206 research outputs found
Hardness of approximation for quantum problems
The polynomial hierarchy plays a central role in classical complexity theory.
Here, we define a quantum generalization of the polynomial hierarchy, and
initiate its study. We show that not only are there natural complete problems
for the second level of this quantum hierarchy, but that these problems are in
fact hard to approximate. Using these techniques, we also obtain hardness of
approximation for the class QCMA. Our approach is based on the use of
dispersers, and is inspired by the classical results of Umans regarding
hardness of approximation for the second level of the classical polynomial
hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of
approximation for include, among others, a quantum version of the Succinct Set
Cover problem, and a variant of the local Hamiltonian problem with hybrid
classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the
39th International Colloquium on Automata, Languages and Programming (ICALP),
pages 387-398, Springer, 201
Quantum Property Testing
A language L has a property tester if there exists a probabilistic algorithm
that given an input x only asks a small number of bits of x and distinguishes
the cases as to whether x is in L and x has large Hamming distance from all y
in L. We define a similar notion of quantum property testing and show that
there exist languages with quantum property testers but no good classical
testers. We also show there exist languages which require a large number of
queries even for quantumly testing
Two queries
AbstractWe consider the question whether two queries SAT are as powerful as one query. We show that if PNP[1]=PNP[2] then: Locally either NP=coNP or NP has polynomial-size circuits; PNP=PNP[1]; Σp2⊆Πp2/1; Σp2=UPNP[1]∩RPNP[1]; PH=BPPNP[1]. Moreover, we extend the work of Hemaspaandra, Hemaspaandra, and Hempel to show that if PΣp2[1]=PΣp2[2] then Σp2=Πp2. We also give a relativized world, where PNP[1]=PNP[2], but NP≠coNP
Counting dependent and independent strings
The paper gives estimations for the sizes of the the following sets: (1) the
set of strings that have a given dependency with a fixed string, (2) the set of
strings that are pairwise \alpha independent, (3) the set of strings that are
mutually \alpha independent. The relevant definitions are as follows: C(x) is
the Kolmogorov complexity of the string x. A string y has \alpha -dependency
with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x_1, \ldots,
x_t} is pairwise \alpha-independent if for all i different from j, C(x_i) -
C(x_i | x_j) \leq \alpha. A tuple of strings (x_1, \ldots, x_t) is mutually
\alpha-independent if C(x_{\pi(1)} \ldots x_{\pi(t)}) \geq C(x_1) + \ldots +
C(x_t) - \alpha, for every permutation \pi of [t]
Impossibility of independence amplification in Kolmogorov complexity theory
The paper studies randomness extraction from sources with bounded
independence and the issue of independence amplification of sources, using the
framework of Kolmogorov complexity. The dependency of strings and is
, where
denotes the Kolmogorov complexity. It is shown that there exists a
computable Kolmogorov extractor such that, for any two -bit strings with
complexity and dependency , it outputs a string of length
with complexity conditioned by any one of the input
strings. It is proven that the above are the optimal parameters a Kolmogorov
extractor can achieve. It is shown that independence amplification cannot be
effectively realized. Specifically, if (after excluding a trivial case) there
exist computable functions and such that for all -bit strings and with , then
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