2,459 research outputs found
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
Quantum and classical strong direct product theorems and optimal time-space tradeoffs
A strong direct product theorem says that if we want to compute
independent instances of a function, using less than times
the resources needed for one instance, then our overall success
probability will be exponentially small in .
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 instances of the disjointness function.
Our direct product theorems imply a time-space tradeoff
T^2S=\Om{N^3} for sorting 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
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
Poisson suspensions and infinite ergodic theory
We investigate ergodic theory of Poisson suspensions. In the process, we
establish close connections between finite and infinite measure preserving
ergodic theory. Poisson suspensions thus provide a new approach to infinite
measure preserving ergodic theory. Fields investigated here are mixing
properties, spectral theory, joinings. We also compare Poisson suspensions to
the apparently similar looking Gaussian dynamical systems.Comment: 18 page
A Lower Bound for Sampling Disjoint Sets
Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}
M\"obius disjointness for models of an ergodic system and beyond
Given a topological dynamical system and an arithmetic function
, we study the strong MOMO
property (relatively to ) which is a strong version of
-disjointness with all observable sequences in . It is
proved that, given an ergodic measure-preserving system
, the strong MOMO property (relatively to
) of a uniquely ergodic model of yields all other
uniquely ergodic models of to be -disjoint. It follows that
all uniquely ergodic models of: ergodic unipotent diffeomorphisms on
nilmanifolds, discrete spectrum automorphisms, systems given by some
substitutions of constant length (including the classical Thue-Morse and
Rudin-Shapiro substitutions), systems determined by Kakutani sequences are
M\"obius (and Liouville) disjoint. The validity of Sarnak's conjecture implies
the strong MOMO property relatively to in all zero entropy
systems, in particular, it makes -disjointness uniform. The
absence of strong MOMO property in positive entropy systems is discussed and,
it is proved that, under the Chowla conjecture, a topological system has the
strong MOMO property relatively to the Liouville function if and only if its
topological entropy is zero.Comment: 35 page
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