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 \emph{noise stability} of the Boolean function $f$. Moreover, we give an \emph{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.

中文翻译：

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布尔函数的量子随机存取码

$n\overset{p}{\mapsto}m$ 随机存取码 (RAC) 是将 $n$ 位编码为 $m$ 位，这样任何初始位都可以以至少 $p$ 的概率恢复，而在量子 RAC (QRAC) 中，$n$ 位被编码为 $m$ 量子位。自提出以来，RAC 的想法以许多不同的方式进行了推广，例如允许使用共享纠缠（称为纠缠辅助随机访问代码，或简称为 EARAC）或恢复多个位而不是一个。在本文中，我们将 RAC 的思想概括为在初始位的固定大小的任何子集上恢复给定的布尔函数 $f$ 的值，我们称之为 $f$-随机访问码。我们研究并给出了 $f$-随机访问代码的协议，使用经典（$f$-RAC）和量子（$f$-QRAC）编码，以及许多不同的资源，例如私有或共享随机性，共享纠缠 ($f$-EARAC) 和 Popescu-Rohrlich 盒 ($f$-PRRAC)。我们协议的成功概率由布尔函数 $f$ 的 \emph{噪声稳定性} 表征。此外，我们给出了任何具有共享随机性的 $f$-QRAC 的成功概率的 \emph{upper bound}，其成功概率匹配乘法常数（以及 $f$-RACs 的扩展），这意味着量子协议与经典的同类产品相比，只能获得有限的优势。