In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.
In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.
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.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.Finally, 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 [19] for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.
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.
Quantum walks play an important role in the area of quantum algorithms. Many interesting problems can be reduced to searching marked states in a quantum Markov chain. In this context, the notion of quantum hitting time is very important, because it quantifies the running time of the algorithms. Markov chain-based algorithms are probabilistic, therefore the calculation of the success probability is also required in the analysis of the computational complexity. Using Szegedy's definition of quantum hitting time, which is a natural extension of the definition of the classical hitting time, we present analytical expressions for the hitting time and success probability of the quantum walk on the complete graph.
There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs. 1
We set the ground for a theory of quantum walks on graphs- the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.