M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to N=2, we might expect an Omega Gamma N) average case complexity as well. However, we will prove that the complexity is nearly p N . For this we need the following result about approximate quantum counting, which follows from Theorem 13 of [6] this is the upcoming journal version of [8] and [17]; see also [18, Theorem 1.10] Theorem 7.1 (Brassard, Hyer, Mosca, Tapp) Let ff 2 [0; 1] There is a quantum algorithm with worst case O(N ff ) queries that outputs an estimate t of the weight t = jXj of its input, such that j t Gamma tj N 1 Gammaff with probability 2=3. Theorem 7.2 ....

M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

....boolean function f ; 0 , given the input as an oracle, makes Omega Gamma p n= Delta p m(n Gamma m) Delta ) queries. This lower bound also holds for the expected query complexity of computing the partial function f ; 0 . Using an approximate counting algorithm of Brassard et al. [5, 14, 6], we show that our query lower bound is optimal to within a constant factor. Theorem 1.3 The quantum query complexity of computing the partial function f ; 0 , given the input as an oracle, is O( p n= Delta p m(n Gamma m) Delta ) The result of Beals et al. mentioned above then ....

....gave an O( 1 ffl log log 1 ffl ) query algorithm for this problem, which is again an almost quadratic improvement over classical algorithms. When the inputs are restricted to be 0 1, the problem reduces to the counting problem. Using the approximate counting algorithm of Brassard et al. [5, 14, 6] mentioned above, we show that the computation of the mean can be made sensitive to the number of ones in the input, resulting in better bounds when jt Gamma n=2j is large. Theorem 1.10 There is a quantum black box algorithm that, given a boolean oracle input X, and an integer Delta 0, ....

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M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. Proceedings of the Workshop on Randomized Algorithms, Mathematical Foundations of Computer Science, 1998.

....probability on the set of X with jXj close to N=2, we might expect an Omega Gamma N) average case complexity. However we will prove that the complexity is nearly p N . For this we need the following result about approximate quantum counting, which follows from [BHT98, Theorem 5] see also [Mos98] or [NW99, Theorem 1.10] Theorem 11 (Brassard, H yer, Tapp; Mosca) Let ff 2 [0; 1] There exists a quantum algorithm with worst case O(N ff ) queries that outputs an estimate t of the weight t = jXj of its input, such that j t Gamma tj N 1 Gammaff with probability 2=3. Theorem 12 ....

....2 GammaN . By Stirling s formula this is O(1= p N ) so the probability of the event D k is O(N 1=2 Gammaff ) In the quantum counting algorithm, Pr[kN 1 Gammaa j t Gamma tj (k 1)N 1 Gammaa ] 2 O(1= k 1) this follows from [BHMT] the upcoming journal version of [BHT98] and [Mos98] Hence also Pr[ t N=2 j D k ] 2 O(1= k 1) We now bound the probability under uniform that the second counting stage is needed by: Pr[ t N=2] N ff =2 X k=0 Pr[ t N=2 j D k ] Delta Pr[D k ] N ff =2 X k=0 O( 1 k 1 ) Delta O(N 1=2 Gammaff ) O(N ....

M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

....our attention on the onesided error case (i.e. when p = 0) and prove bounds on quantum amplification by translating them to bounds on quantum search. In this case, for any x 2 f0; 1g n , f(x) 1 iff (9r 2 S) A(x; r) 1) Grover s quantum search algorithm [15] and some refinements of it [6, 7, 8, 29, 16]) can be cast as a quantum amplification method that is provably more efficient than any classical method. It amplifies a (0; q) algorithm to a (0; 1 2 ) quantum computer with O(1= p q) executions of A, whereas classically Theta(1=q) executions of A would be required to achieve this. It is ....

M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

....on the one sided error case (i.e. when p = 0) and prove bounds on quantum amplification by translating them to bounds on quantum search. In this case, for any x 2 f0; 1g n , f(x) 1 iff (9r 2 S) A(x; r) 1) Grover s quantum search algorithm [Gro96] and some refinements of it [BBHT98, BH97, Mos98, Gro98] can be cast as a quantum amplification method that is provably more efficient than any classical method. It amplifies a classical (0; 1 N ) computer A to a (0; 1 2 ) quantum computer with O( p N) executions of A. More generally, it amplifies a (0; q) algorithm to a (0; 1 2 ....

M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

....for every symmetric Boolean function f : Theorem 8 Let f be symmetric. For (0) 1) 1=2 Gamma and 2 O(1= p N) we have Q (f) 2 O( p N) Proof There is a bounded error quantum algorithm that counts t = jXj exactly using an expected number of O( p (t 1)N ) queries [BHT98, Mos98] Hence an algorithm A that counts t and outputs f(Y ) for any Y of weight t, is a bounded error algorithm for f and has T A (f) N X t=0 (t) q (t 1)N p N N X t=2 (t) q (t 1)N p N 2 N 2 O( p N) 2 6 Average Case Complexity of MAJORITY Here we examine the ....

....probability on the set of X with jXj close to N=2, we might expect an Omega Gamma N) average case complexity. However we will prove that the complexity is nearly p N . For this we need the following result about approximate quantum counting, which follows from [BHT98, Theorem 5] see also [Mos98] or [NW99, Theorem 1.10] Theorem 9 (Brassard, H yer, Tapp; Mosca) Let ff 2 [0; 1] There exists a quantum algorithm with worst case O(N ff ) queries that outputs an estimate t of the weight t = jXj of its input, such that j t Gamma tj N 1 Gammaff with probability 2=3. Theorem 10 ....

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M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

....do much better than the classical computer: it would also need at least ck applications of the machine to obtain error 1=2 k (when treating the machine as a black box) Here c 0 does not depend on k. We interpret this as follows: general results on amplitude amplification [BHT98, Gro98b, Mos98] show that a quantum computer can achieve a square root speed up when amplifying a very small success probability to a constant one, but our result shows that it can achieve at most a linear speed up when amplifying a constant success probability to a probability very close to 1. Finally, in ....

M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

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M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

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M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

....the function f : f0; 1g N f0; 1g is partial; it is not defined on all X 2 f0; 1g N . In the previous example of OR, the function is total; however, the quantum speed up is only quadratic. Some other quantum algorithms that are naturally expressed in the black box model are described in [10, 4, 19, 5, 6, 17, 22, 9, 7, 21, 8]. Of course, upper bounds in the black box model immediately yield upper bounds for the circuit description model in which the function X is succinctly described as a (log N) O(1) sized circuit computing x i from i. On the other hand, lower bounds in the black box model do not imply lower ....

....3. 3 gives a lower bound for symmetric functions in the bounded error setting: if f is non constant and symmetric, then Q 2 (f) Omega Gamma p N(N Gamma Gamma(f ) We can in fact prove a matching upper bound, using the following result, which follows immediately from [7] as noted by Mosca [21]. It shows that we can count the number of 1s in X exactly, with bounded error probability: Theorem 4.9 (Brassard, H yer, Tapp; Mosca) There exists a quantum algorithm that returns t = jX j with probability at least 3=4 using expected time Theta( p (t 1) N Gamma t 1) for all X 2 f0; ....

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M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector analysis. In MFCS'98 workshop on Randomized Algorithms, 1998.

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