Approximating the noise sensitivity of a monotone Boolean function

Rubinfeld, Ronitt; Vasilyan, Arsen

doi:10.48550/arxiv.1904.06745

Cited by 1 publication

(1 citation statement)

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“…Moreover, a randomized algorithm for approximating the noise stability of monotone Boolean functions up to relative error was proposed in [51].…”

Section: Our Resultsmentioning

confidence: 99%

Quantum Random Access Codes for Boolean Functions

Doriguello,

Montanaro

2020

Preprint

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An n p → m random access code (RAC) is an encoding of n bits into m bits such that any initial bit can be recovered with probability at least p, while in a quantum RAC (QRAC), the n bits are encoded into m qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function f on any subset of fixed size of the initial bits, which we call f -random access codes. We study and give protocols for f -random access codes with classical (f -RAC) and quantum (f -QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement (f -EARAC) and Popescu-Rohrlich boxes (f -PRRAC). The success probability of our protocols is characterized by the noise stability of the Boolean function f . Moreover, we give an upper bound on the success probability of any f -QRAC with shared randomness that matches its success probability up to a multiplicative constant (and f -RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.

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“…Moreover, a randomized algorithm for approximating the noise stability of monotone Boolean functions up to relative error was proposed in [51].…”

Section: Our Resultsmentioning

confidence: 99%