2,546 research outputs found
On Local Testability in the Non-Signaling Setting
Non-signaling strategies are a generalization of quantum strategies that have been studied in physics for decades, and have recently found applications in theoretical computer science. These applications motivate the study of local-to-global phenomena for non-signaling functions.
We prove that low-degree testing in the non-signaling setting is possible, assuming that the locality of the non-signaling function exceeds a threshold. We additionally show that if the locality is below the threshold then the test fails spectacularly, in that there exists a non-signaling function which passes the test with probability 1 and yet is maximally far from being low-degree.
Along the way, we present general results about the local testability of linear codes in the non-signaling setting. These include formulating natural definitions that capture the condition that a non-signaling function "belongs" to a given code, and characterizing the sets of local constraints that imply membership in the code. We prove these results by formulating a logical inference system for linear constraints on non-signaling functions that is complete and sound
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
High rate locally-correctable and locally-testable codes with sub-polynomial query complexity
In this work, we construct the first locally-correctable codes (LCCs), and
locally-testable codes (LTCs) with constant rate, constant relative distance,
and sub-polynomial query complexity. Specifically, we show that there exist
binary LCCs and LTCs with block length , constant rate (which can even be
taken arbitrarily close to 1), constant relative distance, and query complexity
. Previously such codes were known to exist
only with query complexity (for constant ), and
there were several, quite different, constructions known.
Our codes are based on a general distance-amplification method of Alon and
Luby~\cite{AL96_codes}. We show that this method interacts well with local
correctors and testers, and obtain our main results by applying it to suitably
constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant
relative distance}.
Along the way, we also construct LCCs and LTCs over large alphabets, with the
same query complexity , which additionally have
the property of approaching the Singleton bound: they have almost the
best-possible relationship between their rate and distance. This has the
surprising consequence that asking for a large alphabet error-correcting code
to further be an LCC or LTC with query
complexity does not require any sacrifice in terms of rate and distance! Such a
result was previously not known for any query complexity.
Our results on LCCs also immediately give locally-decodable codes (LDCs) with
the same parameters
High Dimensional Expanders and Property Testing
We show that the high dimensional expansion property as defined by Gromov,
Linial and Meshulam, for simplicial complexes is a form of testability. Namely,
a simplicial complex is a high dimensional expander iff a suitable property is
testable. Using this connection, we derive several testability results
Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness
An error-correcting code C subseteq F^n is called (q,epsilon)-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords x not in C with probability at least epsilon * delta(x,C), where delta(x,C) denotes the relative Hamming distance between the word x and the code C. The parameter q is called the query complexity and the parameter epsilon is called soundness.
Goldreich and Sudan (J.ACM 2006) asked about the existence of strong LTCs with constant query complexity, constant relative distance, constant soundness and inverse polylogarithmic rate. They also asked about the explicit constructions of these codes.
Strong LTCs with the required range of parameters were obtained recently in the works of Viderman (CCC 2013, FOCS 2013) based on the papers of Meir (SICOMP 2009) and Dinur (J.ACM 2007). However, the construction of these codes was probabilistic.
In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. Previously, such codes were proven to be weak LTCs. Using the results of Viderman (CCC 2013, FOCS 2013) we prove that such codes are, in fact, strong LTCs
Every locally characterized affine-invariant property is testable
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a
property of functions on F^n that is closed under taking affine transformations
of the domain. We prove that all affine-invariant property having local
characterizations are testable. In fact, we show a proximity-oblivious test for
any such property P, meaning that there is a test that, given an input function
f, makes a constant number of queries to f, always accepts if f satisfies P,
and rejects with positive probability if the distance between f and P is
nonzero. More generally, we show that any affine-invariant property that is
closed under taking restrictions to subspaces and has bounded complexity is
testable.
We also prove that any property that can be described as the property of
decomposing into a known structure of low-degree polynomials is locally
characterized and is, hence, testable. For example, whether a function is a
product of two degree-d polynomials, whether a function splits into a product
of d linear polynomials, and whether a function has low rank are all examples
of degree-structural properties and are therefore locally characterized.
Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low
characteristic fields that decomposes any polynomial with large Gowers norm
into a function of low-degree non-classical polynomials. We establish a new
equidistribution result for high rank non-classical polynomials that drives the
proofs of both the testability results and the local characterization of
degree-structural properties
Locally Testable Codes and Cayley Graphs
We give two new characterizations of (\F_2-linear) locally testable
error-correcting codes in terms of Cayley graphs over \F_2^h:
\begin{enumerate} \item A locally testable code is equivalent to a Cayley
graph over \F_2^h whose set of generators is significantly larger than
and has no short linear dependencies, but yields a shortest-path metric that
embeds into with constant distortion. This extends and gives a
converse to a result of Khot and Naor (2006), which showed that codes with
large dual distance imply Cayley graphs that have no low-distortion embeddings
into .
\item A locally testable code is equivalent to a Cayley graph over \F_2^h
that has significantly more than eigenvalues near 1, which have no short
linear dependencies among them and which "explain" all of the large
eigenvalues. This extends and gives a converse to a recent construction of
Barak et al. (2012), which showed that locally testable codes imply Cayley
graphs that are small-set expanders but have many large eigenvalues.
\end{enumerate}Comment: 22 page
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