1,506 research outputs found
From Graphs to Keyed Quantum Hash Functions
We present two new constructions of quantum hash functions: the first based
on expander graphs and the second based on extractor functions and estimate the
amount of randomness that is needed to construct them. We also propose a keyed
quantum hash function based on extractor function that can be used in quantum
message authentication codes and assess its security in a limited attacker
model
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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