2,369 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
Quantum Locally Testable Code with Exotic Parameters
In this paper, we present a few simple constructions of quantum locally
testable codes that achieve interesting parameters which were previously
unknown. We introduce an operation which we give the name check product, and
show how this operation gives rise to quantum locally testable codes of
constant soundness and linear rate, with varying distance and locality
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
The Physics of (good) LDPC Codes I. Gauging and dualities
Low-depth parity check (LDPC) codes are a paradigm of error correction that
allow for spatially non-local interactions between (qu)bits, while still
enforcing that each (qu)bit interacts only with finitely many others. On
expander graphs, they can give rise to ``good codes'' that combine a finite
encoding rate with an optimal scaling of the code distance, which governs the
code's robustness against noise. Such codes have garnered much recent attention
due to two breakthrough developments: the construction of good quantum LDPC
codes and good locally testable classical LDPC codes, using similar methods.
Here we explore these developments from a physics lens, establishing
connections between LDPC codes and ordered phases of matter defined for systems
with non-local interactions and on non-Euclidean geometries. We generalize the
physical notions of Kramers-Wannier (KW) dualities and gauge theories to this
context, using the notion of chain complexes as an organizing principle. We
discuss gauge theories based on generic classical LDPC codes and make a
distinction between two classes, based on whether their excitations are
point-like or extended. For the former, we describe KW dualities, analogous to
the 1D Ising model and describe the role played by ``boundary conditions''. For
the latter we generalize Wegner's duality to obtain generic quantum LDPC codes
within the deconfined phase of a Z_2 gauge theory. We show that all known
examples of good quantum LDPC codes are obtained by gauging locally testable
classical codes. We also construct cluster Hamiltonians from arbitrary
classical codes, related to the Higgs phase of the gauge theory, and formulate
generalizations of the Kennedy-Tasaki duality transformation. We use the chain
complex language to discuss edge modes and non-local order parameters for these
models, initiating the study of SPT phases in non-Euclidean geometries
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
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
Lower bounds for constant query affine-invariant LCCs and LTCs
Affine-invariant codes are codes whose coordinates form a vector space over a
finite field and which are invariant under affine transformations of the
coordinate space. They form a natural, well-studied class of codes; they
include popular codes such as Reed-Muller and Reed-Solomon. A particularly
appealing feature of affine-invariant codes is that they seem well-suited to
admit local correctors and testers.
In this work, we give lower bounds on the length of locally correctable and
locally testable affine-invariant codes with constant query complexity. We show
that if a code is an -query
locally correctable code (LCC), where is a finite field and
is a finite alphabet, then the number of codewords in is
at most . Also, we show that if
is an -query locally testable
code (LTC), then the number of codewords in is at most
. The dependence on in these
bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan
(ITCS `13) construct affine-invariant codes via lifting that have the same
asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas
previously, Ben-Sasson and Sudan (RANDOM `11) assumed linearity to derive
similar results.
Our analysis uses higher-order Fourier analysis. In particular, we show that
the codewords corresponding to an affine-invariant LCC/LTC must be far from
each other with respect to Gowers norm of an appropriate order. This then
allows us to bound the number of codewords, using known decomposition theorems
which approximate any bounded function in terms of a finite number of
low-degree non-classical polynomials, upto a small error in the Gowers norm
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