We make use of the Open Quantum Random Walk setting due to S. Attal, F. Petruccione, C. Sabot and I. Sinayskiy in order to discuss hitting times and a quantum version of the Mean Hitting Time Formula from classical probability theory. We study an open quantum notion of hitting probability on a finite collection of sites and with this we are able to describe the problem in terms of linear maps and its matrix representations. After setting an open quantum version of the fundamental matrix for ergodic Markov chains we are able to prove our main result and as consequence a version of the Random Target Lemma. We also study a mean hitting time formula in terms of the minimal polynomial associated to the matrix representation of the quantum walk. We discuss applications of the results to open quantum dynamics on graphs together with open questions.
We make use of the Open Quantum Random Walk setting due to S. Attal, F. Petruccione, C. Sabot and I. Sinayskiy in order to discuss hitting times and a quantum version of the Mean Hitting Time Formula from classical probability theory. We study an open quantum notion of hitting probability on a finite collection of sites and with this we are able to describe the problem in terms of linear maps and its matrix representations. After setting an open quantum version of the fundamental matrix for ergodic Markov chains we are able to prove our main result and as consequence a version of the Random Target Lemma. We also study a mean hitting time formula in terms of the minimal polynomial associated to the matrix representation of the quantum walk. We discuss applications of the results to open quantum dynamics on graphs together with open questions.
We consider a model of open quantum random walk and together with a quantum trajectory approach we are able to examine a notion of hitting time. We see that many constructions, such as minimal solutions to hitting time problems, are variations of well-known classical probability results, but the density matrix degree of freedom on each site gives rise to systems which are seen to be nonclassical. As a more specific application we study the collection of walks induced by normal commuting contractions, for which the corresponding probability expressions are obtained. We examine quantum versions of the gambler's ruin, birth-and-death chain and a basic theorem on potential theory.
We make use of matrix representations of completely positive maps in order to study open quantum dynamics on graphs, with emphasis on quantum walks and the associated trajectories obtained via a monitoring of the position. We discuss the discrete and continuous time setting and consider the statistics of particles located on a vertex with some internal degree of freedom. Other classes of semigroup generators, such as graph-induced generators are also considered. In the case of primitive quantum channels we present expressions of the mean hitting time for a particle to reach a vertex in terms of a quantum version of the fundamental matrix, thus extending recent results in the setting of open quantum random walks. An open quantum version of Kac's Lemma for the expected return time is discussed. By considering appropriate block matrix expressions we are able to make computations in terms of matrix-valued transition rates, in analogy with the notion of rates from classical continuous-time Markov chains. Also inspired by the classical case a recurrence criterion for the continuous time open walk setting is presented.
Random walks (or Markov chains) are models extensively used in theoretical computer science. Several tools, including analysis of quantities such as hitting and mixing times, are helpful for devising randomized algorithms. A notable example is Schöning's algorithm for the satisfiability (SAT) problem. In this work, we use the density-matrix formalism to define a quantum Markov chain model which directly generalizes classical walks, and we show that a common tools such as hitting times can be computed with a similar formula as the one found in the classical theory, which we then apply to known quantum settings such as Grover's algorithm.
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.