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119
Computational Complexity and Feasibility of Data Processing and Interval Computations, With Extension to Cases When We Have Partial Information about Probabilities
, 2003
"... In many reallife situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easiertomeasure quantities x 1 ; : : : ; xn which are related to y by a known relation y = f(x 1 ; : : : ; xn ). Measurements a ..."
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Cited by 216 (128 self)
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In many reallife situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easiertomeasure quantities x 1 ; : : : ; xn which are related to y by a known relation y = f(x 1 ; : : : ; xn ). Measurements are never 100% accurate; hence, the measured values e x i are different from x i , and the resulting estimate e y = f(ex 1 ; : : : ; e xn ) is different from the desired value y = f(x 1 ; : : : ; xn ). How different? Traditional engineering to error estimation in data processing assumes that we know the probabilities of different measurement error \Deltax i = e x i \Gamma x i . In many practical situations, we only know the upper bound \Delta i for this error; hence, after the measurement, the only information that we have about x i is that it belongs to the interval x i = [ex i \Gamma \Delta i ; e x i + \Delta i ]. In this case, it is important to find the range y of all possible values of y = f(x 1 ; : : : ; xn ) when x i 2 x i . We start the paper with a brief overview of the computational complexity of the corresponding interval computation problems.
Quantum amplitude amplification and estimation
, 2002
"... Abstract. Consider a Boolean function χ: X → {0, 1} that partitions set X between its good and bad elements, where x is good if χ(x) = 1 and bad otherwise. Consider also a quantum algorithm A such that A0 〉 = � x∈X αxx 〉 is a quantum superposition of the elements of X, and let a denote the proba ..."
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Cited by 174 (14 self)
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Abstract. Consider a Boolean function χ: X → {0, 1} that partitions set X between its good and bad elements, where x is good if χ(x) = 1 and bad otherwise. Consider also a quantum algorithm A such that A0 〉 = � x∈X αxx 〉 is a quantum superposition of the elements of X, and let a denote the probability that a good element is produced if A0 〉 is measured. If we repeat the process of running A, measuring the output, and using χ to check the validity of the result, we shall expect to repeat 1/a times on the average before a solution is found. Amplitude amplification is a process that allows to find a good x after an expected number of applications of A and its inverse which is proportional to 1 / √ a, assuming algorithm A makes no measurements. This is a generalization of Grover’s searching algorithm in which A was restricted to producing an equal superposition of all members of X and we had a promise that a single x existed such that χ(x) = 1. Our algorithm works whether or not the value of a is known ahead of time. In case the value of a is known, we can find a good x after a number of applications of A and its inverse which is proportional to 1 / √ a even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover’s and Shor’s quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of a. We apply amplitude estimation to the problem of approximate counting, in which we wish to estimate the number of x ∈ X such that χ(x) = 1. We obtain optimal quantum algorithms in a variety of settings. 1.
An Introduction to Quantum Computing for NonPhysicists
 Los Alamos Physics Preprint Archive http://xxx.lanl.gov/abs/quantph/9809016
, 2000
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On quantum algorithms for noncommutative hidden subgroups
, 2000
"... Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum ..."
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Cited by 81 (3 self)
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Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum algorithm for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function. We also explore the difficulties of developing an algorithm to process the data to explicitly calculate a generating set for the subgroup. A general framework for the noncommutative hidden subgroup problem is discussed and we indicate future research directions.
Quantum Algorithms for Element Distinctness
 SIAM Journal of Computing
, 2001
"... 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 c ..."
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Cited by 75 (9 self)
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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. This contrasts 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. 1
The quantum query complexity of approximating the median and related statistics
 STOC'99
, 1999
"... Let X = (z,, , z,,) be a sequence of n numbers. For 6> 0, we say that 5; is an eapproximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):. We consider the quantum query complexity of computing an capprox ..."
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Cited by 74 (1 self)
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Let X = (z,, , z,,) be a sequence of n numbers. For 6> 0, we say that 5; is an eapproximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):. We consider the quantum query complexity of computing an capproximate median, given the sequence X as an oracle. We prove a lower bound of n(min{t,n}) queries for any quantum algorithm that computes an rapproximate median with any constant probability greater than l/2. We also show how an capproximate median may be computed with 0 ( $ log(t) log log ( $)) oracle queries, which rep resents an improvement over an earlier algorithm due to Grover [ll, 121. Thus, the lower bound we obtain is essentially optimal. The upper and the lower bound both hold in the comparison tree model as well. Our lower bound result is an application of the polynomial paradigm recently introduced to quantum complexity theory by Be & et ol. [l]. The main ingredient in the proof is a polynomial degree lower bound far real multilinear polynomials that “approximate” symmetric partial boolean functions. The degree bound extends a result of Patti [15] and also immediately yields lower bounds for the problems of approximating the kthsmallest element, approximating the mean of a sequence of numbers, and approximately counting the number of ones of a boolean function. All bounds obtained come within a polylogarithmic factor of the optimal (as we show by presenting algorithms where no such optimal or near optimal algorithms were known), thus demonstrating the power of the polynomial method.
Fast parallel circuits for the quantum Fourier transform
 PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00)
, 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
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Cited by 72 (1 self)
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We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomialsize, in combination with classical polynomialtime pre and postprocessing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with boundederror probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the
Quantum summation with an application to integration
, 2001
"... We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a psummability ( condition and for d integration of functions from Lebesgue spaces Lp [0, 1] ) and analyze their convergence rates. We ..."
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Cited by 43 (11 self)
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We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a psummability ( condition and for d integration of functions from Lebesgue spaces Lp [0, 1] ) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Brassard, Høyer, Mosca, and Tapp (2000) on computing the mean for bounded sequences and complements results of Novak (2001) on integration of functions from Hölder classes.
Quantum complexity of integration
 J. COMPLEXITY
, 2001
"... It is known that quantum computers yield a speedup for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on [0, 1] d and define γ by γ = (k + ..."
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Cited by 37 (4 self)
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It is known that quantum computers yield a speedup for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on [0, 1] d and define γ by γ = (k + α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are and comp(F k,α