35 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
The Communication Complexity of the Hamming Distance Problem
We investigate the randomized and quantum communication complexity of the
Hamming Distance problem, which is to determine if the Hamming distance between
two n-bit strings is no less than a threshold d. We prove a quantum lower bound
of \Omega(d) qubits in the general interactive model with shared prior
entanglement. We also construct a classical protocol of O(d \log d) bits in the
restricted Simultaneous Message Passing model, improving previous protocols of
O(d^2) bits (A. C.-C. Yao, Proceedings of the Thirty-Fifth Annual ACM Symposium
on Theory of Computing, pp. 77-81, 2003), and O(d\log n) bits (D. Gavinsky, J.
Kempe, and R. de Wolf, quant-ph/0411051, 2004).Comment: 8 pages, v3, updated reference. to appear in Information Processing
Letters, 200
Efficient quantum protocols for XOR functions
We show that for any Boolean function f on {0,1}^n, the bounded-error quantum
communication complexity of XOR functions satisfies that
, where d is the F2-degree of f, and
.
This implies that the previous lower bound by Lee and Shraibman \cite{LS09} is tight
for f with low F2-degree. The result also confirms the quantum version of the
Log-rank Conjecture for low-degree XOR functions. In addition, we show that the
exact quantum communication complexity satisfies , where is the number of nonzero Fourier coefficients of
f. This matches the previous lower bound
by Buhrman and de Wolf \cite{BdW01} for low-degree XOR functions.Comment: 11 pages, no figur
A Note on Shared Randomness and Shared Entanglement in Communication
We consider several models of 1-round classical and quantum communication,
some of these models have not been defined before. We "almost separate" the
models of simultaneous quantum message passing with shared entanglement and the
model of simultaneous quantum message passing with shared randomness. We define
a relation which can be efficiently exactly solved in the first model but
cannot be solved efficiently, either exactly or in 0-error setup in the second
model. In fact, our relation is exactly solvable even in a more restricted
model of simultaneous classical message passing with shared entanglement.
As our second contribution we strengthen a result by Yao that a "very short"
protocol from the model of simultaneous classical message passing with shared
randomness can be simulated in the model of simultaneous quantum message
passing: for a boolean function f, QII(f) \in exp(O(RIIp(f))) log n.
We show a similar result for protocols from a (stronger) model of 1-way
classical message passing with shared randomness: QII(f) \in exp(O(RIp(f))) log
n.
We demonstrate a problem whose efficient solution in the model of
simultaneous quantum message passing follows from our result but not from
Yao's.Comment: Stronger separation, minor changes and fixe
Classical and quantum fingerprinting with shared randomness and one-sided error
Within the simultaneous message passing model of communication complexity,
under a public-coin assumption, we derive the minimum achievable worst-case
error probability of a classical fingerprinting protocol with one-sided error.
We then present entanglement-assisted quantum fingerprinting protocols
attaining worst-case error probabilities that breach this bound.Comment: 10 pages, 1 figur
Unbounded-Error Classical and Quantum Communication Complexity
Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86},
the unbounded-error classical communication complexity of a Boolean function
has been studied based on the arrangement of points and hyperplanes. Recently,
\cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum}
communication complexity in the {\em one-way communication} model can also be
investigated using the arrangement, and showed that it is exactly (without a
difference of even one qubit) half of the classical one-way communication
complexity. In this paper, we extend the arrangement argument to the {\em
two-way} and {\em simultaneous message passing} (SMP) models. As a result, we
show similarly tight bounds of the unbounded-error two-way/one-way/SMP
quantum/classical communication complexities for {\em any} partial/total
Boolean function, implying that all of them are equivalent up to a
multiplicative constant of four. Moreover, the arrangement argument is also
used to show that the gap between {\em weakly} unbounded-error quantum and
classical communication complexities is at most a factor of three.Comment: 11 pages. To appear at Proc. ISAAC 200