A. Nayak and F. Wu, The quantum query complexity of approximating the median and related statistics, Proc. Symp. on Theory of Computing STOC'99, May 1999, 384--393.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....can compute the desired minimum. Quantum algorithm for computing the mean. The above algorithms can be used to compute the average of several numbers, and, in general, the mean of a given random variable. The first such algorithm was proposed by Grover in [7] for further developments, see, e.g. [8, 15, 18]. The traditional Monte Carlo method for computing the mean consists of picking M random values and averaging them. It is a well known fact [19, 21] that the accuracy of this method is 1= M , so, to achieve the given accuracy , we need M iterations. Another way to compute the average ....

A. Nayak and F. Wu, The quantum query complexity of approximating the median and related statistics, Proc. Symp. on Theory of Computing STOC'99, May 1999, 384--393.

....worst case average performance with respect to all Boolean functions and the probabilistic performance with respect to outcomes of the QS algorithm. It turns out that the QS algorithm is optimal in these two settings. The corresponding lower bounds for the Boolean summation problem were shown in [5] for the worst probabilistic setting, and in [7] for the average probabilistic setting. In particular, we know that the QS algorithm with M quantum queries, M N , has the error bound of order M in the worst probabilistic setting. In this paper we study the worst average setting. In this ....

A. Nayak and F. Wu, The quantum query complexity of approximating the median and related statistics, Proceedings of the 31th Annual ACM Symposium on the Theory of Computing (STOC), 384-393, 1999, http://arXiv.org/quantph /9804066, 1998.

....the classical protocols. An example is given where a quantum procedure with 10 queries returns a string of which 80 of the bits are correct. Any classical protocol would need queries to establish such a correctness ratio. 19 3. 2 Known Quantum Query Complexity Bounds Various articles [10, 40, 72] have determined several lower bounds on the capability of quantum computers to outperform classical computers in the black box setting. These bounds refer to the required amount of queries to a black box or oracle (with a domain size n) in order to decide some general property of this black box. ....

....required amount of queries to a black box or oracle (with a domain size n) in order to decide some general property of this black box. For example, if we want to know (with bounded error) the parity of the n values, then it is still necessary for a quantum computer to call the black box times[10, 72]. It has also been shown that for the exact calculation of certain functions (the bitwise OR for example) all n calls are required[10] Here, we present an upper bound on the number of black box queries that is sufficient to compute any function over the n bits provided that we allow a small ....

Ashwin Nayak and Felix Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 384--393. ACM Press, 1999. quantph report no. 9804066.

....for additive approximation of the mean; Dagum et al. 11] and also, implicitly, Schulman and Vazirani [25] give lower bounds for relative approximation of the mean on any input x. Charikar et al. 9] prove a lower bound for ratio approximation of the frequency moment of order 0. Nayak and Wu [22] give a lower bound on the quantum query complexity of the median and some other statistics. Statistical decision theory [4] studies the process of making decisions under incomplete information. The decision procedure can gain information about the environment by sampling from a sample space that ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing (STOC), pages 384--393, 1999.

....approximation of the mean; Dagum et al. DKLR95] and also, implicitly, Schulman and Vazirani [SV99] give lower bounds for relative approximation of the mean on any input x. Charikar et al. CCMN00] prove a lower bound for ratio approximation of the frequency moment of order 0. Nayak and Wu [NW99] give a lower bound on the quantum query complexity of the median and some other statistics. Sampling algorithms can be viewed as a special case of the general framework studied in statistical 28 decision theory [Ber85] Classical decision theory takes a point of view, which is somewhat ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing (STOC), pages 384-393, 1999.

....because, given x 2 X (or y 2 Y ) and i 2 f1; Ng, there is only one input that di ers from x only in the i th position. Therefore, q mm 0 ll 0 = p N and the result follows from Theorem 5.1. Our theorem can be also used to give another proof for the following theorem of Nayak and Wu[18]. Theorem 5.4. 18] Let f : f0; 1; n 1g f0; 1g be a Boolean function that is equal to 1 either at exactly n=2 points of the domain or at exactly (1 )n=2 points. Then, any quantum algorithm that determines whether the number of points where f(x) 1 is n=2 or (1 )n=2 uses 1 ....

....X (or y 2 Y ) and i 2 f1; Ng, there is only one input that di ers from x only in the i th position. Therefore, q mm 0 ll 0 = p N and the result follows from Theorem 5.1. Our theorem can be also used to give another proof for the following theorem of Nayak and Wu[18] Theorem 5.4. [18] Let f : f0; 1; n 1g f0; 1g be a Boolean function that is equal to 1 either at exactly n=2 points of the domain or at exactly (1 )n=2 points. Then, any quantum algorithm that determines whether the number of points where f(x) 1 is n=2 or (1 )n=2 uses 1 ) queries. This ....

[Article contains additional citation context not shown here]

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of STOC'99, pages 384-393. Also quant-ph/9804066.

