exceptional configuration of quantum walk search in some sense.

Quantum walks are a kind of basic quantum computation model. Quantum walks with memory(QWM) are types of modified quantum walks that record the walker’s latest path. In this paper we present QWM-P, a kind of QWM whose evolution depends on the parity of memory. QWM-P has an identity coin shift function which helps in analyzing and designing algorithms. Then we discuss some properties of QWM-P. We find that the parity of memory length affects the appearance of QWM-P.

Quantum walk has been successfully used to search for targets on graphs with vertices identified as the elements of a database. This spacial search on a two-dimensional periodic grid takes O N log N oracle consultations to find a target vertex from N number of vertices with O ( 1 ) success probability, while reaching optimal speed of O ( N ) on d ≥ 3 dimensional square lattice. Our numerical analysis based on lackadaisical quantum walks searches M vertices on a 2-dimensional grid with optimal speed of O ( N / M ) , provided the grid is attached with additional long range edges. Based on the numerical analysis performed with multiple sets of randomly generated targets for a wide range of N and M we suggest that the optimal time complexity of O ( N / M ) with constant success probability can be achieved for quantum search on a two-dimensional periodic grid with long-range edges.

We study the usefulness of the broken-line-type decoherence in quantum Hash functions based on discrete-time quantum walks on a cycle. We first observe that the time-evolution of probability distribution of quantum walks on a cycle is of high sensitivity to such broken-line-type decoherence, increasing with the number of steps in the walk. Based on this observation, we further propose a quantum Hash function based on the broken-line quantum walk on a cycle. Numerical simulation and performance analyses show that decoherence can be useful to improve the performances of quantum hash functions such as better diffusion and confusion, better collision resistance and more uniform distribution of hash values in the hash space. Moreover, our results promote the practical use of quantum-walk-based hash functions in realistic situations.

We study whether the probability distribution of a discrete quantum walk can get arbitrarily close to uniform, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We establish a characterization of this phenomenon on regular non-bipartite graphs in terms of their adjacency eigenvalues and eigenprojections. Using theory from association schemes, we show this phenomenon happens on a strongly regular graph X if and only if X or X ¯ has parameters ( 4 m 2 , 2 m 2 ± m , m 2 ± m , m 2 ± m ) where m ≥ 2 .

Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter δ≃0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)≃1150 compared to the rectangular billiard (regular) case, where the average PR≃1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.

Prediction of missing links is an important part of many applications, such as friends’ recommendations on social media, reduction of economic cost of protein functional modular mining, and implementation of accurate recommendations in the shopping platform. However, the existing algorithms for predicting missing links fall short in the accuracy and the efficiency. To ameliorate these, we propose a simplified quantum walk model whose Hilbert space dimension is only twice the number of nodes in a complex network. This property facilitates simultaneous consideration of the self-loop of each node and the common neighbour information between arbitrary pair of nodes. These effects decrease the negative effect generated by the interference effect in quantum walks while also recording the similarity between nodes and its neighbours. Consequently, the observed probability after the two-step walk is utilised to represent the score of each link as a missing link, by which extensive computations are omitted. Using the AUC index as a performance metric, the proposed model records the highest average accuracy in the prediction of missing links compared to 14 competing algorithms in nine real complex networks. Furthermore, experiments using the precision index show that our proposed model ranks in the first echelon in predicting missing links. These performances indicate the potential of our simplified quantum walk model for applications in network alignment and functional modular mining of protein–protein networks.

Role embedding aims to embed role-similar nodes into similar representations. Role embedding is significant in graph mining, providing a key bridge between traditional role analysis and machine learning. However, current methods suffer from information loss due to the inherent drawbacks, thus failing to capture role information comprehensively from both global and local perspectives. This paper proposes RED (Role Embedding via Discrete-time quantum walk) to address the above issue via quantum walks, whose characters are naturally applicable to role embedding. Based on the superposition, RED simultaneously learns global role representations by evolving features in a global evolution. Besides, RED uses the quasi-periodicity to capture long-term evolving features within steps. To represent local role information, RED simulates a wave-like diffusion by biased walks, where it learns the closeness from accumulated probabilities for local role representations. To the best of our knowledge, RED is the first to apply quantum walks to the role embedding. Substantial experiments demonstrate that RED significantly outperforms state-of-the-art methods by up to 2300.00% in role detection, 90.93% in equivalency identification, and is overwhelmingly superior in robustness.

Node proximity estimation studies structural similarity between nodes and is the key issue of network analysis. It can exist as the node recommendation task and is a fundamental basis of other graph mining techniques. Although Discrete-time quantum walk (DTQW), a promising new technique with distinctive characters, is widely used in many graph mining problems such as graph isomorphism and graph kernel, there are only a few works estimating proximity via DTQW, limiting the further application of DTQW in graph mining. In this paper, we study the capability of DTQW for proximity estimation and propose QSIM to estimate node proximity by DTQW. By analyzing the diffusion process of biased walks, we discover two influential effects that are beneficial to proximity estimation. The Diminishing Effect shows that a node close to the starting node can generally have a high average probability during the diffusion process, which serves as the basis of QSIM. The Returning Effect shows the probability has a tendency to stay around the starting node during the diffusion, which enhances the capability for mining local information especially in densely-connected structures. Benefited from the two effects, QSIM faithfully reveals node proximity and comprehensively unifies different kinds of node proximity. QSIM is the first mature quantum-walk-based method for proximity estimation. Extensive experiments validate the effectiveness of QSIM and show that QSIM outperforms state-of-the-art methods in the node recommendation task, significantly surpassing Refex, Node2vec, and Role2vec, by up to 1094.2% in the first-order node proximity and 18.8% in the second-order node proximity.

Smart systems and technologies have become integral parts of modern society. Their ubiquity makes it paramount to prioritise securing the privacy of data transferred between smart devices. Visual encryption is a technique employed to obscure images by rendering them meaningless to evade attention during transmission. However, the astounding computing power ascribed to quantum technology implies that even the best visually encrypted systems can be effortlessly violated. Consequently, the physical realisation quantum hardware portends great danger for visually encrypted date on smart systems. To circumvent this, our study proposes the integration of quantum walks (QWs) as a cryptographic mechanism to forestall violation of the integrity of images on smart systems. Specifically, we use QW first to substitute the original image and to subsequently permutate and embed it onto the reference image. Based on this structure, our proposed quantum walks visually meaningful cryptosystem facilities confidential transmission of visual information. Simulation-based experiments validate the performance of the proposed system in terms of visual quality, efficiency, robustness, and key space sensitivity, and by that, its potential to safeguard smart systems now and as we transition to the quantum era.