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arXiv:quant-ph/9804066 · ACM Symposium on Theory of Computing (STOC)

The quantum query complexity of approximating the median and related statistics

Ashwin Nayak, Felix Wu

Published May 1, 1999• 178 citations
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Abstract

Let X = (z,, , z,-,) be a sequence of n numbers.For 6 > 0, we say that 5; is an e-approximate 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 c-approximate 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 r-approximate median with any constant probability greater than l/2.We also show how an c-approximate 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 kth-smallest 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.

DOI
10.1145/301250.301349

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Let X = (z,, , z,-,) be a sequence of n numbers.For 6 > 0, we say that 5; is an e-approximate 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 c-approximate 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 r-approximate median with any constant probability greater than l/2.We also show how an c-approximate median may be computed with 0( log(t)loglog(log(t) log log(log(t)loglog()) 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 kth-smallest 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.
The approximate degree of a Boolean function f(x1,x2,…,xn)f(x_{1},x_{2},\ldots,x_{n})f(x1​,x2​,…,xn​) is the minimum degree of a real polynomial that approximates fff pointwise within 1/31/31/3. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: - O(n34−14(2k−1))O\bigl(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}}\bigr)O(n43​−4(2k−1)1​) for the kkk-element distinctness problem; - O(n1−1k+1)O(n^{1-\frac{1}{k+1}})O(n1−k+11​) for the kkk-subset sum problem; - O(n1−1k+1)O(n^{1-\frac{1}{k+1}})O(n1−k+11​) for any kkk-DNF or kkk-CNF formula; - O(n3/4)O(n^{3/4})O(n3/4) for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the Θ(n)Θ(n)Θ(n) quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree Ω(n)Ω(n)Ω(n). In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.
One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by epsilon<1/2 iff f can be approximated by a degree-2 polynomial with error bounded by epsilon'<1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [Aaronson/Ambainis, STOC 2015]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms. We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires ~Omega(n) quantum queries but can be represented by a block-multilinear polynomial of degree ~O(sqrt(n)). Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms. Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [Aaronson/Ambainis, STOC 2015], showing that one can estimate the value of any bounded degree-k polynomial p:{0,1}^n -> [-1,1] with O(n^{1-1/(2k)) queries.