We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walkà la Szegedy [25] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [6] and Szegedy [25]. Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in Section

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set M consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time HT(P, M ) of any reversible random walk P on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity HT + (P , M ) which we call extended hitting time.Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk P and the absorbing walk P ′ , whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not statetransitive. We also provide algorithms in the cases when only approximations or bounds on parameters p M (the probability of picking a marked vertex from the stationary distribution) and HT + (P , M ) are known.

We present three semi-streaming algorithms for Maximum Bipartite Matching with one and two passes. Our one-pass semi-streaming algorithm is deterministic and returns a matching of size at least 1/2 + 0.005 times the optimal matching size in expectation, assuming that edges arrive one by one in (uniform) random order. Our first two-pass algorithm is randomized and returns a matching of size at least 1/2 + 0.019 times the optimal matching size in expectation (over its internal random coin flips) for any arrival order. These two algorithms apply the simple Greedy matching algorithm several times on carefully chosen subgraphs as a subroutine. Furthermore, we present a two-pass deterministic algorithm for any arrival order returning a matching of size at least 1/2 + 0.019 times the optimal matching size. This algorithm is built on ideas from the computation of semi-matchings.

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N ) quantum upper bound for the element distinctness problem in the comparison complexity model (contrasting with Θ(N log N ) classical complexity). We also prove a lower bound of Ω( √ N ) comparisons for this problem and derive bounds for a number of related 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, A preliminary version of this article appeared in 92 Algorithmica (2012) 63:91-116 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.

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

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) with space O( √ n log n ) bits, time per letter polylog(n), and one-sided error. We prove that this one-pass algorithm is optimal, up to a log n factor, even when twosided error is allowed, and conjecture that a similar bound holds for any constant number of passes over the input.Surprisingly, the space requirement shrinks drastically if we have access to the input stream in reverse. We present a two-pass randomized streaming algorithm for Dyck(2) with space O((log n)2 ), time polylog(n) and one-sided error, where the second pass is in the reverse direction. Both algorithms can be extended to Dyck(s) since this problem is reducible to Dyck(2) for a suitable notion of reduction in the streaming model. Except for an extra O( √ log s ) multiplicative overhead in the space required in the one-pass algorithm, the resource requirements are of the same order.For the lower bound, we exhibit hard instances Ascension(m) of Dyck(2) with length Θ(mn). We embed these in what we call a "one-pass" communication problem with 2m-players, where m =Õ(n). To establish the hardness of Ascension(m), we prove a direct sum result by following the "information cost" approach, but with a few *