In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm for parity learning on $n$ bits requires either $\Omega(n^2)$ bits of classical memory or an exponential number (in~$n$) of random samples. A line of recent works continued that research direction and showed that for a large collection of classical learning tasks, either super-linear classical memory size or super-polynomially many samples are needed. However, these results do not capture all physical computational models, remarkably, quantum computers and the use of quantum memory. It leaves the possibility that a small piece of quantum memory could significantly reduce the need for classical memory or samples and thus completely change the nature of the classical learning task. In this work, we prove that any quantum algorithm with both, classical memory and quantum memory, for parity learning on $n$ bits, requires either $\Omega(n^2)$ bits of classical memory or $\Omega(n)$ bits of quantum memory or an exponential number of samples. In other words, the memory-sample lower bound for parity learning remains qualitatively the same, even if the learning algorithm can use, in addition to the classical memory, a quantum memory of size $c n$ (for some constant $c>0$). Our results refute the possibility that a small amount of quantum memory significantly reduces the size of classical memory needed for efficient learning on these problems. Our results also imply improved security of several existing cryptographical protocols in the bounded-storage model (protocols that are based on parity learning on $n$ bits), proving that security holds even in the presence of a quantum adversary with at most $c n^2$ bits of classical memory and $c n$ bits of quantum memory (for some constant $c>0$).

翻译：在Raz的工作（J. ACM和FOCS 16）中，证明了任何用于$n$比特奇偶性学习的算法，要么需要$\Omega(n^2)$比特的经典记忆，要么需要指数级（关于$n$）的随机样本。近期一系列工作延续了这一研究方向，表明对于大量经典学习任务而言，要么需要超线性的经典记忆大小，要么需要超多项式的样本数量。然而，这些结果并未涵盖所有物理计算模型，尤其是量子计算机和量子记忆的使用。这留下了可能性：一小块量子记忆可能显著减少对经典记忆或样本的需求，从而彻底改变经典学习任务的性质。在本工作中，我们证明任何同时使用经典记忆和量子记忆的量子算法，用于$n$比特奇偶性学习时，要么需要$\Omega(n^2)$比特的经典记忆，要么需要$\Omega(n)$比特的量子记忆，要么需要指数级数量的样本。换言之，即使学习算法除了经典记忆外还能使用大小为$c n$（对于某个常数$c>0$）的量子记忆，奇偶性学习的记忆-样本下界在本质上仍保持不变。我们的结果否定了少量量子记忆能显著减少这些问题的有效学习中所需经典记忆大小的可能性。我们的结果还意味着，在有限存储模型下（基于$n$比特奇偶性学习的协议），若干现有密码协议的安全性得到增强，证明即使面对最多拥有$c n^2$比特经典记忆和$c n$比特量子记忆（对于某个常数$c>0$）的量子敌手，这些协议的安全性依然成立。