一种后量子联想记忆
A Post-Quantum Associative Memory
AI总结：
本文在一般概率理论框架下研究联想记忆的状态维数极限，证明其可实现对经典与量子理论的指数级优势，并给出数值求解方法。
摘要：
联想记忆是存储信息的装置，当该信息被部分披露时，便可将其完整检索出来。我们在一般概率理论（GPTs）框架内考察了一个联想记忆的玩具模型及其所受的根本限制；GPTs代表满足若干基本操作公理的最一般物理理论类别。我们自问，一个GPT的维度应当多大，才能容纳$2^m$个状态，并使其中任意$N$个都能被完美区分。将这一最小维度记为$d(N,m)$。借助Danzer和Grünbaum的一个旧结果，我们证明$d(2,m)=m+1$；相比之下，当GPT被要求为经典或量子理论时，该维度为$O(2^m)$。这给出了一个GPT在任务上指数级优于经典理论和量子理论的例子。更一般地，我们解决了固定$N$而$m$渐近增大的情形，证明对每个$N\geq 2$都有$d(N,m) \leq m^{1+o_N(1)}$（当$m\to\infty$），这再次带来相对于经典理论和量子理论的指数级改进。最后，我们针对给定GPT寻找最大的N元两两可区分集合这一一般问题，发展了一种数值方法；该问题可视为N-正则超图上最大团问题的一个实例。
英文摘要：
Associative memories are devices storing information that can be fully retrieved given partial disclosure of it. We examine a toy model of associative memory and the ultimate limitations it is subjected to within the framework of general probabilistic theories (GPTs), which represent the most general class of physical theories satisfying some basic operational axioms. We ask ourselves how large the dimension of a GPT should be so that it can accommodate $2^m$ states with the property that any $N$ of them are perfectly distinguishable. Call $d(N,m)$ the minimal such dimension. Invoking an old result by Danzer and Grünbaum, we prove that $d(2,m)=m+1$, to be compared with $O(2^m)$ when the GPT is required to be either classical or quantum. This yields an example of a task where GPTs outperform both classical and quantum theory exponentially. More generally, we resolve the case of fixed $N$ and asymptotically large $m$, proving that $d(N,m) \leq m^{1+o_N(1)}$ (as $m\to\infty$) for every $N\geq 2$, which yields again an exponential improvement over classical and quantum theories. Finally, we develop a numerical approach to the general problem of finding the largest $N$-wise mutually distinguishable set for a given GPT, which can be seen as an instance of the maximum clique problem on $N$-regular hypergraphs.
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