108 research outputs found
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
Strengths and Weaknesses of Quantum Fingerprinting
We study the power of quantum fingerprints in the simultaneous message
passing (SMP) setting of communication complexity. Yao recently showed how to
simulate, with exponential overhead, classical shared-randomness SMP protocols
by means of quantum SMP protocols without shared randomness
(-protocols). Our first result is to extend Yao's simulation to
the strongest possible model: every many-round quantum protocol with unlimited
shared entanglement can be simulated, with exponential overhead, by
-protocols. We apply our technique to obtain an efficient
-protocol for a function which cannot be efficiently solved
through more restricted simulations. Second, we tightly characterize the power
of the quantum fingerprinting technique by making a connection to arrangements
of homogeneous halfspaces with maximal margin. These arrangements have been
well studied in computational learning theory, and we use some strong results
obtained in this area to exhibit weaknesses of quantum fingerprinting. In
particular, this implies that for almost all functions, quantum fingerprinting
protocols are exponentially worse than classical deterministic SMP protocols.Comment: 13 pages, no figures, to appear in CCC'0
Exponential Separation of Quantum and Classical One-Way Communication Complexity for a Boolean Function
We give an exponential separation between one-way quantum and classical
communication complexity for a Boolean function. Earlier such a separation was
known only for a relation. A very similar result was obtained earlier but
independently by Kerenidis and Raz [KR06]. Our version of the result gives an
example in the bounded storage model of cryptography, where the key is secure
if the adversary has a certain amount of classical storage, but is completely
insecure if he has a similar amount of quantum storage.Comment: 8 pages, no figure
Generalized Performance of Concatenated Quantum Codes -- A Dynamical Systems Approach
We apply a dynamical systems approach to concatenation of quantum error
correcting codes, extending and generalizing the results of Rahn et al. [1] to
both diagonal and nondiagonal channels. Our point of view is global: instead of
focusing on particular types of noise channels, we study the geometry of the
coding map as a discrete-time dynamical system on the entire space of noise
channels. In the case of diagonal channels, we show that any code with distance
at least three corrects (in the infinite concatenation limit) an open set of
errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise
characterization of that set. We show how to incorporate noise in the gates,
thus completing the framework. We derive some general bounds for noise
channels, which allows us to analyze several codes in detail.Comment: 12 pages two-column format, no figures, slightly revised versio
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