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.
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.
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.
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 .
Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a three-dimensional subspace but most have no global symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs, nevertheless, achieves the full quantum speed-up of $\mathrm{\ensuremath{\Theta}}(\sqrt{N})$, disproving the intuition that fast quantum search requires global symmetry.
Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.
Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size Θ(N), for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.
Abstract 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.