Existence of good quantum locally testable codes
In plain words
A locally testable code lets a checker detect a corrupted message by looking at a few randomly chosen checks, with failure probability growing with the amount of corruption. Quantum versions with good size and error tolerance would be a step toward the quantum PCP conjecture, and their existence is open.
Precise statement
Does a family of quantum stabilizer codes exist with $O(1)$-weight checks, rate $k/n = \Omega(1)$, relative distance $d/n = \Omega(1)$, and constant soundness: the fraction of violated checks is at least c times the relative distance of the state from the code space? Dinur, Lin and Vidick (arXiv:2402.07476) give constant rate with inverse-polylogarithmic distance and soundness.
What would settle it
An explicit good qLTC family with proofs of all three parameters, or a no-go theorem.
Status in the literature
Unverified note
Almost-good qLTCs (constant rate, inverse-polylog distance and soundness) were constructed in 2024, with the rate analysis corrected in a 2025 revision.