335 research outputs found
Local tests of global entanglement and a counterexample to the generalized area law
We introduce a technique for applying quantum expanders in a distributed
fashion, and use it to solve two basic questions: testing whether a bipartite
quantum state shared by two parties is the maximally entangled state and
disproving a generalized area law. In the process these two questions which
appear completely unrelated turn out to be two sides of the same coin.
Strikingly in both cases a constant amount of resources are used to verify a
global property.Comment: 21 pages, to appear FOCS 201
Variations on Classical and Quantum Extractors
Many constructions of randomness extractors are known to work in the presence
of quantum side information, but there also exist extractors which do not
[Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors
with a bound on the second largest eigenvalue
are quantum-proof. We then discuss fully
quantum extractors and call constructions that also work in the presence of
quantum correlations decoupling. As in the classical case we show that spectral
extractors are decoupling. The drawback of classical and quantum spectral
extractors is that they always have a long seed, whereas there exist classical
extractors with exponentially smaller seed size. For the quantum case, we show
that there exists an extractor with extremely short seed size
, where denotes the quality of the
randomness. In contrast to the classical case this is independent of the input
size and min-entropy and matches the simple lower bound
.Comment: 7 pages, slightly enhanced IEEE ISIT submission including all the
proof
07411 Abstracts Collection -- Algebraic Methods in Computational Complexity
From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
The parallel approximability of a subclass of quadratic programming
In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version
Exponential Quantum Speed-ups are Generic
A central problem in quantum computation is to understand which quantum
circuits are useful for exponential speed-ups over classical computation. We
address this question in the setting of query complexity and show that for
almost any sufficiently long quantum circuit one can construct a black-box
problem which is solved by the circuit with a constant number of quantum
queries, but which requires exponentially many classical queries, even if the
classical machine has the ability to postselect.
We prove the result in two steps. In the first, we show that almost any
element of an approximate unitary 3-design is useful to solve a certain
black-box problem efficiently. The problem is based on a recent oracle
construction of Aaronson and gives an exponential separation between quantum
and classical bounded-error with postselection query complexities.
In the second step, which may be of independent interest, we prove that
linear-sized random quantum circuits give an approximate unitary 3-design. The
key ingredient in the proof is a technique from quantum many-body theory to
lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction
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