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arXiv:1511.01937 · ACM Symposium on Theory of Computing (STOC)

Separations in query complexity using cheat sheets

Scott Aaronson, Shalev Ben-David, Robin Kothari

Published Jun 10, 2016• 74 citations
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Abstract

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.

DOI
10.1145/2897518.2897644

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We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.
While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2.5 gap. We further present a 1.5 power separation between exact quantum and randomized communication complexity, improving on the previous ≅ 1.15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1.5 separation due to Goos, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.
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 that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( 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 ) = Õ(√ 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 ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic 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.
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.
We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F)), randomized zero error query complexity (R_0(F)) and randomized one-sided error query complexity (R_1(F)): R_1(F) = ~O(sqrt{D(F)) and R_0(F)=~O(D(F))^3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f, R_0(f)=Omega(D(f))^0.753.. . This also shows widest separation between R_1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R_0(f), and polynomial separation between R_0(f) and R_1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F, in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity.
We study how large the certificate complexity C(f){C}(f)C(f) of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function fff whose randomized query complexity with one-sided error satisfies R1(f)=Θ(C(f)){R}_1(f) = Θ(\sqrt{{C}(f)})R1​(f)=Θ(C(f)​). This separation is optimal even when two-sided error is allowed. From this construction, we obtain another total Boolean function FFF with bounded-error quantum query complexity Q(F)=O~(C(F)1/4){Q}(F) = \widetilde O({C}(F)^{1/4})Q(F)=O(C(F)1/4), attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are O~(C(F))\widetilde O(\sqrt{{C}(F)})O(C(F)​). We also prove matching lower bounds for these two measures on FFF.
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.