Zero-Knowledge Proofs of Proximity

Berman, Itay; Rothblum, Ron D.; Vaikuntanathan, Vinod

doi:10.4230/lipics.itcs.2018.19

Cited by 2 publications

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“…14 The threshold degree of f , denoted thrdeg(f ), is the minimal degree of an n-variate polynomial p that sign-represents f , i.e., such that f (x) = sgn(p(x)) for all x ∈ X . 15 Note that no constraints are placed on the behaviour of p(x) at inputs in {1, −1} n \ X .…”

Section: A Hard Class Of Problems For Qmapsmentioning

confidence: 99%

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Quantum Proofs of Proximity

Dall'Agnol

Gur

Moulik

et al. 2022

Quantum

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

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Section: A Hard Class Of Problems For Qmapsmentioning

confidence: 99%

“…For notational convenience, we consider Boolean functions with codomain {1, −1}, noting that this is equivalent to the usual codomain {0, 1} by mapping 0 → 1, 1 → −1, and ⊕ to multiplication 15. Here, sgn(t) is defined to equal 1 if t > 0, −1 if t < 0, and 0 if t = 0.…”

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confidence: 99%