14,829 research outputs found
Communication Complexity Lower Bounds by Polynomials
The quantum version of communication complexity allows the two communicating
parties to exchange qubits and/or to make use of prior entanglement (shared
EPR-pairs). Some lower bound techniques are available for qubit communication
complexity, but except for the inner product function, no bounds are known for
the model with unlimited prior entanglement. We show that the log-rank lower
bound extends to the strongest model (qubit communication + unlimited prior
entanglement). By relating the rank of the communication matrix to properties
of polynomials, we are able to derive some strong bounds for exact protocols.
In particular, we prove both the "log-rank conjecture" and the polynomial
equivalence of quantum and classical communication complexity for various
classes of functions. We also derive some weaker bounds for bounded-error
quantum protocols.Comment: 16 pages LaTeX, no figures. 2nd version: rewritten and some results
adde
Probabilistic communication complexity over the reals
Deterministic and probabilistic communication protocols are introduced in
which parties can exchange the values of polynomials (rather than bits in the
usual setting). It is established a sharp lower bound on the communication
complexity of recognizing the -dimensional orthant, on the other hand the
probabilistic communication complexity of its recognizing does not exceed 4. A
polyhedron and a union of hyperplanes are constructed in \RR^{2n} for which a
lower bound on the probabilistic communication complexity of recognizing
each is proved. As a consequence this bound holds also for the EMPTINESS and
the KNAPSACK problems
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
The degrees of polynomials representing or approximating Boolean functions
are a prominent tool in various branches of complexity theory. Sherstov
recently characterized the minimal degree deg_{\eps}(f) among all polynomials
(over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to
worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) +
\sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the
log-factors hidden in the ~\Theta-notation), can be derived quite easily using
the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie
Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
A strong direct product theorem says that if we want to compute k independent
instances of a function, using less than k times the resources needed for one
instance, then our overall success probability will be exponentially small in
k. We establish such theorems for the classical as well as quantum query
complexity of the OR function. This implies slightly weaker direct product
results for all total functions. We prove a similar result for quantum
communication protocols computing k instances of the Disjointness function.
Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for
sorting N items on a quantum computer, which is optimal up to polylog factors.
They also give several tight time-space and communication-space tradeoffs for
the problems of Boolean matrix-vector multiplication and matrix multiplication.Comment: 22 pages LaTeX. 2nd version: some parts rewritten, results are
essentially the same. A shorter version will appear in IEEE FOCS 0
A Nearly Optimal Lower Bound on the Approximate Degree of AC
The approximate degree of a Boolean function is the least degree of a real polynomial that
approximates pointwise to error at most . We introduce a generic
method for increasing the approximate degree of a given function, while
preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function with
approximate degree into a function on variables with approximate degree at least . In particular, if , then
is polynomially larger than . Moreover, if is computed by a
polynomial-size Boolean circuit of constant depth, then so is .
By recursively applying our transformation, for any constant we
exhibit an AC function of approximate degree . This
improves over the best previous lower bound of due to
Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
that holds for any function. Our lower bounds also apply to
(quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
* For any constant , an lower bound on the
quantum communication complexity of a function in AC.
* A Boolean function with approximate degree at least ,
where is the certificate complexity of . This separation is optimal
up to the term in the exponent.
* Improved secret sharing schemes with reconstruction procedures in AC.Comment: 40 pages, 1 figur
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
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