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We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the <italic>Hadamard walk</italic>. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the(More)
We consider the possibility of encoding <i>m</i> classical bits into many fewer <i>n</i> quantum bits (qubits) so that an arbitrary bit from the original <i>m</i> bits can be recovered with good probability. We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand, we show that quantum encoding cannot save more than(More)
Let X = (x0, . . . , xn−1) be a sequence of n numbers. For ǫ > 0, we say that xi is an ǫapproximate median if the number of elements strictly less than xi, and the number of elements strictly greater than xi are each less than (1+ǫ)n/2. We consider the quantum query complexity of computing an ǫ-approximate median, given the sequence X as an oracle. We prove(More)
We propose a new method for designing quantum search algorithms forfinding a "marked" element in the state space of a classical Markovchain. The algorithm is based on a quantum walk &#224; la Szegedy [25] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantumwalk in order to implement an(More)
We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that nontrivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared(More)
Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider quantum walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the “Hadamard walk”. In particular, we show that after t time steps, the probability distribution on(More)
We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in $\tilde{O}(k^{2/3})$ . The(More)
Motivated by a concrete problem and with the goal of understanding the relationship between the complexity of streaming algorithms and the computational complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with s different types of parenthesis. We present a one-pass randomized streaming algorithm for Dyck(2)(More)
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(More)