Quantum Locally Testable Codes

Abstract

We initiate the study of quantum locally testable codes (qLTCs). Classical LTCs are very important in computational complexity. These codes are defined as the linear subspace satisfying a set of local constraints, with the additional requirement that their soundness, R(δ), which is the probability that a randomly chosen constraint is violated, is proportional to the proximity δ, where δn is the distance of a word from the code. Excellent LTCs exist in the classical world, and they are tightly related to the celebrated PCP (probabilistically checkable proof) theorem. In quantum complexity, quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement, and they have played a crucial role in many computational complexity results. We provide a definition of the quantum analogue of LTCs and motivate it by connecting its central notions in the study of both entanglement and quantum Hamiltonian complexity. A natural question is whether such codes exist, and how good can their soundness be. To the best of our knowledge all quantum codes known today exhibit poor soundness. Moreover, we show that the soundness of CSS codes (which are commonly used quantum codes defined by two classical codes) is governed by the minimal soundness of the two classical codes; in the most natural CSS code we examined as a candidate qLTC, namely, the Reed--Muller code, there is a tradeoff between the parameters of the two codes, which prevents the resulting quantum code from being qLTC. These facts seem to suggest a more general phenomenon, by which the soundness of qLTCs is inherently restricted due to multiparticle entanglement. Our main technical contribution consists of two complementary results regarding qLTCs which are stabilizer codes (denoted sLTCs). We first prove a surprising, inherently quantum property of sLTCs. For small constant values of proximity, the better the local expansion of the interaction graph of the constraints, the less sound the sLTC becomes. This stands in sharp contrast to the classical setting. The complementary, more intuitive result also holds (and is actually much more involved technically to prove in the quantum case): an upper bound on the soundness when the code is defined on bad local expanders. Together we arrive at a quantum upper bound on the soundness of sLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable qLTCs, and their relation to other objects of interest in quantum information theory. In the appendix we also define a quantum PCPs of proximity (PCPPs) and point out that the result of [E. Ben-Sasson et al., SIAM J. Comput., 36 (2006), pp. 889--974] by which PCPPs imply LTCs with related parameters carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs [D. Aharonov, I. Arad, and T. Vidick, ACM SIGACT News Archive, 44 (2013), pp. 47--79].