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arXiv:0805.1237 · Physical Review A

Quantum searches on highly symmetric graphs

Daniel Reitzner, Mark S. Hillery, Edgar Feldman, Vladimír Bužek

Published Jan 26, 2009• 72 citations
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Abstract

We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.

DOI
10.1103/physreva.79.012323

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We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an MMM-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.
We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has NNN vertices and the subgraph has KKK vertices, the particle becomes localized on the subgraph in O(N/K)O(N/K)O(N/K) steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resources needed for a quantitative comparison of the efficiency of classical and quantum searches---the number of oracle calls.
Quantum walk is a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. Two specific cases of complete multipartite graphs are probed in this paper, and in both cases, each set consists of an equal number of vertices. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs. We also provide the numerical simulation and circuit implementation of our quantum algorithm.
Quantum walks are a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex in two specific cases of complete multipartite graphs. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. To establish this conclusion, we employ approximation and reduction methods to conduct theoretical analysis and proofs, and provide an intuitive demonstration through numerical simulations. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs and may offer insights for robust quantum search on broader graph structures. Finally, we design a circuit for the algorithm that future researchers can use to conduct experiments.
Quantum walks are a powerful framework for simulating complex quantum systems and designing quantum algorithms, particularly for spatial search on graphs, where the goal is to find a marked vertex efficiently. In this work, we present efficient quantum circuits that implement the evolution operator of continuous-time quantum-walk-based search algorithms for three graph families: complete graphs, complete bipartite graphs, and hypercubes. For complete and complete bipartite graphs, our circuits exactly implement the evolution operator. For hypercubes, we propose an approximate implementation that closely matches the exact evolution operator as the number of vertices increases. Our Qiskit simulations demonstrate that even for low-dimensional hypercubes, the algorithm effectively identifies the marked vertex. Furthermore, the approximate implementation developed for hypercubes can be extended to a broad class of graphs, enabling efficient quantum search in scenarios where exact implementations are impractical.
The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of NNN vertices for one of kkk marked vertices with different initial states. We prove an intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one time step to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches 1/21/21/2, but if they are in different partite sets, it instead reaches 1. This differs from the complete graph, which is the quantum walk formulation of Grover's algorithm, where the success probability with two marked vertices is 8/98/98/9. This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schr\"odinger's equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.
Quantum walks, being the quantum analog of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the ``glued trees'' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed and discuss potential applications to quantum search for marked vertices along these graphs.