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.
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 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 walks have been proven to be a useful framework in designing new quantum algorithms, of which the search algorithm is the most notable. Besides a general search for a special vertex, recent researches have shown that quantum walks can also be used to find structural anomalies. Suppose a vertex of complete graph KN is attached to a second graph G, then the kind of structure anomaly will break the symmetry of the complete graph. The search algorithm based on scattering quantum walk model is presented to speed up locating this kind of structure anomaly. The concepts of scattering quantum walk model and collapsed graphs are presented. The definition of the evolutionary operator, which is different from that of a general search, is given. Based on the specific definition of evolutionary operator and the obvious symmetry of complete graph, it is shown that the search space is confined to a low-dimensional collapsed space, and the initial state is chosen to lie in this subspace. To illustrate the evolutionary process of the search algorithm, an example is given in the case that G is a single vertex. Taking advantage of our earlier study on the evolutionary operator of coined quantum walks with Grover coin, calculations of the unitary operator in the collapsed space are greatly simplified. To quantify the evolutionary process of the algorithm, we use the matrix perturbation theory involving a perturbative approach to find the eigenvalues and eigenstates. It is the degenerate zeroth-order eigenvalue 0 = 1 that leads to the interesting parts of the Hilbert space. Most of the recent researches of searching the structure anomalies focus on star graph SN with an unspecified graph G attached to one of its external vertices, where the overall graph is divided into two parts by the central vertex. It is shown that quantum speedup will occur if and only if the eigenvalues associated with these two parts in the N limit are the same. In this paper, we find that the collapsed graph of complete graphs can also be divided into two parts by a single collapsed vertex. As these two parts roughly correspond to the initial state and the desired state respectively, the techniques and results in star graphs can be generalized to the collapse graph on complete graph. What is more, under our definition of unitary evolution operator these two parts in the N limit will always share the same eigenvalue, i.e. 0 = 1, no matter what the structure of graph G is. Based on this, we prove that the search algorithm can find the target vertex in O(N) time steps with a success probability of 1-O(1N). That is to say, the quantum search algorithm gains a quadratic speedup over classical counterpart.
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.
The search for structural anomalies in a complete graph, [Formula: see text], using scattering quantum walks (SQWs) is investigated in this study. A complete graph with a second graph, [Formula: see text], attached to one of its vertices is referred to as external structural anomaly. The structural change within a complete graph is called internal structural anomaly. First, an example with [Formula: see text] being a triangle is presented to illustrate the problem. Then, a general proof is provided to show that the vertex to which [Formula: see text] is attached can be found in [Formula: see text] time steps as long as the connectivity between [Formula: see text] and [Formula: see text] is far less than [Formula: see text]. Finally, two types of internal structural anomalies, namely, a complete graph with a missing edge and that with an extra loop, are considered. These two anomalies can be solved in a similar manner as external anomalies in [Formula: see text] time steps. These examples demonstrate that Cottrell’s formalism in star graphs can be applied more generally.
We develop a general theory for a quantum-walk search on a star graph. A star graph has N edges each of which is attached to a central vertex. A graph G is attached to one of these edges, and we would like to find out to which edge it is attached. This is done by means of a quantum walk, a quantum version of a random walk. This walk contains O(sqrt[N]) steps, which represents a speedup over a classical search, which would require O(N) steps. The overall graph, star plus G, is divided into two parts, and we find that for a quantum speedup to occur, the eigenvalues associated with these two parts in the N→∞ limit must be the same. Our theory tells us how the initial state of the walk should be chosen, and how many steps the walk must make in order to find G.