Continuous-time Quantum Walks Research Articles

Quantum search by continuous-time quantum walk on t-designs

Quantum search by continuous-time quantum walk on t-designs

Enhanced quantum transport in chiral quantum walks

Quantum transport across discrete structures is a relevant topic that can be suitably studied in the context of continuous-time quantum walks. The addition of phase degrees of freedom, leading to chiral quantum walks, can also account for directional transport on graphs with loops. We discuss criteria for quantum transport and study the enhancement that can be achieved with chiral quantum walks on chain-like graphs, exploring different topologies for the chain units and optimizing over the phases. We select three candidate structures with optimal performances and we investigate their transport behaviour with Krylov reduction. While one of them can be reduced to a weighted line with minor couplings modulation, the other two are truly chiral quantum walks, with enhanced transport probability over long chain structures.

Parrondo's effect in continuous-time quantum walks

Parrondo's effect in continuous-time quantum walks

Search algorithm on strongly regular graph by lackadaisical quantum walk

Quantum walk is a widely used method in designing quantum algorithms. In this article, we consider the lackadaisical discrete-time quantum walk (DTQW) on strongly regular graphs (SRG). When there is a single marked vertex in a SRG, we prove that lackadaisical DTQW can find the marked vertex with asymptotic success probability 1, with a quadratic speedup compared to classical random walk. This paper deals with any parameter family of SRG and argues that, by adding self-loops with proper weights, the asymptotic success probability can reach 1. The running time and asymptotic success probability matches the result of continuous-time quantum walk, and improves the result of DTQW.

Quantum hash function based on continuous quantum walks

A method of constructing a quantum hash function (QHF) based on continuous-time quantum walk is proposed, in which the time of quantum walk is controlled by the binary string message, namely, a binary string message as the input of the constructed QHF and the probability value of finally walking to the nodes on the cycle is used as the output of the QHF. Numerical simulation and performance analysis show that our QHF has obvious advantages in the efficiency of the QHF compared with the existing QHF based on the discrete-time quantum walk, that is, the efficiency of our scheme is nearly 4 times faster in computing the hash value of the same size file compared with the most efficient scheme. Besides, the QHF has also better collision resistance compared with the existing QHF based on the discrete-time quantum walk.

Continuous-time open quantum walks in one dimension: matrix-valued orthogonal polynomials and Lindblad generators

Continuous-time open quantum walks in one dimension: matrix-valued orthogonal polynomials and Lindblad generators

Identifying network topologies via quantum walk distributions

Control and characterization of networks are paramount steps in the development of many quantum technologies. Even for moderate-sized networks, this amounts to exploring an extremely vast parameter space in search for the couplings defining the network topology. Here, we explore the use of a genetic algorithm to retrieve the topology of a network from the measured probability distribution obtained from the evolution of a continuous-time quantum walk on the network. We show that we can successfully retrieve the topology of different networks with efficiencies above 70% in all the examined scenarios and that the algorithm is capable of efficiently retrieving the required information even in the presence of noise.

Quantum transport on multilayer generalized scale-free networks

Abstract We study single-particle quantum transport on multilayer generalized scale-free networks using the continuous-time quantum walk model. Our focus is directed at the average return probability and its long-time average value as measures for the transport efficiency. In the continuous-time model these quantities are completely determined by all the eigenvalues and eigenvectors of the connectivity matrix. For all multilayer networks a nontrivial interplay between good spreading and localization effects is observed. The spreading is enhanced by increasing the number of layers L or the power-law exponent γ of the degree distribution. For our choice of the parameters, namely L (1 ≤ L ≤ 50) or γ (1 ≤ γ ≤ 4), the quantum efficiency is increased by at least one order of magnitude. The topological transition between networks without loops, which corresponds to a single scale-free network layer (L = 1), and networks with loops (L = 2) is the most impactful. Another important change occurs when L gets higher than the average diameter d of the layers, namely a new scaling behavior for random walks and lower fluctuations around the long-time average value for quantum walks. The quantum transport is more sensitive to changes of the minimum allowed degree, K min, than to the maximum allowed degree, K max. The same quantum efficiency is found by varying at least one of the parameters: L, γ, K min, or K max, although the network’s topology is different. The quantum efficiency of all multilayer scale-free networks shows a universal behavior for any size of the layers, more precise, is inversely proportional to the number of layers.

Experimental implementation of quantum-walk-based portfolio optimization

The application of quantum algorithms has attracted much attention as it holds the promise of solving practical problems that are intractable to classical algorithms. One such application is the recent development of a quantum-walk-based optimization algorithm approach to portfolio optimization under the modern portfolio theory framework. In this paper, we demonstrate an experimental realization of the alternating phase-shift and continuous-time quantum walk unitaries that underpin this quantum algorithm using optical networks and single photons. The experimental analysis confirms that the probability of states corresponding to high-quality solutions is efficiently amplified by increasing the number of phase-shift and quantum walk iterations. This work provides strong evidence for practical applications of quantum-walk-based algorithms such as financial portfolio optimization.

