19,842 research outputs found
New bounds on classical and quantum one-way communication complexity
In this paper we provide new bounds on classical and quantum distributional
communication complexity in the two-party, one-way model of communication. In
the classical model, our bound extends the well known upper bound of Kremer,
Nisan and Ron to include non-product distributions. We show that for a boolean
function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and
epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where
D_{epsilon}^{1, mu}(f) represents the one-way distributional communication
complexity of f with error at most epsilon under mu; vc(f) represents the
Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual
information, under mu, between the random inputs of the two parties. For a
non-boolean function f:X x Y ->[k], we show a similar upper bound on
D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f' =
f/k. In the quantum one-way model we provide a lower bound on the
distributional communication complexity, under product distributions, of a
function f, in terms the well studied complexity measure of f referred to as
the rectangle bound or the corruption bound of f . We show for a non-boolean
total function f : X x Y -> Z and a product distribution mu on XxY,
Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where
Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional
communication complexity of f with error at most epsilon^3/8 under mu and rec_
epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at
most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z
U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x
15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).Comment: ver 1, 19 page
Classical and quantum partition bound and detector inefficiency
We study randomized and quantum efficiency lower bounds in communication
complexity. These arise from the study of zero-communication protocols in which
players are allowed to abort. Our scenario is inspired by the physics setup of
Bell experiments, where two players share a predefined entangled state but are
not allowed to communicate. Each is given a measurement as input, which they
perform on their share of the system. The outcomes of the measurements should
follow a distribution predicted by quantum mechanics; however, in practice, the
detectors may fail to produce an output in some of the runs. The efficiency of
the experiment is the probability that the experiment succeeds (neither of the
detectors fails).
When the players share a quantum state, this gives rise to a new bound on
quantum communication complexity (eff*) that subsumes the factorization norm.
When players share randomness instead of a quantum state, the efficiency bound
(eff), coincides with the partition bound of Jain and Klauck. This is one of
the strongest lower bounds known for randomized communication complexity, which
subsumes all the known combinatorial and algebraic methods including the
rectangle (corruption) bound, the factorization norm, and discrepancy.
The lower bound is formulated as a convex optimization problem. In practice,
the dual form is more feasible to use, and we show that it amounts to
constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for
eff*). We give an example of a quantum distribution where the violation can be
exponentially bigger than the previously studied class of normalized Bell
inequalities.
For one-way communication, we show that the quantum one-way partition bound
is tight for classical communication with shared entanglement up to arbitrarily
small error.Comment: 21 pages, extended versio
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
Non-locality and Communication Complexity
Quantum information processing is the emerging field that defines and
realizes computing devices that make use of quantum mechanical principles, like
the superposition principle, entanglement, and interference. In this review we
study the information counterpart of computing. The abstract form of the
distributed computing setting is called communication complexity. It studies
the amount of information, in terms of bits or in our case qubits, that two
spatially separated computing devices need to exchange in order to perform some
computational task. Surprisingly, quantum mechanics can be used to obtain
dramatic advantages for such tasks.
We review the area of quantum communication complexity, and show how it
connects the foundational physics questions regarding non-locality with those
of communication complexity studied in theoretical computer science. The first
examples exhibiting the advantage of the use of qubits in distributed
information-processing tasks were based on non-locality tests. However, by now
the field has produced strong and interesting quantum protocols and algorithms
of its own that demonstrate that entanglement, although it cannot be used to
replace communication, can be used to reduce the communication exponentially.
In turn, these new advances yield a new outlook on the foundations of physics,
and could even yield new proposals for experiments that test the foundations of
physics.Comment: Survey paper, 63 pages LaTeX. A reformatted version will appear in
Reviews of Modern Physic
Quantum Communication Cannot Simulate a Public Coin
We study the simultaneous message passing model of communication complexity.
Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently
showed that a large class of efficient classical public-coin protocols can be
turned into efficient quantum protocols without public coin. This raises the
question whether this can be done always, i.e. whether quantum communication
can always replace a public coin in the SMP model. We answer this question in
the negative, exhibiting a communication problem where classical communication
with public coin is exponentially more efficient than quantum communication.
Together with a separation in the other direction due to Bar-Yossef et al.,
this shows that the quantum SMP model is incomparable with the classical
public-coin SMP model.
In addition we give a characterization of the power of quantum fingerprinting
by means of a connection to geometrical tools from machine learning, a
quadratic improvement of Yao's simulation, and a nearly tight analysis of the
Hamming distance problem from Yao's paper.Comment: 12 pages LaTe
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