1,811 research outputs found
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
Lower bounds for adaptive linearity tests
Linearity tests are randomized algorithms which have oracle access to the
truth table of some function f, and are supposed to distinguish between linear
functions and functions which are far from linear. Linearity tests were first
introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP
theorem, among other applications. The quality of a linearity test is described
by its correctness c - the probability it accepts linear functions, its
soundness s - the probability it accepts functions far from linear, and its
query complexity q - the number of queries it makes. Linearity tests were
studied in order to decrease the soundness of linearity tests, while keeping
the query complexity small (for one reason, to improve PCP constructions).
Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the
Complete Graph Test, and prove that no Hyper Graph Test can perform better than
the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove,
among other results, that no non-adaptive linearity test can perform better
than the Complete Graph Test. Their proof uses the algebraic machinery of the
Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to
generalize this lower bound also to adaptive linearity tests. We also prove the
same optimal lower bound for adaptive linearity test, but our proof technique
is arguably simpler and more direct than the one used in (Samorodnitsky and
Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the
behavior of linearity tests on quadratic functions. However, instead of
analyzing the Gowers Norm of certain functions, we provide a more direct
combinatorial proof, studying the behavior of linearity tests on random
quadratic functions..
A Hypergraph Dictatorship Test with Perfect Completeness
A hypergraph dictatorship test is first introduced by Samorodnitsky and
Trevisan and serves as a key component in their unique games based \PCP
construction. Such a test has oracle access to a collection of functions and
determines whether all the functions are the same dictatorship, or all their
low degree influences are Their test makes queries and has
amortized query complexity but has an inherent loss of
perfect completeness. In this paper we give an adaptive hypergraph dictatorship
test that achieves both perfect completeness and amortized query complexity
.Comment: Some minor correction
Constant-Soundness Interactive Proofs for Local Hamiltonians
We give a quantum multiprover interactive proof
system for the local Hamiltonian problem in which there is a constant number of
provers, questions are classical of length polynomial in the number of qubits,
and answers are of constant length. The main novelty of our protocol is that
the gap between completeness and soundness is directly proportional to the
promise gap on the (normalized) ground state energy of the Hamiltonian. This
result can be interpreted as a concrete step towards a quantum PCP theorem
giving entangled-prover interactive proof systems for QMA-complete problems.
The key ingredient is a quantum version of the classical linearity test of
Blum, Luby, and Rubinfeld, where the function is
replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C),
the set of -dimensional Hermitian matrices that square to identity. The test
enforces that (i) each function is exactly linear,
\Xlin(a)\Xlin(b)=\Xlin(a+b) and \Zlin(a) \Zlin(b)=\Zlin(a+b), and (ii) the
two functions are approximately complementary, \Xlin(a)\Zlin(b)\approx
(-1)^{a\cdot b} \Zlin(b)\Xlin(a).Comment: 33 page
Robust self-testing of many-qubit states
We introduce a simple two-player test which certifies that the players apply
tensor products of Pauli and observables on the tensor
product of EPR pairs. The test has constant robustness: any strategy
achieving success probability within an additive of the optimal
must be -close, in the appropriate distance
measure, to the honest -qubit strategy. The test involves -bit questions
and -bit answers. The key technical ingredient is a quantum version of the
classical linearity test of Blum, Luby, and Rubinfeld.
As applications of our result we give (i) the first robust self-test for
EPR pairs; (ii) a quantum multiprover interactive proof system for the local
Hamiltonian problem with a constant number of provers and classical questions
and answers, and a constant completeness-soundness gap independent of system
size; (iii) a robust protocol for delegated quantum computation.Comment: 36 pages. Improves upon and supersedes our earlier submission
arXiv:1512.0209
Load curve data cleansing and imputation via sparsity and low rank
The smart grid vision is to build an intelligent power network with an
unprecedented level of situational awareness and controllability over its
services and infrastructure. This paper advocates statistical inference methods
to robustify power monitoring tasks against the outlier effects owing to faulty
readings and malicious attacks, as well as against missing data due to privacy
concerns and communication errors. In this context, a novel load cleansing and
imputation scheme is developed leveraging the low intrinsic-dimensionality of
spatiotemporal load profiles and the sparse nature of "bad data.'' A robust
estimator based on principal components pursuit (PCP) is adopted, which effects
a twofold sparsity-promoting regularization through an -norm of the
outliers, and the nuclear norm of the nominal load profiles. Upon recasting the
non-separable nuclear norm into a form amenable to decentralized optimization,
a distributed (D-) PCP algorithm is developed to carry out the imputation and
cleansing tasks using networked devices comprising the so-termed advanced
metering infrastructure. If D-PCP converges and a qualification inequality is
satisfied, the novel distributed estimator provably attains the performance of
its centralized PCP counterpart, which has access to all networkwide data.
Computer simulations and tests with real load curve data corroborate the
convergence and effectiveness of the novel D-PCP algorithm.Comment: 8 figures, submitted to IEEE Transactions on Smart Grid - Special
issue on "Optimization methods and algorithms applied to smart grid
Quantum Locally Testable Codes
We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a
definition together with a simplification, denoted sLTCs, for the special case
of stabilizer codes, together with some basic results using those definitions.
The most crucial parameter of such codes is their soundness, ,
namely, the probability that a randomly chosen constraint is violated as a
function of the distance of a word from the code (, the relative
distance from the code, is called the proximity). We then proceed to study
limitations on qLTCs. In our first main result we prove a surprising,
inherently quantum, property of sLTCs: for small values of proximity, the
better the small-set expansion of the interaction graph of the constraints, the
less sound the qLTC becomes. This phenomenon, which can be attributed to
monogamy of entanglement, stands in sharp contrast to the classical setting.
The complementary, more intuitive, result also holds: an upper bound on the
soundness when the code is defined on poor small-set expanders (a bound which
turns out to be far more difficult to show in the quantum case). Together we
arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any
graph, which does not hold in the classical case. Many open questions are
raised regarding what possible parameters are achievable for qLTCs. In the
appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and
point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with
related parameters, carries over to the sLTCs. This creates a first link
between qLTCs and quantum PCPs.Comment: Some of the results presented here appeared in an initial form in our
quant-ph submission arXiv:1301.3407. This is a much extended and improved
version. 30 pages, no figure
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