发表机构
Mie University（三重大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文严格证明了在随机RSA模数分解困难性假设下，两个特定哈密顿量函数学习问题的平均情况经典困难性，即高效经典学习器将导致经典多项式时间分解RSA模数。
AI 中文摘要
Morohoshi、Nakayama、Manabe和Mitarai提出了一个具有物理动机的量子机器学习问题，其目标是从哈密顿量H和量子态ρ的经典描述中预测形如Tr[f(H)ρ]的量，其中f是未知函数。在本文中，我们将此问题称为哈密顿量函数学习。他们在适当条件下构建了一个高效的量子学习算法，但留下了平均情况经典困难性的严格证明未解决。在本文中，我们在随机RSA模数分解的平均情况困难性假设下，严格证明了Morohoshi等人论文中讨论的两个特定分布哈密顿量函数学习问题（针对f_{cos,π}(λ)=cos(πλ)和f_{exp,β}(λ)=e^{-βλ}）的平均情况经典困难性。更具体地说，我们证明了对于任一问题，若存在一个在平方损失下高效且输出假设可在经典多项式时间内评估的经典随机化学习器，则将产生一个用于分解随机RSA模数的经典随机化多项式时间算法。
英文摘要
Morohoshi, Nakayama, Manabe, and Mitarai proposed a physically motivated quantum machine learning problem in which the goal is to predict quantities of the form $\operatorname{Tr}[f(H)ρ]$ from classical descriptions of a Hamiltonian $H$ and a quantum state $ρ$, where $f$ is an unknown function. We call this problem Hamiltonian function learning in this paper. They constructed an efficient quantum learning algorithm under suitable conditions, while leaving a rigorous proof of average-case classical hardness open. In this paper, we rigorously prove the average-case classical hardness for two distribution-specific Hamiltonian function learning problems for $f_{\cos,π}(λ)=\cos(πλ)$ and $f_{\exp,β}(λ)=e^{-βλ}$ discussed in the paper of Morohoshi et al. under the assumption of the average-case hardness of factoring random RSA moduli. More specifically, we show that an efficient classical randomized learner under squared loss whose output hypotheses are evaluable in classical polynomial time for either problem would yield a classical randomized polynomial-time algorithm for factoring random RSA moduli.
Comments11 pages, 1 figure
