1,545 research outputs found
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
New Bounds for the Garden-Hose Model
We show new results about the garden-hose model. Our main results include
improved lower bounds based on non-deterministic communication complexity
(leading to the previously unknown bounds for Inner Product mod 2
and Disjointness), as well as an upper bound for the
Distributed Majority function (previously conjectured to have quadratic
complexity). We show an efficient simulation of formulae made of AND, OR, XOR
gates in the garden-hose model, which implies that lower bounds on the
garden-hose complexity of the order will be
hard to obtain for explicit functions. Furthermore we study a time-bounded
variant of the model, in which even modest savings in time can lead to
exponential lower bounds on the size of garden-hose protocols.Comment: In FSTTCS 201
Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
Let be a fixed graph on vertices. Let iff the input
graph on vertices contains as a (not necessarily induced) subgraph.
Let denote the cardinality of a maximum independent set of . In
this paper we show:
where
denotes the quantum query complexity of .
As a consequence we obtain a lower bounds for in terms of several
other parameters of such as the average degree, minimum vertex cover,
chromatic number, and the critical probability.
We also use the above bound to show that for any
, improving on the previously best known bound of . Until
very recently, it was believed that the quantum query complexity is at least
square root of the randomized one. Our bound for
matches the square root of the current best known bound for the randomized
query complexity of , which is due to Gr\"oger.
Interestingly, the randomized bound of for
still remains open.
We also study the Subgraph Homomorphism Problem, denoted by , and
show that .
Finally we extend our results to the -uniform hypergraphs. In particular,
we show an bound for quantum query complexity of the Subgraph
Isomorphism, improving on the previously known bound. For the
Subgraph Homomorphism, we obtain an bound for the same.Comment: 16 pages, 2 figure
Random tree recursions: which fixed points correspond to tangible sets of trees?
Let be the set of rooted trees containing an infinite binary
subtree starting at the root. This set satisfies the metaproperty that a tree
belongs to it if and only if its root has children and such that the
subtrees rooted at and belong to it. Let be the probability that a
Galton-Watson tree falls in . The metaproperty makes satisfy a
fixed-point equation, which can have multiple solutions. One of these solutions
is , but what is the meaning of the others? In particular, are they
probabilities of the Galton-Watson tree falling into other sets satisfying the
same metaproperty? We create a framework for posing questions of this sort, and
we classify solutions to fixed-point equations according to whether they admit
probabilistic interpretations. Our proofs use spine decompositions of
Galton-Watson trees and the analysis of Boolean functions.Comment: 41 pages; small changes in response to referees' comments; to appear
in Random Structures & Algorithm
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