484 research outputs found
On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost
of a factor of two in the total communication, Alice could just as well have
given the two possible replies without listening and have Bob select which
applies to him. Motivated by a conjecture stating that this form of "round
elimination" is impossible in exact quantum communication complexity, we study
the orthogonal rank and a symmetric variant thereof for a certain family of
Cayley graphs. The orthogonal rank of a graph is the smallest number for
which one can label each vertex with a nonzero -dimensional complex vector
such that adjacent vertices receive orthogonal vectors.
We show an exp lower bound on the orthogonal rank of the graph on
in which two strings are adjacent if they have Hamming distance at
least . In combination with previous work, this implies an affirmative
answer to the above conjecture.Comment: 13 page
Failure of the trilinear operator space Grothendieck theorem
We give a counterexample to a trilinear version of the operator space
Grothendieck theorem. In particular, we show that for trilinear forms on
, the ratio of the symmetrized completely bounded norm and the
jointly completely bounded norm is in general unbounded, answering a question
of Pisier. The proof is based on a non-commutative version of the generalized
von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi
Revisiting the Sanders-Freiman-Ruzsa Theorem in and its Application to Non-malleable Codes
Non-malleable codes (NMCs) protect sensitive data against degrees of
corruption that prohibit error detection, ensuring instead that a corrupted
codeword decodes correctly or to something that bears little relation to the
original message. The split-state model, in which codewords consist of two
blocks, considers adversaries who tamper with either block arbitrarily but
independently of the other. The simplest construction in this model, due to
Aggarwal, Dodis, and Lovett (STOC'14), was shown to give NMCs sending k-bit
messages to -bit codewords. It is conjectured, however, that the
construction allows linear-length codewords. Towards resolving this conjecture,
we show that the construction allows for code-length . This is achieved
by analysing a special case of Sanders's Bogolyubov-Ruzsa theorem for general
Abelian groups. Closely following the excellent exposition of this result for
the group by Lovett, we expose its dependence on for the
group , where is a prime
Gaussian width bounds with applications to arithmetic progressions in random settings
Motivated by problems on random differences in Szemer\'{e}di's theorem and on
large deviations for arithmetic progressions in random sets, we prove upper
bounds on the Gaussian width of point sets that are formed by the image of the
-dimensional Boolean hypercube under a mapping
, where each coordinate is a constant-degree
multilinear polynomial with 0-1 coefficients. We show the following
applications of our bounds. Let be the random
subset of containing each element independently with
probability .
A set is -intersective if
any dense subset of contains a proper -term
arithmetic progression with common difference in . Our main result implies
that is -intersective with probability provided for . This gives a polynomial improvement for all
of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and
reproves more directly the same improvement shown recently by the authors and
Dvir.
Let be the number of -term arithmetic progressions in
and consider the large deviation rate
. We give quadratic
improvements of the best-known range of for which a highly precise estimate
of due to Bhattacharya, Ganguly, Shao and Zhao is valid for
all odd .
We also discuss connections with error correcting codes (locally decodable
codes) and the Banach-space notion of type for injective tensor products of
-spaces.Comment: 18 pages, some typos fixe
Effects of changing mosquito host searching behaviour on the cost effectiveness of a mass distribution of long-lasting, insecticidal nets : a modelling study
The effectiveness of long-lasting, insecticidal nets (LLINs) in preventing malaria is threatened by the changing biting behaviour of mosquitoes, from nocturnal and endophagic to crepuscular and exophagic, and by their increasing resistance to insecticides.; Using epidemiological stochastic simulation models, we studied the impact of a mass LLIN distribution on Plasmodium falciparum malaria. Specifically, we looked at impact in terms of episodes prevented during the effective life of the batch and in terms of net health benefits (NHB) expressed in disability adjusted life years (DALYs) averted, depending on biting behaviour, resistance (as measured in experimental hut studies), and on pre-intervention transmission levels.; Results were very sensitive to assumptions about the probabilistic nature of host searching behaviour. With a shift towards crepuscular biting, under the assumption that individual mosquitoes repeat their behaviour each gonotrophic cycle, LLIN effectiveness was far less than when individual mosquitoes were assumed to vary their behaviour between gonotrophic cycles. LLIN effectiveness was equally sensitive to variations in host-searching behaviour (if repeated) and to variations in resistance. LLIN effectiveness was most sensitive to pre-intervention transmission level, with LLINs being least effective at both very low and very high transmission levels, and most effective at around four infectious bites per adult per year. A single LLIN distribution round remained cost effective, except in transmission settings with a pre-intervention inoculation rate of over 128 bites per year and with resistant mosquitoes that displayed a high proportion (over 40%) of determined crepuscular host searching, where some model variants showed negative NHB. Shifts towards crepuscular host searching behaviour can be as important in reducing LLIN effectiveness and cost effectiveness as resistance to pyrethroids. As resistance to insecticides is likely to slow dow the development of behavioural resistance and vice versa, the two types of resistance are unlikely to occur within the same mosquito population. LLINs are likely cost effective interventions against malaria, even in areas with strong resistance to pyrethroids or where a large proportion of host-mosquito contact occurs during times when LLIN users are not under their nets
Locally Decodable Quantum Codes
We study a quantum analogue of locally decodable error-correcting codes. A
q-query locally decodable quantum code encodes n classical bits in an m-qubit
state, in such a way that each of the encoded bits can be recovered with high
probability by a measurement on at most q qubits of the quantum code, even if a
constant fraction of its qubits have been corrupted adversarially. We show that
such a quantum code can be transformed into a classical q-query locally
decodable code of the same length that can be decoded well on average (albeit
with smaller success probability and noise-tolerance). This shows, roughly
speaking, that q-query quantum codes are not significantly better than q-query
classical codes, at least for constant or small q.Comment: 15 pages, LaTe
A generalized Grothendieck inequality and entanglement in XOR games
Suppose Alice and Bob make local two-outcome measurements on a shared
entangled state. For any d, we show that there are correlations that can only
be reproduced if the local dimension is at least d. This resolves a conjecture
of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and establishes that the
amount of entanglement required to maximally violate a Bell inequality must
depend on the number of measurement settings, not just the number of
measurement outcomes. We prove this result by establishing the first lower
bounds on a new generalization of Grothendieck's constant.Comment: Version submitted to QIP on 10-20-08. See also arxiv:0812.1572 for
related results, obtained independentl
Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for
decoding of a single message bit using a small number of queries to a corrupted
encoding. Despite decades of study, the optimal trade-off between query
complexity and codeword length is far from understood. In this work, we give a
new characterization of LDCs using distributions over Boolean functions whose
expectation is hard to approximate (in~~norm) with a small number of
samples. We coin the term `outlaw distributions' for such distributions since
they `defy' the Law of Large Numbers. We show that the existence of outlaw
distributions over sufficiently `smooth' functions implies the existence of
constant query LDCs and vice versa. We give several candidates for outlaw
distributions over smooth functions coming from finite field incidence
geometry, additive combinatorics and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only
required to work on average over a random message and a random message index
can be turned into true LDCs at the cost of only constant factors in the
parameters.Comment: A preliminary version of this paper appeared in the proceedings of
ITCS 201
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
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