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arXiv:1208.1122

Optimal quantum query bounds for almost all Boolean functions

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

Published Aug 6, 2012• 5 citations
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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, and shows that van Dam's oracle interrogation 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
10.48550/arxiv.1208.1122

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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, and shows that van Dam's oracle interrogation 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.
An oracle with bias ε is an oracle that answers queries correctly with a probability of at least 1/2+ε. In this paper, we study the upper and lower bounds of quantum query complexity of oracles with bias ε. For general upper bounds, we show that for any quantum algorithm solving some problem with high probability using T queries of perfect quantum oracles, i.e., oracles with ε =1/2, there exists a quantum algorithm solving the same problem, also with high probability, using O(T/ε) queries of the corresponding biased quantum oracles. As corollaries we can show robust quantum algorithms and gaps between biased quantum and classical oracles, e.g., by showing a problem where the quantum query complexity is O(N/ε) but the classical query complexity is lower bounded by Ω(N logN/ε2). For general lower bounds, we generalize Ambainis' quantum adversary argument to biased quantum oracles and obtain the first lower bounds with explicit bias factor. As one of its applications we can provide another proof of the optimality of quantum algorithms for the so-called quantum Goldreich-Levin problem which was proved before by Adcock, et al. using different and more complicated methods.
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.