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Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
Limitations of Quantum Advice and OneWay Communication
 Theory of Computing
, 2004
"... Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. ..."
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Cited by 59 (15 self)
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Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
Element Distinctness, Frequency Moments, and Sliding Windows
"... Abstract — We derive new timespace tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. In particular, we develop a randomized algorithm for the ele ..."
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Abstract — We derive new timespace tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. In particular, we develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T ∈ Õ(n3/2/S1/2), smaller than previous lower bounds for comparisonbased algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and spaceefficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows [18], where the task is to take an input of length 2n − 1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a timespace tradeoff lower bound of T ∈ Ω(n2/S) for randomized multiway branching programs, and hence standard RAM and wordRAM models, to compute the number of distinct elements, F0, over sliding windows. The same lower bound holds for computing the loworder bit of F0 and computing any frequency moment Fk for k 6 = 1. This shows that frequency moments Fk 6 = 1 and even the decision problem F0 mod 2 are strictly harder than element distinctness. We provide even stronger separations on average for inputs from [n]. We complement this lower bound with a T ∈ Õ(n2/S) comparisonbased deterministic RAM algorithm for exactly computing Fk over sliding windows, nearly matching both our general lower bound for the slidingwindow version and the comparisonbased lower bounds for a single instance of the problem. We also consider the computations of order statistics over sliding windows.
Quantum vs. classical readonce branching programs
, 504
"... Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum readonce branching programs. It is shown that the computational power of quantum and classical readonce branching programs is incomparable in the following strict sense: (i) A simple, explicit boole ..."
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Abstract. The paper presents the first nontrivial upper and lower bounds for (nonoblivious) quantum readonce branching programs. It is shown that the computational power of quantum and classical readonce branching programs is incomparable in the following strict sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by errorfree quantum readonce branching programs of size O ( n 3) , while each classical randomized readonce branching program for this function with bounded twosided error requires size 2 Ω(n). (ii) Quantum readonce branching programs with twosided error bounded by a constant smaller than 1/2 − 2 √ 3/7 ≈ 0.005 are shown to require size 2 Ω(n) for the setdisjointness function DISJn from communication complexity theory. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multipartition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented. 1.
Abstract
, 2001
"... We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [30] to the quantum case. Applying this ..."
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We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [30] to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other Fourier based lower bound methods, notably showing that ¯s(f) / log n, for the average sensitivity ¯s(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f(x ∧ y ⊕ z), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(log n). 1
Classical vs. Quantum ReadOnce Branching Programs
"... Abstract. A simple, explicit boolean function on 2n input bits is presented that is computable by errorfree quantum readonce branching programs of size O(n 3), while each classical randomized readonce branching program and each quantum OBDD for this function with bounded twosided error requires si ..."
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Abstract. A simple, explicit boolean function on 2n input bits is presented that is computable by errorfree quantum readonce branching programs of size O(n 3), while each classical randomized readonce branching program and each quantum OBDD for this function with bounded twosided error requires size 2 Ω(n).
Basic Quantum Algorithms and Applications
"... Quantum computation, the ultimate goal of future computing, is an interesting field for researchers. The concept of quantum computation is based on basics of quantum mechanics. A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena such as superposition ..."
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Quantum computation, the ultimate goal of future computing, is an interesting field for researchers. The concept of quantum computation is based on basics of quantum mechanics. A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena such as superposition and entanglement, to perform operations on data. The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data. A quantum computer operates by manipulating the qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. The field of quantum computation algorithm is fast moving and the scope is vast. Major quantum algorithms are summarized in this paper along with their applications.