Efficient quantum protocols for XOR functions

We show that for any Boolean function f on {0,1}^n, the bounded-error quantum communication complexity of XOR functions $f\circ \oplus$ satisfies that $Q_\epsilon(f\circ \oplus) = O(2^d (\log\|\hat f\|_{1,\epsilon} + \log \frac{n}{\epsilon}) \log(1/\epsilon))$, where d is the F2-degree of f, and $\|\hat f\|_{1,\epsilon} = \min_{g:\|f-g\|_\infty \leq \epsilon} \|\hat f\|_1$. This implies that the previous lower bound $Q_\epsilon(f\circ \oplus) = \Omega(\log\|\hat f\|_{1,\epsilon})$ by Lee and Shraibman \cite{LS09} is tight for f with low F2-degree. The result also confirms the quantum version of the Log-rank Conjecture for low-degree XOR functions. In addition, we show that the exact quantum communication complexity satisfies $Q_E(f) = O(2^d \log \|\hat f\|_0)$, where $\|\hat f\|_0$ is the number of nonzero Fourier coefficients of f. This matches the previous lower bound $Q_E(f(x,y)) = \Omega(\log rank(M_f))$ by Buhrman and de Wolf \cite{BdW01} for low-degree XOR functions.Comment: 11 pages, no figur

Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $\Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget)

Exponential Separation between Quantum Communication and Logarithm of Approximate Rank

Chattopadhyay, Mande and Sherif (ECCC 2018) recently exhibited a total Boolean function, the sink function, that has polynomial approximate rank and polynomial randomized communication complexity. This gives an exponential separation between randomized communication complexity and logarithm of the approximate rank, refuting the log-approximate-rank conjecture. We show that even the quantum communication complexity of the sink function is polynomial, thus also refuting the quantum log-approximate-rank conjecture. Our lower bound is based on the fooling distribution method introduced by Rao and Sinha (ECCC 2015) for the classical case and extended by Anshu, Touchette, Yao and Yu (STOC 2017) for the quantum case. We also give a new proof of the classical lower bound using the fooling distribution method.Comment: The same lower bound has been obtained independently and simultaneously by Anurag Anshu, Naresh Goud Boddu and Dave Touchett

Exponential separation between quantum communication and logarithm of approximate rank

Chattopadhyay, Mande and Sherif (ECCC 2018) recently exhibited a total Boolean function, the sink function, that has polynomial approximate rank and polynomial randomized communication complexity. This gives an exponential separation between randomized communication complexity and logarithm of the approximate rank, refuting the log-approximate-rank conjecture. We show that even the quantum communication complexity of the sink function is polynomial, thus also refuting the quantum log-approximate-rank conjecture. Our lower bound is based on the fooling distribution method introduced by Rao and Sinha (ECCC 2015) for the classical case and extended by Anshu, Touchette, Yao and Yu (STOC 2017) for the quantum case. We also give a new proof of the classical lower bound using the fooling distribution method.</p