Quantum Locally Testable Codes

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition to-gether with a simplification, denoted sLTCs, for the special case of stabilizer codes, and providesome basic results using those definitions. The most crucial parameter of such codes is their sound-ness, R(δ), namely, the probability that a randomly chosen constraint is violated as a function ofthe distance of a word from the code (δ, 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 ex-pansion of the interaction graph of the constraints, the less sound the qLTC becomes. This stands insharp contrast to the classical setting. The complementary, more intuitive, result also holds: an up-per bound on the soundness when the code is defined on bad small-set expanders (a bound whichturns out to be far more difficult to show in the quantum case). Together we arrive at a quantumupper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in theclassical case. Many open questions are raised regarding what possible parameters are achievablefor qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) andpoint out that the result of [15] by which PCPPs imply LTCs with related parameters, carries overto the sLTCs. This creates a first link between qLTCs and quantum PCPs [6].