Abstract
An n↦pm 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.
Highlights
One of the possible origins of quantum computers’ power is the exponential size of the Hilbert space: a n-qubit quantum state is a unit vector in a 2n dimensional complex vector space
One of these scenarios is the concept of quantum random access codes (QRACs), where a number of bits are encoded into a smaller number of qubits such that any one of the initial bits can be recovered with some probability of success
A QRAC is normally denoted by n →p m, meaning that n bits are encoded into m qubits such that any initial bit can be recovered with probability at least p > 1/2, and a classical version, called random access code (RAC), is defined, with the encoding message being m bits
Summary
One of the possible origins of quantum computers’ power is the exponential size of the Hilbert space: a n-qubit quantum state is a unit vector in a 2n dimensional complex vector space. Interesting scenarios arise when allowing a small chance of transmitting the wrong message or/and obtaining partial information at the expense of losing information about the rest of the system. One of these scenarios is the concept of quantum random access codes (QRACs), where a number of bits are encoded into a smaller number of qubits such that any one of the initial bits can be recovered with some probability of success. In this paper we further generalize the idea of (quantum) random access codes to recovering not just an initial bit, but the value of a fixed Boolean function on any subset of the initial bits with fixed size. The case of the Parity function was already considered in [8], and here we generalize to arbitrary Boolean functions f : {−1, 1}k → {−1, 1}
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
