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arXiv:1601.04212 · Journal of Physics A Mathematical and Theoretical

Quantum walk search on Johnson graphs

Thomas G. Wong

Published Apr 12, 2016• 23 citations
arXivHTMLPDF

Abstract

Abstract The Johnson graph J ( n , k ) is defined by n symbols, where vertices are k -element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, J ( n , 1 ) is the complete graph K n , and J ( n , 2 ) is the strongly regular triangular graph T n , both of which are known to support fast spatial search by continuous-time quantum walk. In this paper, we prove that J ( n , 3 ) , which is the n -tetrahedral graph, also supports fast search. In the process, we show that a change of basis is needed for degenerate perturbation theory to accurately describe the dynamics. This method can also be applied to general Johnson graphs J ( n , k ) with fixed k .

DOI
10.1088/1751-8113/49/19/195303

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Abstract The Johnson graph J ( n , k ) is defined by n symbols, where vertices are k -element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, J ( n , 1 ) is the complete graph K n , and J ( n , 2 ) is the strongly regular triangular graph T n , both of which are known to support fast spatial search by continuous-time quantum walk. In this paper, we prove that J ( n , 3 ) , which is the n -tetrahedral graph, also supports fast search. In the process, we show that a change of basis is needed for degenerate perturbation theory to accurately describe the dynamics. This method can also be applied to general Johnson graphs J ( n , k ) with fixed k .
Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix M , the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of M . We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is O(N)O(\sqrt {N})O(N​) with an asymptotic probability of 1+ o (1), where N is the number of vertices.
In 1996, Farhi & Gutmann presented a continuous time analogue of Grover’s algorithm which was later interpreted as search by a quantum random walk on the complete graph. Similar work has since then been done on hypercubes and periodic lattices. We investigate the same search algorithm but for a new type of graphs called strongly regular graphs. The problem is solved by a, in this context, new technique involving degenerate perturbation theory. The result is that the search is optimal on all strongly regular graphs when the number of vertices N is large. However, different kinds of strongly regular graphs perform different when N is not large. Analyzing this provides some insight into what properties of graphs that are important for quantum search. Apart from the results, an important takeaway is the degenerate perturbation theory technique that was used to solve the problem and might be useful in solving other similar problems. In the thesis, an alternative solution to the complete graph case is given using this method. Also, the fact that strongly regular graphs proved to be convenient to analyze in these kinds of settings suggests that they can be used in other similar problems.
Continuous-time quantum walks are typically effected by either the discrete Laplacian or the adjacency matrix. In this paper, we explore a third option: the signless Laplacian, which has applications in algebraic graph theory and may arise in layered antiferromagnetic materials. We explore spatial search on the complete bipartite graph, which is generally irregular and breaks the equivalence of the three quantum walks for regular graphs, and where the search oracle breaks the equivalence of the Laplacian and signless Laplacian quantum walks on bipartite graphs without the oracle. We prove that a uniform superposition over all the vertices of the graph partially evolves to the marked vertices in one partite set, with the choice of set depending on the jumping rate of the quantum walk. We boost this success probability to 1 by proving that a particular nonuniform initial state completely evolves to the marked vertices in one partite set, again depending on the jumping rate. For some parameter regimes, the signless Laplacian yields the fastest search algorithm of the three, suggesting that it could be a new tool for developing faster quantum algorithms.
Spatial search by continuous-time quantum walks on complex networks is focused on using a quantum walk in continuous time in order to find a single or multiple marked vertices within the complex network. The specific formalism used here is to consider a coupling constant that shifts the state of the quantum walker from the initial state to the target state, which is the marked vertex. The thesis begins with establishing the mathematical framework of network theory, quantum walks and numerical methods that will be used in the remainder of the thesis. Then spatial search by continuous-time quantum walk is studied on regular and semi-regular graphs, where most analytical results can be found. This will get us acquainted with spatial search by quantum walk. The complex networks studied are Barabasi-Albert graphs and the Internet network on the level of autonomous systems. Different renormalized and pruned versions of the Internet network are studied. The parameters of the quantum walk that are focused on are the optimal values for the coupling constant, success probability, time and search time.
Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs (SRGs) with parameters ( N , k , λ , μ ) achieves full quantum speedup. The problem is reconsidered in terms of scattering quantum walk, a type of discrete-time quantum walks. Here, the search space is confined to a low-dimensional subspace corresponding to the collapsed graph of SRGs. To quantify the algorithm's performance, we leverage the fundamental pairing theorem, a general theory developed by Cottrell for quantum search of structural anomalies in star graphs. The search algorithm on the SRGs with k scales as N satisfies the theorem, and results can be immediately obtained, while search on the SRGs with k scales as N does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in O ( N ) time steps with a success probability close to 1. The analytical conclusions are verified by simulation results on two SRGs. These examples show that the formalism on star graphs can be applied more generally.