Quantum Proofs of Proximity

We initiate the systematic study of QMA algorithms in setting property testing, to which we refer as proofs proximity (QMAPs). These are quantum query that receive explicit access a sublinear-size untrusted proof and required accept inputs having xmlns:mml="http://www.w3.org/1998/Math/MathML">Π reject xmlns:mml="http://www.w3.org/1998/Math/MathML">ε-far from mathvariant="normal">Π, while only probing minuscule portion their input.We investigate complexity landscape this model, showing QMAPs can be xmlns:mml="http://www.w3.org/1998/Math/MathML">exponentially stronger than both classical proofs proximity testers. To end, extend methodology Blais, Brody, Matulef (Computational Complexity, 2012) prove testing lower bounds via reductions communication complexity. This also resolves question raised 2013 by Montanaro de Wolf (cf. Theory Computing, 2016).Our algorithmic results include purpose an framework enables speedups for expressive class properties, namely, those succinctly xmlns:mml="http://www.w3.org/1998/Math/MathML">decomposable. A consequence is algorithm verify Parity xmlns:mml="http://www.w3.org/1998/Math/MathML">n-bit string with xmlns:mml="http://www.w3.org/1998/Math/MathML">O(n2/3) queries length. propose graph bipartitneness, lies outside family, there speedup.