Perfect state transfer, equitable partition and continuous-time quantum walk based search

Perfect state transfer, equitable partition and continuous-time quantum walk based search

Continuous-time quantum walks for MAX-CUT are hot

By exploiting the link between time-independent Hamiltonians and thermalisation, heuristic predictions on the performance of continuous-time quantum walks for MAX-CUT are made. The resulting predictions depend on the number of triangles in the underlying MAX-CUT graph. We extend these results to the time-dependent setting with multi-stage quantum walks and Floquet systems. The approach followed here provides a novel way of understanding the role of unitary dynamics in tackling combinatorial optimisation problems with continuous-time quantum algorithms.

Witnessing edge modes in trimerized circuit quantum electrodynamic lattice

We propose a scheme to investigate and witness edge modes of general one-dimensional photonic trimers in a circuit quantum electrodynamic lattice. These in-gap edge modes are strictly and analytically solved and the criteria for their emergence are indicated respectively. Moreover, the energy spectrum of the system shows two different regimes characterized by a discrepancy in the number of edge modes. Specifically, while there are always a couple of edge modes only present at one single boundary in both the regimes, one of the regimes also shows another pair of edge modes localized on the opposite boundary. Furthermore, these edge modes are witnessed with the aid of continuous-time quantum walks and average photon number measurements. Our scheme provides a comprehensive method for studying the edge modes of matter.

Quantitative approach to Grover’s quantum walk on graphs

In this paper, we study Grover’s search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover’s algorithm on graphs that can be summarized as follows: Instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph Laplacians. As a result, we search for the most appropriate analytical structure on graphs endowed with fixed topologies yielding better search outcomes. We discuss strategies to investigate the optimality of Grover’s algorithm and provide an example with an easy tunable graph Laplacian to investigate our ideas.

A parameter-independent algorithm of finding maximum clique with Seidel continuous-time quantum walks

A parameter-independent algorithm of finding maximum clique with Seidel continuous-time quantum walks

On the complexity of quantum link prediction in complex networks

Link prediction methods use patterns in known network data to infer which connections may be missing. Previous work has shown that continuous-time quantum walks can be used to represent path-based link prediction, which we further study here to develop a more optimized quantum algorithm. Using a sampling framework for link prediction, we analyze the query access to the input network required to produce a certain number of prediction samples. Considering both well-known classical path-based algorithms using powers of the adjacency matrix as well as our proposed quantum algorithm for path-based link prediction, we argue that there is a polynomial quantum advantage on the dependence on N, the number of nodes in the network. We further argue that the complexity of our algorithm, although sub-linear in N, is limited by the complexity of performing a quantum simulation of the network’s adjacency matrix, which may prove to be an important problem in the development of quantum algorithms for network science in general.

Disorder-free localization in continuous-time quantum walks: Role of symmetries

Disorder-free localization in continuous-time quantum walks: Role of symmetries

Lackadaisical discrete-time quantum walk on Johnson graph

Quantum walk is widely used in search problems, and it has an ideal acceleration effect compared with classical algorithm. Suppose there is a marked vertex w on Johnson graph J(n,k), we hope to find w using quantum walk. We adopt the coined quantum walk model, and we have proved that, for fixed order k, lackadaisical discrete-time quantum walk (DTQW) can find w with success probability 1−o(1), keeping a quadratic speedup. Before this paper, the best result of DTQW is that discrete-time quantum walk finds w with success probability 12−o(1). We improve the success probability to 1−o(1) by adding self-loops with proper weights and choose the appropriate oracle operator. Our result also matches the result of continuous-time quantum walk.

A Novel Image Segmentation Algorithm based on Continuous-Time Quantum Walk using Superpixels

A Novel Image Segmentation Algorithm based on Continuous-Time Quantum Walk using Superpixels

Continuous-time quantum walk based on cycle under broken-line decoherent noise

Continuous-time quantum walk based on cycle under broken-line decoherent noise

How to Sample From the Limiting Distribution of a Continuous-Time Quantum Walk

We introduce ε-projectors, using which we can sample from limiting distributions of continuous-time quantum walks. The standard algorithm for sampling from a distribution that is close to the limiting distribution of a given quantum walk is to run the quantum walk for a time chosen uniformly at random from a large interval, and measure the resulting quantum state. This approach usually results in an exponential running time. We show that, using ε-projectors, we can sample exactly from the limiting distribution. In the black-box setting, where we only have query access to the adjacency matrix of the graph, our sampling algorithm runs in time proportional to Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> , where Δ is the minimum spacing between the distinct eigenvalues of the graph. In the non-black-box setting, we give examples of graphs for which our algorithm runs exponentially faster than the standard sampling algorithm.