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Exceptional Configurations for Search by Szegedy's Quantum Walk

Thomas G. Wong, Raqueline A. M. Santos

Published Oct 19, 2016• 0 citations
OpenAlex

Abstract

In seminal work, Szegedy gave a method for quantizing a classical Markov chain that has been especially useful for searching for marked vertices in graphs. We prove, however, that this algorithm fails when searching on the two-dimensional periodic square lattice with a marked diagonal, in the sense that the state of the system only evolves by acquiring minus signs. Then the state is a uniform probability distribution over the vertices for all time, which is equivalent to classically guessing and checking. Furthermore, we prove that this failure also occurs for any configuration of marked vertices on the one-dimensional cycle and for any configuration that reduces to it. Thus, Szegedy's quantum walk has a family of exceptional configurations for which it does not spread. Despite this, we exploit this stationary behavior to construct a quantum walk with an arbitrary polynomial speedup over the classical random walk, and it is the first example of a greater-than-quadratic speedup.

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In seminal work, Szegedy gave a method for quantizing a classical Markov chain that has been especially useful for searching for marked vertices in graphs. We prove, however, that this algorithm fails when searching on the two-dimensional periodic square lattice with a marked diagonal, in the sense that the state of the system only evolves by acquiring minus signs. Then the state is a uniform probability distribution over the vertices for all time, which is equivalent to classically guessing and checking. Furthermore, we prove that this failure also occurs for any configuration of marked vertices on the one-dimensional cycle and for any configuration that reduces to it. Thus, Szegedy's quantum walk has a family of exceptional configurations for which it does not spread. Despite this, we exploit this stationary behavior to construct a quantum walk with an arbitrary polynomial speedup over the classical random walk, and it is the first example of a greater-than-quadratic speedup.
A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years.
We will give an introduction into the quantum search algorithms on the Markov chains introduced by Szegedy and recent modifications based on partially absorbing Markov chains due to Krovi et al.Algorithmica 74, 851 (2016) It has been shown that a quantum search can find a set of marked vertices quadratically faster (in units of the hitting time) for any reversible Markov chain.The proofs are based on certain properties of the stationary state of the partially absorbing walk and rely on quantum phase estimation techniques.We will offer an alternative view of the underlying mechanism of the quantum search based on spectral properties of the quantum walk operator.By considering the complete graph as an example, we identify the relevant two-level quantum subspace leading to a Grover-like rotation in the underlying vector space.
There exist several types of configurations of marked vertices, referred to as the exceptional configurations, on one- and two-dimensional periodic lattices with additional long-range edges of the Hanoi network of degree four (HN4), which are challenging to find using discrete-time quantum walk algorithms. In this article, we conduct a comparative analysis of the discrete-time quantum walk algorithm utilizing various coin operators to search for these exceptional configurations. First, we study the emergence of several new exceptional configurations/vertices due to the additional long-range edges of the HN4 on both one- and two-dimensional lattices. Second, our study shows that the diagonal configuration on a two-dimensional lattice, which is exceptional in the case without long-range edges, no longer remains an exceptional configuration. Third, it is also shown that a recently proposed modified coin can search all these configurations, including any other configurations in one- and two-dimensional lattices with very high success probability. Additionally, we construct stationary states for the exceptional configurations caused by the additional long-range edges, which explains why the standard and lackadaisical quantum walks with the Grover coin cannot search these configurations.
When searching for a marked vertex in a graph, Szegedy's usual search operator is defined by using the transition probability matrix of the random walk with absorbing barriers at the marked vertices. Instead of using this operator, we analyze searching with Szegedy's quantum walk by using reflections around the marked vertices, that is, the standard form of quantum query. We show we can boost the probability to 1 of finding a marked vertex in the complete graph. Numerical simulations suggests that the success probability can be improved for other graphs, like the two-dimensional grid. We also prove that, for a certain class of graphs, we can express Szegedy's search operator, obtained from the absorbing walk, using the standard query model.
When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if nonbipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.
The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.