14 research outputs found
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 satisfies that
, where d is the F2-degree of f, and
.
This implies that the previous lower bound 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 , where is the number of nonzero Fourier coefficients of
f. This matches the previous lower bound
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 and with monotone
functions , where and are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions which
alternate values for a relatively small number of times on any monotone path
from to . 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 input bits, each of which has approximate Fourier
sparsity at most and randomized parity decision tree complexity
. 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