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 g
ates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537ldots})$. 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.
Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is polynomially r
elated to other major complexity measures. Despite much attention to the problem and major advances in analysis of Boolean functions in the past decade, the problem remains wide open with no positive result toward the conjecture since the work of Kenyon and Kutin from 2004. In this work, we present new upper bounds for various complexity measures in terms of sensitivity improving the bounds provided by Kenyon and Kutin. Specifically, we show that deg(f)^{1-o(1)}=O(2^{s(f)}) and C(f) < 2^{s(f)-1} s(f); these in turn imply various corollaries regarding the relation between sensitivity and other complexity measures, such as block sensitivity, via known results. The gap between sensitivity and other complexity measures remains exponential but these results are the first improvement for this difficult problem that has been achieved in a decade.
It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-b
it Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.
We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succ
eed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.
Sensitivity cite{CD82,CDR86} and block sensitivity cite{Nisan91} are two important complexity measures of Boolean functions. A longstanding open problem in decision tree complexity, the Sensitivity versus Block Sensitivity question, proposed by Nisan
and Szegedy cite{Nisan94} in 1992, is whether these two complexity measures are polynomially related, i.e., whether $bs(f)=O(s(f)^{O(1)})$. We prove an new upper bound on block sensitivity in terms of sensitivity: $bs(f) leq 2^{s(f)-1} s(f)$. Previously, the best upper bound on block sensitivity was $bs(f) leq (frac{e}{sqrt{2pi}}) e^{s(f)} sqrt{s(f)}$ by Kenyon and Kutin cite{KK}. We also prove that if $min{s_0(f),s_1(f)}$ is a constant, then sensitivity and block sensitivity are linearly related, i.e. $bs(f)=O(s(f))$.
We present three new quantum algorithms in the quantum query model for textsc{graph-collision} problem: begin{itemize} item an algorithm based on tree decomposition that uses $Oleft(sqrt{n}t^{sfrac{1}{6}}right)$ queries where $t$ is the treewidth of
the graph; item an algorithm constructed on a span program that improves a result by Gavinsky and Ito. The algorithm uses $O(sqrt{n}+sqrt{alpha^{**}})$ queries, where $alpha^{**}(G)$ is a graph parameter defined by [alpha^{**}(G):=min_{VCtext{-- vertex cover of}G}{max_{substack{Isubseteq VCItext{-- independent set}}}{sum_{vin I}{deg{v}}}};] item an algorithm for a subclass of circulant graphs that uses $O(sqrt{n})$ queries. end{itemize} We also present an example of a possibly difficult graph $G$ for which all the known graphs fail to solve graph collision in $O(sqrt{n} log^c n)$ queries.
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 Dams 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.
In this note we give a new separation between sensitivity and block sensitivity of Boolean functions: $bs(f)=(2/3)s(f)^2-(1/3)s(f)$.
We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also pr
ove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph structure.
We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quan
tum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .
