1,979 research outputs found
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
Succinct Representation of Codes with Applications to Testing
Motivated by questions in property testing, we search for linear
error-correcting codes that have the "single local orbit" property: i.e., they
are specified by a single local constraint and its translations under the
symmetry group of the code. We show that the dual of every "sparse" binary code
whose coordinates are indexed by elements of F_{2^n} for prime n, and whose
symmetry group includes the group of non-singular affine transformations of
F_{2^n} has the single local orbit property. (A code is said to be "sparse" if
it contains polynomially many codewords in its block length.) In particular
this class includes the dual-BCH codes for whose duals (i.e., for BCH codes)
simple bases were not known. Our result gives the first short (O(n)-bit, as
opposed to the natural exp(n)-bit) description of a low-weight basis for BCH
codes. The interest in the "single local orbit" property comes from the recent
result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that
have the single local orbit property under the affine symmetry group are
locally testable. When combined with our main result, this shows that all
sparse affine-invariant codes over the coordinates F_{2^n} for prime n are
locally testable. If, in addition to n being prime, if 2^n-1 is also prime
(i.e., 2^n-1 is a Mersenne prime), then we get that every sparse cyclic code
also has the single local orbit. In particular this implies that BCH codes of
Mersenne prime length are generated by a single low-weight codeword and its
cyclic shifts
05291 Abstracts Collection -- Sublinear Algorithms
From 17.07.05 to 22.07.05, the Dagstuhl Seminar
05291 ``Sublinear Algorithms\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
Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings [Freedman and Hastings, 2014] - which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states - identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes.
For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally-testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems
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
Robust Quantum Entanglement at (Nearly) Room Temperature
We formulate a mixed-state analog of the NLTS conjecture [FH14] by asking
whether there exist topologically-ordered systems for which the thermal Gibbs
state for constant temperature is globally-entangled in the sense that it
cannot even be approximated by shallow quantum circuits. We then prove this
conjecture holds for nearly optimal parameters: when the "inverse temperature"
is almost a constant (temperature decays as 1/loglog(n))) and the Hamiltonian
is nearly local (log(n)-local). The construction and proof combine quantum
codes that arise from high-dimensional manifolds [Has17, LLZ19], the
local-decoding approach to quantum codes [LTZ15, FGL18] and quantum
locally-testable codes [AE15].Comment: Strengthened main theorem, small modifications to the proof, revised
introductio
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