Separations in query complexity based on pointer functions

Abstract

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k, which achieves R 0 (f ) = O(D(f ) 0.7537... ). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/ log(n)) while its zero-error randomized query complexity is O( √ n). We further show that the quantum query complexity of the same function is O(n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/ log(n)) and bounded-error randomized query complexity R(g) = O( √ n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E (g) = O( √ n). These two functions show that the relations D(f ) = O(R 1 (f ) 2 ) and R 0 (f ) = O(R(f ) 2 ) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity.All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

Understanding Quantum Algorithms via Query Complexity

“…This basic construction can be modified in several ways [12]. To separate two models of computation, we should make the certificate for f = 1 easy to find in one of them but difficult in the other model.…”

“…The first of them, the pointer function method was first invented by Göös, Pitassi and Watson [28] for solving the communication vs. partition number problem in classical communication complexity. It was then quickly adapted to separating query complexity measures by Ambainis et al [12]:…”

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“…For example, Nisan and Szegedy [14] proved bs( f ) ≤ 2 deg( f ), and also proved a result in the other direction, which was improved in [3] to deg( f ) ≤ 3 bs( f ). For a survey of such results, see [4] and [9]; some recent results include [1,8].…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] By construction, Π is order-oblivious. Also for any permutation σ there is a permutation τ for which…”

“…We can now state the information-theoretic analog of Question 1.3, which was independently posed by Andy Drucker [6]. 1 Here we restrict attention to the uniform distribution on σ , b; see the remark following the proof of Theorem 3.5.…”