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 (J. ACM 和 FOCS 16) 的一篇文章中,事实证明,任何以美元位数进行等同学习的算法,只要0元位数的计算模型、显著的量子计算机和使用量子内存,就有可能需要一小块量子内存大大减少对古典记忆或随机样本的需求,或者需要以美元位数进行大量古典学习任务,或者需要超线古典内存大小或超球体积样本。然而,这些结果并不包含所有物理计算模型、显著的量子计算机和量子内存的使用情况。这让小块量子内存大大减少对古典内存或样本的需求,从而可能大大减少对古典内存或样本的需求,从而彻底改变古典学习任务的性质。 在这项工作中,任何带有古典内存和量内存的量值的量值计算方法,只要以美元进行等值学习,则需要以美元位内存的内存的内存价值内存为等值。</s>