Quantum Random Walk

Shenvi, Kempe and Whaley's quantum random-walk search (SKW) algorithm [Phys. Rev. A 67, 052307 (2003)] is known to require $O(\sqrt N)$ number of oracle queries to find the marked element, where $N$ is the size of the search space. The overall time complexity of the SKW algorithm differs from the best achievable on a quantum computer only by a constant factor. We present improvements to the SKW algorithm which yield significant increase in success probability, and an improvement on query complexity such that the theoretical limit of a search algorithm succeeding with probability close to one is reached. We point out which improvement can be applied if there is more than one marked element to find.

In this paper we extend the concept of persistence, well defined for classical stochastic dynamics,
to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a
successive measurement scheme, where persistence is defined as the time during which a given site
remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the
first-passage time and the succession probability (newly defined), etc. The study reveals power law
scaling behavior of these quantities with new exponents. Comparable features of the classical and
the quantum walks are discussed.

We study the motion of two non-interacting quantum particles performing a random walk on a line and analyze the probability that the two particles are detected at a particular position after a certain number of steps (meeting problem). The results are compared to the corresponding classical problem and differences are pointed out. Analytic formulas for the meeting probability and its asymptotic behavior are derived. The decay of the meeting probability for distinguishable particles is faster then in the classical case, but not quadratically faster. Entangled initial states and the bosonic or fermionic nature of the walkers are considered.

Entropy rate is a real valued functional on the space of discrete random sources which lacks a closed formula even for subclasses of sources which have intuitive parameterizations. A good way to overcome this problem is to examine its analytic properties relative to some reasonable topology. A canonical choice of a topology is that of the norm of total variation

While two hidden Markov process (HMP) resp. quantum random walk (QRW) parametrizations can differ from one another, the stochastic processes arising from them can be equivalent. Here a polynomial-time algorithm is presented which can determine equivalence of two HMP parametrizations M1, M2 resp. two QRW parametrizations Q1, Q2 in time O(|Σ| max(N1, N2)4), where N1,N2 are the number of hidden states in M1, M2 resp. the dimension of the state spaces associated with Q1, Q2, and Σ is the set of output symbols. Previously avail able algorithms for testing equivalence of HMPs were exponential in the number of hidden states. In case of QRWs, algorithms for testing equivalence had not yet been presented. The core subrou tines of this algorithm can also be used to efficiently test hidden Markov processes and quantum random walks for ergodicity.