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48
A new quantum lower bound method, with an application to strong direct product theorem for quantum search
, 2005
"... We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2sided error quantum algorithms computing ..."
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We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the timespace tradeoff of this algorithm is close to optimal. Categories and Subject Descriptors F.1.2 [Computation by Abstract Devices]: Modes of Computation; F.1.3 [Computation by Abstract Devices]: Complexity Measures and Classes—Relations among complexity
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 24 (2 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
Quantum speedup of classical mixing processes. ArXiv: quantph/0609204. Kendon 52
 Physica A
, 2004
"... It is known that repeated measurements performed at uniformly random times enable the continuoustime quantum walk on a finite set S (using a stochastic transition matrix P as the timeindependent Hamiltonian) to sample almost uniformly from S provided that P does. Here we show that the same phenome ..."
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Cited by 14 (1 self)
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It is known that repeated measurements performed at uniformly random times enable the continuoustime quantum walk on a finite set S (using a stochastic transition matrix P as the timeindependent Hamiltonian) to sample almost uniformly from S provided that P does. Here we show that the same phenomenon holds for other (discretetime) walk variants and more general measurements types, then focus our attention on two questions: How are these repeatedlymeasured walks related to the decohering quantum walks proposed by Kendon/Tregenna and Alagic/Russell? And, when do they yield a speedup over their classical counterparts? We answer the first question with a proof that the two quantum walk models are essentially equivalent (in that they sample almost uniformly from S with nearly the same efficiency) by relating the spectral gaps of the Markov chains describing their action on S. We answer the second question (in part) by showing that these quantum walks sample almost uniformly from the torus Z d n in time O(n log ǫ−1). This represents a quadratic speedup over classical and for d = 1 confirms a conjecture of Kendon and Tregenna based on numerical experiments. 1
Quantum and Classical CommunicationSpace Tradeoffs from Rectangle Bounds
"... We derive bounds on the product of the communication C and space S for communicating circuits. The first bound applies to quantum circuits and follows from a "bipartite product" result for the discrepancy of communication problems. If for any problem f : XY the multicolor discrepancy ..."
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Cited by 13 (6 self)
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We derive bounds on the product of the communication C and space S for communicating circuits. The first bound applies to quantum circuits and follows from a "bipartite product" result for the discrepancy of communication problems. If for any problem f : XY the multicolor discrepancy of the communication matrix of f is 1/2 then the problem in which Alice receives some l inputs, Bob r inputs, and their task is to compute f(x i , y j ) for the l r pairs of inputs (x i , y j ), has a quantum communicationspace tradeo# CS (lrd log Z).
Recent Progress in Quantum Algorithms  What quantum algorithms outperform classical computation and how do they do it?
, 2010
"... It is impossible to imagine today’s technological world without algorithms: sorting, searching, calculating, and simulating are being used everywhere to make our everyday lives better. But what are the benefits of the more philosophical endeavor of studying the notion of an algorithm through the per ..."
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Cited by 12 (0 self)
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It is impossible to imagine today’s technological world without algorithms: sorting, searching, calculating, and simulating are being used everywhere to make our everyday lives better. But what are the benefits of the more philosophical endeavor of studying the notion of an algorithm through the perspective of the physical laws of the universe? This simple idea, that we desire an understanding of the algorithm based upon physics seems, upon first reflection, to be nothing more than mere plumbing in the basement of computer science. That is, until one realizes
Finding is as easy as detecting for quantum walks
, 2010
"... We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph. Our approach is new, simpler and more g ..."
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Cited by 12 (2 self)
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We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the walk P and the absorbing walk P ′ , whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of the interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not statetransitive, and in the presence of multiple marked vertices. As a consequence we make a progress on an open problem related to the spatial search on the 2Dgrid.
QUANTUM SEARCH WITH VARIABLE TIMES
, 2008
"... Since Grover’s seminal work, quantum search has been studied in great detail. In the usual search problem, we have a collection of n items x1,..., xn and we would like to find i: xi = 1. We consider a new variant of this problem in which evaluating xi for different i may take a different number of ..."
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Cited by 10 (2 self)
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Since Grover’s seminal work, quantum search has been studied in great detail. In the usual search problem, we have a collection of n items x1,..., xn and we would like to find i: xi = 1. We consider a new variant of this problem in which evaluating xi for different i may take a different number of time steps. Let ti be the number of time steps required to evaluate xi. If the numbers ti are known in advance, we give an algorithm that solves the problem in O ( p t 2 1 + t2 2 +... + t2 n) steps. This is optimal, as we also show a matching lower bound. The case, when ti are not known in advance, can be solved with a polylogarithmic overhead. We also give an application of our new search algorithm to computing readonce functions.
Nested Quantum Walks with Quantum Data Structures
"... We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via ..."
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Cited by 7 (1 self)
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We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the O(n 35/27) and O(n 9/7) learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.