Separations in Query Complexity Based on Pointer Functions

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 $R_0(f) = O(D(f)^{0.7537\ldots})$. We show this is false by giving an example of a total boolean function $f$ on $n$ bits whose deterministic query complexity is $\Omega(n/\log(n))$ while its zero-error randomized query complexity is $\tilde O(\sqrt{n})$. We further show that the quantum query complexity of the same function is $\tilde O(n^{1/4})$, 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 $\Omega(n/\log(n))$ and bounded-error randomized query complexity $R(g) = \tilde O(\sqrt{n})$. This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is $Q_E(g) = \tilde O(\sqrt{n})$. These two functions show that the relations $D(f) = O(R_1(f)^2)$ and $R_0(f) = \tilde O(R(f)^2)$ 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 \goos, 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.Comment: 25 pages, 6 figures. Version 3 improves separation between Q_E and R_0 and updates reference

Discrete-Query Quantum Algorithm for NAND Trees

This is a comment on the article “A Quantum Algorithm for the Hamiltonian NAND Tree” by Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, Theory of Computing 4 (2008) 169--190. That paper gave a quantum algorithm for evaluating NAND trees with running time O(√N) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using N^[1/2 + o(1)] queries in the conventional (discrete) quantum query model

A lower bound on the quantum query complexity of read-once functions

We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions. Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' of initially coherently superposed inputs in the query register, having different values of the function. The number of queries is bounded by comparing the required total amount of decoherence of a judiciously selected set of input-output pairs to an upper bound on the amount achievable in a single query step. We use an extension of this result to general weights on input pairs, and general superpositions of inputs.Comment: 12 pages, LaTe

DNF Sparsification and a Faster Deterministic Counting Algorithm

Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f). It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be $\epsilon$-approximated by a width $w$ DNF with at most $(w\log(1/\epsilon))^{O(w)}$ terms. We combine our sparsification result with the work of Luby and Velikovic to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic $n^{\tilde{O}(\log \log(n))}$ time algorithm that computes an additive $\epsilon$ approximation to the fraction of satisfying assignments of f for \epsilon = 1/\poly(\log n). The previous best result due to Luby and Velickovic from nearly two decades ago had a run-time of $n^{\exp(O(\sqrt{\log \log n}))}$.Comment: To appear in the IEEE Conference on Computational Complexity, 201

Towards Better Separation between Deterministic and Randomized Query Complexity

We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and $R_0(F)=\widetilde{O}(D(F))^{3/4}$. 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 G{\"{o}}{\"{o}}s, 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 considerd in the work of Ambainis et al are different variants of $F$, we work with the original function $F$ itself.Comment: Reference adde