27,485 research outputs found
On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model
We study the communication complexity of linear algebraic problems over
finite fields in the multi-player message passing model, proving a number of
tight lower bounds. Specifically, for a matrix which is distributed among a
number of players, we consider the problem of determining its rank, of
computing entries in its inverse, and of solving linear equations. We also
consider related problems such as computing the generalized inner product of
vectors held on different servers. We give a general framework for reducing
these multi-player problems to their two-player counterparts, showing that the
randomized -player communication complexity of these problems is at least
times the randomized two-player communication complexity. Provided the
problem has a certain amount of algebraic symmetry, which we formally define,
we can show the hardest input distribution is a symmetric distribution, and
therefore apply a recent multi-player lower bound technique of Phillips et al.
Further, we give new two-player lower bounds for a number of these problems. In
particular, our optimal lower bound for the two-player version of the matrix
rank problem resolves an open question of Sun and Wang.
A common feature of our lower bounds is that they apply even to the special
"threshold promise" versions of these problems, wherein the underlying
quantity, e.g., rank, is promised to be one of just two values, one on each
side of some critical threshold. These kinds of promise problems are
commonplace in the literature on data streaming as sources of hardness for
reductions giving space lower bounds
Quantum Information Complexity and Amortized Communication
We define a new notion of information cost for quantum protocols, and a
corresponding notion of quantum information complexity for bipartite quantum
channels, and then investigate the properties of such quantities. These are the
fully quantum generalizations of the analogous quantities for bipartite
classical functions that have found many applications recently, in particular
for proving communication complexity lower bounds. Our definition is strongly
tied to the quantum state redistribution task.
Previous attempts have been made to define such a quantity for quantum
protocols, with particular applications in mind; our notion differs from these
in many respects. First, it directly provides a lower bound on the quantum
communication cost, independent of the number of rounds of the underlying
protocol. Secondly, we provide an operational interpretation for quantum
information complexity: we show that it is exactly equal to the amortized
quantum communication complexity of a bipartite channel on a given state. This
generalizes a result of Braverman and Rao to quantum protocols, and even
strengthens the classical result in a bounded round scenario. Also, this
provides an analogue of the Schumacher source compression theorem for
interactive quantum protocols, and answers a question raised by Braverman.
We also discuss some potential applications to quantum communication
complexity lower bounds by specializing our definition for classical functions
and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new
evidence suggesting that the bounded round quantum communication complexity of
the disjointness function is \Omega (n/M + M), for M-message protocols. This
would match the best known upper bound.Comment: v1, 38 pages, 1 figur
Improved Quantum Communication Complexity Bounds for Disjointness and Equality
We prove new bounds on the quantum communication complexity of the
disjointness and equality problems. For the case of exact and non-deterministic
protocols we show that these complexities are all equal to n+1, the previous
best lower bound being n/2. We show this by improving a general bound for
non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^*
n})-qubit bounded-error protocol for disjointness, modifying and improving the
earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an
Omega(sqrt{n}) lower bound for a large class of protocols that includes the
BCW-protocol as well as our new protocol.Comment: 11 pages LaTe
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
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
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
Towards Optimal Moment Estimation in Streaming and Distributed Models
One of the oldest problems in the data stream model is to approximate the p-th moment ||X||_p^p = sum_{i=1}^n X_i^p of an underlying non-negative vector X in R^n, which is presented as a sequence of poly(n) updates to its coordinates. Of particular interest is when p in (0,2]. Although a tight space bound of Theta(epsilon^-2 log n) bits is known for this problem when both positive and negative updates are allowed, surprisingly there is still a gap in the space complexity of this problem when all updates are positive. Specifically, the upper bound is O(epsilon^-2 log n) bits, while the lower bound is only Omega(epsilon^-2 + log n) bits. Recently, an upper bound of O~(epsilon^-2 + log n) bits was obtained under the assumption that the updates arrive in a random order.
We show that for p in (0, 1], the random order assumption is not needed. Namely, we give an upper bound for worst-case streams of O~(epsilon^-2 + log n) bits for estimating |X |_p^p. Our techniques also give new upper bounds for estimating the empirical entropy in a stream. On the other hand, we show that for p in (1,2], in the natural coordinator and blackboard distributed communication topologies, there is an O~(epsilon^-2) bit max-communication upper bound based on a randomized rounding scheme. Our protocols also give rise to protocols for heavy hitters and approximate matrix product. We generalize our results to arbitrary communication topologies G, obtaining an O~(epsilon^2 log d) max-communication upper bound, where d is the diameter of G. Interestingly, our upper bound rules out natural communication complexity-based approaches for proving an Omega(epsilon^-2 log n) bit lower bound for p in (1,2] for streaming algorithms. In particular, any such lower bound must come from a topology with large diameter
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
- …