19 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
Spectral Norm of Symmetric Functions
The spectral norm of a Boolean function is the sum
of the absolute values of its Fourier coefficients. This quantity provides
useful upper and lower bounds on the complexity of a function in areas such as
learning theory, circuit complexity, and communication complexity. In this
paper, we give a combinatorial characterization for the spectral norm of
symmetric functions. We show that the logarithm of the spectral norm is of the
same order of magnitude as where ,
and and are the smallest integers less than such that
or is constant for all with . We mention some applications to the decision tree and communication
complexity of symmetric functions
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
A composition theorem for parity kill number
In this work, we study the parity complexity measures
and .
is the \emph{parity kill number} of , the
fewest number of parities on the input variables one has to fix in order to
"kill" , i.e. to make it constant. is the depth
of the shortest \emph{parity decision tree} which computes . These
complexity measures have in recent years become increasingly important in the
fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and
pseudorandomness \cite{BK12, Sha11, CT13}.
Our main result is a composition theorem for .
The -th power of , denoted , is the function which results
from composing with itself times. We prove that if is not a parity
function, then In other words, the parity kill number of
is essentially supermultiplicative in the \emph{normal} kill number of
(also known as the minimum certificate complexity).
As an application of our composition theorem, we show lower bounds on the
parity complexity measures of and . Here is the sort function due to Ambainis \cite{Amb06},
and is Kushilevitz's hemi-icosahedron function \cite{NW95}. In
doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which
had applications to communication complexity and computational learning theory.
In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and
\cite{TWXZ13}
Fourier sparsity, spectral norm, and the Log-rank conjecture
We study Boolean functions with sparse Fourier coefficients or small spectral
norm, and show their applications to the Log-rank Conjecture for XOR functions
f(x\oplus y) --- a fairly large class of functions including well studied ones
such as Equality and Hamming Distance. The rank of the communication matrix M_f
for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree
of f and D^CC(f) stand for the deterministic communication complexity for
f(x\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\hat f||_1). In
particular, the Log-rank conjecture holds for XOR functions with constant
F2-degree. 2. D^CC(f) = O(d ||\hat f||_1) = O(\sqrt{rank(M_f)}\logrank(M_f)).
We obtain our results through a degree-reduction protocol based on a variant of
polynomial rank, and actually conjecture that its communication cost is already
\log^{O(1)}rank(M_f). The above bounds also hold for the parity decision tree
complexity of f, a measure that is no less than the communication complexity
(up to a factor of 2).
Along the way we also show several structural results about Boolean functions
with small F2-degree or small spectral norm, which could be of independent
interest. For functions f with constant F2-degree: 1) f can be written as the
summation of quasi-polynomially many indicator functions of subspaces with
\pm-signs, improving the previous doubly exponential upper bound by Green and
Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having
a small parity decision tree complexity; 3) f depends only on polylog||\hat
f||_1 linear functions of input variables. For functions f with small spectral
norm: 1) there is an affine subspace with co-dimension O(||\hat f||_1) on which
f is a constant; 2) there is a parity decision tree with depth O(||\hat f||_1
log ||\hat f||_0).Comment: v2: Corollary 31 of v1 removed because of a bug in the proof. (Other
results not affected.
Bounds on oblivious multiparty quantum communication complexity
The main conceptual contribution of this paper is investigating quantum
multiparty communication complexity in the setting where communication is
\emph{oblivious}. This requirement, which to our knowledge is satisfied by all
quantum multiparty protocols in the literature, means that the communication
pattern, and in particular the amount of communication exchanged between each
pair of players at each round is fixed \emph{independently of the input} before
the execution of the protocol. We show, for a wide class of functions, how to
prove strong lower bounds on their oblivious quantum -party communication
complexity using lower bounds on their \emph{two-party} communication
complexity. We apply this technique to prove tight lower bounds for all
symmetric functions with \textsf{AND} gadget, and in particular obtain an
optimal lower bound on the oblivious quantum -party
communication complexity of the -bit Set-Disjointness function. We also show
the tightness of these lower bounds by giving (nearly) matching upper bounds.Comment: 13 pages, an accepted paper of LATIN 202