Quantum Random Access Codes for Boolean Functions

João F. Doriguello1,2 and Ashley Montanaro2,3

1Quantum Engineering Centre for Doctoral Training, University of Bristol, United Kingdom
2School of Mathematics, University of Bristol, United Kingdom
3Phasecraft Ltd.

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An $n\overset{p}{\mapsto}m$ 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 $\textit{noise stability}$ of the Boolean function $f$. Moreover, we give an $\textit{upper bound}$ 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.

Random Access Code is a fancy name for a scheme that compresses an initial amount of data into a smaller amount, such that any initial bit can be recovered with high probability. In this work we generalise random access codes to recovering not just one bit, but the value of a Boolean function on any constant size set of the initial bits. This is done by considering a variety of resources, both classical and quantum.

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