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Leibniz international proceedings in informatics

Optimal quantum query bounds for almost all Boolean functions

Andris Ambainis, Artūrs Bačkurs, Juris Smotrovs, Ronald de Wolf

Published Jan 1, 2013• 7 citations
Publisher

Abstract

We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis (A. Ambainis, 1999), and shows that van Dam's oracle interrogation (W. van Dam, 1998) is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.

DOI
10.4230/lipics.stacs.2013.446

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IEEE Conference on Computational Complexity (CCC)•Jun 1, 2013•3 citations
We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis (A. Ambainis, 1999), and shows that van Dam's oracle interrogation (W. van Dam, 1998) is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambai- nis [5]. He constructed functions with polynomial degree M and number of variables Omega(M2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.
We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1) show that the workspace of a quantum computer can be limited to at most n+k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model; 2) give an algorithm that on input the truth table of a partial Boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries; 3) use semidefinite programming duality to formulate a dual SDP P/spl circ/(f, t, /spl epsi/) that is feasible if and only if f cannot be evaluated within error /spl epsi/ by a t-step quantum query algorithm. Using this SDP, we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations; 4) give an interpretation of a generalized form of branching in quantum computation.
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.
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.