954 research outputs found
Information Theory and Noisy Computation
We report on two types of results. The first is a study of the rate of decay of information carried by a signal which is being propagated over a noisy channel. The second is a series of lower bounds on the depth, size, and component reliability of noisy logic circuits which are required to compute some function reliably. The arguments used for the circuit results are information-theoretic, and in particular, the signal decay result is essential to the depth lower bound. Our first result can be viewed as a quantified version of the data processing lemma, for the case of Boolean random variables
Signal Propagation, with Application to a Lower Bound on the Depth of Noisy Formulas
We study the decay of an information signal propagating through a series of noisy channels. We obtain exact bounds on such decay, and as a result provide a new lower bound on the depth of formulas with noisy components. This improves upon previous work of N. Pippenger and significantly decreases the gap between his lower bound and the classical upper bound of von Neumann. We also discuss connections between our work and the study of mixing rates of Markov chains
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Tight Limits on Nonlocality from Nontrivial Communication Complexity; a.k.a. Reliable Computation with Asymmetric Gate Noise
It has long been known that the existence of certain superquantum nonlocal
correlations would cause communication complexity to collapse. The absurdity of
a world in which any nonlocal binary function could be evaluated with a
constant amount of communication in turn provides a tantalizing way to
distinguish quantum mechanics from incorrect theories of physics; the statement
"communication complexity is nontrivial" has even been conjectured to be a
concise information-theoretic axiom for characterizing quantum mechanics. We
directly address the viability of that perspective with two results. First, we
exhibit a nonlocal game such that communication complexity collapses in any
physical theory whose maximal winning probability exceeds the quantum value.
Second, we consider the venerable CHSH game that initiated this line of
inquiry. In that case, the quantum value is about 0.85 but it is known that a
winning probability of approximately 0.91 would collapse communication
complexity. We show that the 0.91 result is the best possible using a large
class of proof strategies, suggesting that the communication complexity axiom
is insufficient for characterizing CHSH correlations. Both results build on new
insights about reliable classical computation. The first exploits our
formalization of an equivalence between amplification and reliable computation,
while the second follows from a rigorous determination of the threshold for
reliable computation with formulas of noise-free XOR gates and
-noisy AND gates.Comment: 64 pages, 6 figure
Average-Case Lower Bounds for Noisy Boolean Decision Trees
We present a new method for deriving lower bounds to the expected number of queries made by noisy decision trees computing Boolean functions. The new method has the feature that expectations are taken with respect to a uniformly distributed random input, as well as with respect to the random noise, thus yielding stronger lower bounds. It also applies to many more functions than do previous results. The method yields a simple proof of the result (previously established by Reischuk and Schmeltz) that almost all Boolean functions of n arguments require \Me(n \log n) queries, and strengthens this bound from the worst-case over inputs to the average over inputs. The method also yields bounds for specific Boolean functions in terms of their spectra (their Fourier transforms). The simplest instance of this spectral bound yields the result (previously established by Feige, Peleg, Raghavan, and Upfal) that the parity function of n arguments requires \Me(n \log n) queries and again strengthens this bound from the worst-case over inputs to the average over inputs. In its full generality, the spectral bound applies to the highly resilient functions introduced by Chor, Friedman, Goldreich, Hastad, Rudich, and Smolensky, and it yields nonlinear lower bounds whenever the resiliency is asymptotic to the number of arguments
Fault-tolerant quantum computation
Recently, it was realized that use of the properties of quantum mechanics
might speed up certain computations dramatically. Interest in quantum
computation has since been growing. One of the main difficulties of realizing
quantum computation is that decoherence tends to destroy the information in a
superposition of states in a quantum computer, thus making long computations
impossible. A futher difficulty is that inaccuracies in quantum state
transformations throughout the computation accumulate, rendering the output of
long computations unreliable. It was previously known that a quantum circuit
with t gates could tolerate O(1/t) amounts of inaccuracy and decoherence per
gate. We show, for any quantum computation with t gates, how to build a
polynomial size quantum circuit that can tolerate O(1/(log t)^c) amounts of
inaccuracy and decoherence per gate, for some constant c. We do this by showing
how to compute using quantum error correcting codes. These codes were
previously known to provide resistance to errors while storing and transmitting
quantum data.Comment: Latex, 11 pages, no figures, in 37th Symposium on Foundations of
Computing, IEEE Computer Society Press, 1996, pp. 56-6
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