34,038 research outputs found
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
Finite-Block-Length Analysis in Classical and Quantum Information Theory
Coding technology is used in several information processing tasks. In
particular, when noise during transmission disturbs communications, coding
technology is employed to protect the information. However, there are two types
of coding technology: coding in classical information theory and coding in
quantum information theory. Although the physical media used to transmit
information ultimately obey quantum mechanics, we need to choose the type of
coding depending on the kind of information device, classical or quantum, that
is being used. In both branches of information theory, there are many elegant
theoretical results under the ideal assumption that an infinitely large system
is available. In a realistic situation, we need to account for finite size
effects. The present paper reviews finite size effects in classical and quantum
information theory with respect to various topics, including applied aspects
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
Quantum Communication Complexity and Nonlocality of Bipartite Quantum Operations.
This dissertation is motivated by the following fundamental questions: (a) are
there any exponential gaps between quantum and classical communication complexities?
(b) what is the role of entanglement in assisting quantum communications? (c)
how to characterize the nonlocality of quantum operations? We study four specific
problems below.
1. The communication complexity of the Hamming Distance problem. The
Hamming Distance problem is for two parties to determine whether or not the
Hamming distance between two n-bit strings is more than a given threshold. We
prove tighter quantum lower bounds in the general two-party, interactive communication
model. We also construct an efficient classical protocol in the more restricted
Simultaneous Message Passing model, improving previous results.
2. The Log-Equivalence Conjecture. A major open problem in communication
complexity is whether or not quantum protocols can be exponentially more efficient
than classical ones for computing a total Boolean function in the two-party, interactive
model. The answer is believed to be No. Razborov proved this conjecture
for the most general class of functions so far. We prove this conjecture for a broader
class of functions that we called block-composed functions. Our proof appears to be
the first demonstration of the dual approach of the polynomial method in proving
new results.
3. Classical simulations of bipartite quantum measurement. We define a new
ix
concept that measures the nonlocality of bipartite quantum operations. From this
measure, we derive an upper bound that shows the limitation of entanglement in
reducing communication costs.
4. The maximum tensor norm of bipartite superoperators. We define a maximum
tensor norm to quantify the nonlocality of bipartite superoperators. We show
that a bipartite physically realizable superoperator is bi-local if and only if its maximum
tensor norm is 1. Furthermore, the estimation of the maximum tensor norm
can also be used to prove quantum lower bounds on communication complexities.Ph.D.Computer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58538/1/yufanzhu_1.pd
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