Quantum soundness of the classical low individual degree test
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Low degree tests play an important role in classical complexity theory, serving as basic ingredients in foundational results such as π¬π¨π― = ππ€π·π― [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 classΒ π¬π¨π―^*. The culmination of this line of work is the result π¬π¨π―^* = π±π€ [arXiv:2001.04383]. One of the key ingredients in the first reported proof of π¬π¨π―^* = π±π€ 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 onΒ π¬π¨π―^* that relied on [arXiv:1302.1242], including π¬π¨π―^* = π±π€.
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