.... p n for all f . Combining Theorems 17 and 18 with Theorems 12 and 13 we obtain the polynomial relations between classical and quantum complexities of [3] Corollary 4 D(f) 2 O(QE (f) 4 ) and D(f) 2 O(Q 2 (f) 6 ) Some other quantum lower bounds via degree lower bounds may be found in [3,1,29,14,8]. The biggest gap that is known between D(f) and QE (f) is only a factor of 2: D(PARITY n ) n and QE (PARITY n ) dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR n ) n and Q 2 (OR n ) 2 Theta( p n) by Grover s quantum search algorithm [19] Also, R 2 (OR n ) 2 ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31st STOC, pages 384--393, 1999. quant-ph/9804066.

.... examples being Shor s algorithm for factoring [Sho97] and Grover s algorithm for searching [Gro96] Whereas the rst so far has remained a somewhat isolated although seminal result, the second has been applied as a building block in quite a few other quantum algorithms [BH97, BHT97, BCW98, BCWZ99, NW99, BHMT00] One of the earliest applications of Grover s algorithm was the algorithm of Brassard, H yer, and Tapp [BHT97] for nding a collision in a 2 to 1 function f . A collision is a pair of distinct elements x; y such that f(x) f(y) Suppose the size of f s domain is N . For a classical ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31st STOC, pages 384-393, 1999. quant-ph/9804066.

....helpful discussions. A Tight Algorithm for Approximate Counting Here we combine the ideas of algorithms Basic Approx Count and Exact Count to obtain an optimal algorithm for approximately counting. That this algorithm is optimal follows readily from Corollary 1.2 and Theorem 1. 13 of Nayak and Wu [14]. Theorem 18 Given a Boolean function f with N and t de ned as above, and any such that 1 3N 1, the following algorithm Approx Count(f; outputs an estimate t such that t t t with probability at least 2 3 , using an expected number of evaluations of f in the order ....

Nayak, Ashwin and Felix Wu, \The quantum query complexity of approximating the median and related statistics", Proceedings of 31st Annual ACM Symposium on Theory of Computing, May 1999, pp. 384 - 393.

.... being Shor s algorithm for factoring [Sho97] and Grover s algorithm for searching [Gro96] Whereas the first so far has remained a somewhat isolated although seminal result, the second has been applied as a building block in quite a few other quantum algorithms [BH97, BHT97, BCW98, BCWZ99, NW99, BHMT00] Research partially supported by the EU fifth framework program QAIP, IST 1999 11234. CWI, P.O. Box 94079, Amsterdam, The Netherlands. email: buhrman cwi.nl. z Universit e Paris Sud, LRI, 91405 Orsay, France. email: durr lri.fr. x NSA, Suite 6111, Fort George G. Meade, MD 20755. ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31st STOC, pages 384--393, 1999. quant-ph/9804066. 11

....given x 2 X (or y 2 Y ) and i 2 f1; ng, there is only one input that differs from x only in the i th position. Therefore, q mm 0 ll 0 = p n and the result follows from Theorem 2. 2 Our theorem can be also used to give another proof for the following theorem of Nayak and Wu[13]. Theorem 5 [13] Let f : f0; 1; n Gamma 1g f0; 1g be a Boolean function that is equal to 1 either at exactly n=2 points of the domain or at exactly (1 ffl)n=2 points. Then, any quantum algorithm that determines whether the number of points where f(x) 1 is n=2 or (1 ffl)n=2 ....

....X (or y 2 Y ) and i 2 f1; ng, there is only one input that differs from x only in the i th position. Therefore, q mm 0 ll 0 = p n and the result follows from Theorem 2. 2 Our theorem can be also used to give another proof for the following theorem of Nayak and Wu[13] Theorem 5 [13] Let f : f0; 1; n Gamma 1g f0; 1g be a Boolean function that is equal to 1 either at exactly n=2 points of the domain or at exactly (1 ffl)n=2 points. Then, any quantum algorithm that determines whether the number of points where f(x) 1 is n=2 or (1 ffl)n=2 uses Omega Gamma 1 ....

[Article contains additional citation context not shown here]

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31th STOC, pages 384--393, 1999. Also quant-ph/9804066.

.... Theorems 17 and 18 with Theorems 12 and 13 we obtain the polynomial relations between classical and quantum complexities of [BBC 98] Corollary 4 D(f) 2 O(QE (f) 4 ) and D(f) 2 O(Q 2 (f) 6 ) Some other quantum lower bounds via degree lower bounds may be found in [BBC 98, Amb99, NW99, FGGS99, BCWZ99] The biggest gap known between D(f) and QE (f) is only a factor of 2: D(PARITY) n and QE (PARITY) dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR) n and Q 2 (OR) 2 Theta( p n) Gro96] Also, R 2 (OR) 2 Theta(n) deg(OR) n, g deg(OR) 2 ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31th STOC, pages 384--393, 1999. quantph /9804066.

....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 For every ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31th STOC, pages 384--393, 1999. quant-ph/9804066.

....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 For every ....

A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31th STOC, 1999. To appear. quant-ph/9804066.

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A. Nayak and F. Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of 31st STOC, pages 384--393, 1999. quant-ph/9804066.

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