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Quantum walk on two-dimensional torus and grid

Mojtaba Komeili

Published May 28, 2018• 1 citation
Publisher

Abstract

Let χ be a function that maps the vertices of a graph to zero or one. We want to find a vertex m, called a marked vertex, such that χ(m) = 1. Random walk is an algorithm for finding such a vertex. The measure of complexity for random walk is hitting time. There is a quantum algorithm that finds a marked vertex, and runs in the square root of the extended hitting time. For some configurations of marked vertices on certain graphs, the extended hitting time is asymptotically larger than the hitting time. This might cause the quantum algorithm for finding to be less efficient than a random walk. Two-dimensional torus and grid are canonical examples of such graphs. This thesis is about efficient quantum algorithms for finding a marked vertex on a two-dimensional torus. We prove the convexity of the extended hitting time, and provide a new upper bound on it. We show that, with a probability that is more than a constant, random walk on a torus is localized, and it is restricted to a sub-graph of the torus. From these observations, we introduce two algorithms that find a marked vertex on a torus. The complexity of these algorithms is in the order of the square root of the hitting time, up to logarithmic factors. They both start by finding an estimate to the hitting time. Using that, they estimate the size of a sub-graph that contains a marked vertex with more than a constant probability, and sample such a sub-graph. Our first algorithm finds a marked vertex on this sub-graph by using a quantum algorithm that runs in extended hitting time, and finds a marked vertex with probability at least a constant. Our second algorithm recursively runs the detection algorithm on cuts of the sub-graph, until reaching a constant sized sub-graph that contains a marked vertex. These algorithms are the first quantum algorithms that achieve close-to-quadratic speed-up in quantum walk on a two-dimensional torus, for all configurations of marked vertices.

DOI
10.11575/prism/31963

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Let χ be a function that maps the vertices of a graph to zero or one. We want to find a vertex m, called a marked vertex, such that χ(m) = 1. Random walk is an algorithm for finding such a vertex. The measure of complexity for random walk is hitting time. There is a quantum algorithm that finds a marked vertex, and runs in the square root of the extended hitting time. For some configurations of marked vertices on certain graphs, the extended hitting time is asymptotically larger than the hitting time. This might cause the quantum algorithm for finding to be less efficient than a random walk. Two-dimensional torus and grid are canonical examples of such graphs. This thesis is about efficient quantum algorithms for finding a marked vertex on a two-dimensional torus. We prove the convexity of the extended hitting time, and provide a new upper bound on it. We show that, with a probability that is more than a constant, random walk on a torus is localized, and it is restricted to a sub-graph of the torus. From these observations, we introduce two algorithms that find a marked vertex on a torus. The complexity of these algorithms is in the order of the square root of the hitting time, up to logarithmic factors. They both start by finding an estimate to the hitting time. Using that, they estimate the size of a sub-graph that contains a marked vertex with more than a constant probability, and sample such a sub-graph. Our first algorithm finds a marked vertex on this sub-graph by using a quantum algorithm that runs in extended hitting time, and finds a marked vertex with probability at least a constant. Our second algorithm recursively runs the detection algorithm on cuts of the sub-graph, until reaching a constant sized sub-graph that contains a marked vertex. These algorithms are the first quantum algorithms that achieve close-to-quadratic speed-up in quantum walk on a two-dimensional torus, for all configurations of marked vertices.
We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound on the extended hitting time of a marked subset, expressed in terms of the hitting times of its members.
We present an adiabatic quantum algorithm for the abstract problem of searching marked vertices in a graph, or spatial search. Given a random walk (or Markov chain) PPP on a graph with a set of unknown marked vertices, one can define a related absorbing walk P′{P}^{'}P′ where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian H(s)H(s)H(s) from the interpolated Markov chain $P(s)=(1\ensuremath{-}s)P+{\mathit{sP}}^{'}$ and use it in an adiabatic quantum algorithm to drive an initial superposition over all vertices to a superposition over marked vertices. The adiabatic condition implies that, for any reversible Markov chain and any set of marked vertices, the running time of the adiabatic algorithm is given by the square root of the classical hitting time. This algorithm therefore demonstrates a novel connection between the adiabatic condition and the classical notion of hitting time of a random walk. It also significantly extends the scope of previous quantum algorithms for this problem, which could only obtain a full quadratic speedup for state-transitive reversible Markov chains with a unique marked vertex.
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.
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 hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find —a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78 :012310 2008 ) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains. Further, we prove that for any reversible ergodic Markov chain P , the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P . We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.
The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find—a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78:012310 2008) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains. Further, we prove that for any reversible ergodic Markov chain P, the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.