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arXiv:2112.03744 · Journal of Physics A Mathematical and Theoretical

Spatial search on Johnson graphs by discrete-time quantum walk

Hajime Tanaka, Mohamed Sabri, Renato Portugal

Published May 12, 2022• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ac6f30

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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.
Spatial search and walk processes. Spatial search by quantum walk is database search with the additional constraint that one is required to move through the search space that obeys some locality structure. For example, the data items may be stored at the vertices of a two-dimensional grid. The requirement of moves along the edges of the grid captures the cost of accessing different items starting from some fixed position in the database. One of possible ways of carrying out spatial search is by performing a random walk on the search space or its quantum analog, a quantum walk. The complexity of spatial search by quantum walk is strongly tied to the quantum hitting time [19] of the walk. Let S , with jS j D n, be a finite set of states. Assume that a subset M S of states are marked. We are given a procedure C that, on input x 2 S and an associated data structure d.x/, checks whether the state x is marked. The goal is either to find a marked state when promised that M ¤ ; (search version) or to determine whether M is nonempty (decision version). The algorithm progresses in stages. In the setup stage, we access some state of S (usually a random state). In the walk stage we move from state to state, performing a spatial walk as described below. The moves are called updates. In addition, in the walk stage we perform checks to see if the current state is marked at steps selected by the algorithm. In the classical setting, the transition probabilities of the spatial walk are described by a stochastic matrix P D .px;y/x;y2S . This makes the walk a Markov chain. In every move the algorithm must perform a random transition according to P . The possible x ! y moves, i.e., those with px;y ¤ 0, form the edges of a (directed) graph G, and we say that the Markov chain P has locality structure G. We define the search problem in the classical setting, which carries over to the quantum case with little modification:
Spatial search is an important problem in quantum computation, which aims to find a marked vertex on a graph. We propose a novel approach for designing deterministic quantum search algorithms on a variety of graphs via alternating quantum walks. Our approach is universal because it does not require an instance-specific analysis for different graphs. We highlight the flexibility of our approach by proving that for Johnson graphs, rook graphs, complete-square graphs and complete bipartite graphs, our quantum algorithms can find the marked vertex with 100%100\%100% success probability and achieve quadratic speedups over classical algorithms. This not only gives an alternative succinct way to prove the existing results, but also leads to new interesting findings on more general graphs.
The task of finding an entry in an unsorted list of NNN elements famously takes O(N)O(N)O(N) queries to an oracle for a classical computer and O(N)O(\sqrt{N})O(N​) queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large NNN, the random walks converge to the same evolution, both taking Nln⁡(1/ϵ)N \ln(1/\epsilon)Nln(1/ϵ) time to reach a success probability of 1−ϵ1-\epsilon1−ϵ. In contrast, the discrete-time quantum walk asymptotically takes πN/22\pi\sqrt{N}/2\sqrt{2}πN​/22​ timesteps to reach a success probability of 1/21/21/2, while the continuous-time quantum walk takes πN/2\pi\sqrt{N}/2πN​/2 time to reach a success probability of 111.
Spatial search is the problem of finding a marked vertex in a graph. A continuous-time quantum walk in the single-excitation subspace of an nnn spin system solves the problem of spatial search by finding the marked vertex in O(n)O(\sqrt{n})O(n​) time. Here, we investigate a natural extension of the spatial search problem, marking multiple vertices of a graph, which are still marked with local fields. We prove that a continuous-time quantum walk in the kkk-excitation subspace of nnn spins can determine the binary string of kkk marked vertices with an asymptotic fidelity in time O(n)O(\sqrt{n})O(n​), despite the size of the state space growing as O(nk)O(n^k)O(nk). Numerically, we show that this algorithm can be implemented with interactions that decay as 1/rα1/r^α1/rα, where rrr is the distance between spins, and an ααα that is readily available in current ion trap systems.
Searching via quantum walk is a topic that has been extensively studied. Most previous results provide approximate solutions, while in this paper we prove an algorithm that can find a marked vertex certainly. We adopt the coined discrete-time quantum walk (DTQW) model with adjusted operators and prove that, on complete bipartite graphs, when parameters are set properly, coined DTQW can deterministically find a marked vertex, i.e., the success probability is exactly 1. Before this paper there have been results of an alternating continuous-time quantum walk method that achieve deterministic spatial search, but this paper provides a deterministic quantum spatial search result via DTQW, while maintaining a quadratic speedup compared to classical algorithms. We also provide the quantum circuit implementation. Published by the American Physical Society 2024