Quantum soundness of the classical low individual degree test

Abstract

Low degree tests play an important role in classical complexity theory, serving as basic ingredients in foundational results such as MIP = NEXP [BFL91] and the PCP theorem [AS98,ALM+98]. Over the last ten years, versions of these tests which are sound against quantum provers have found increasing applications to the study of nonlocal games and the complexity class MIP^*. The culmination of this line of work is the result MIP^* = RE [arXiv:2001.04383]. One of the key ingredients in the first reported proof of MIP^* = RE is a two-prover variant of the low degree test, initially shown to be sound against multiple quantum provers in [arXiv:1302.1242]. Unfortunately a mistake was recently discovered in the latter result, invalidating the main result of [arXiv:1302.1242] as well as its use in subsequent works, including [arXiv:2001.04383]. We analyze a variant of the low degree test called the low individual degree test. Our main result is that the two-player version of this test is sound against quantum provers. This soundness result is sufficient to re-derive several bounds on MIP^* that relied on [arXiv:1302.1242], including MIP^* = RE.