发表机构
University of Innsbruck（因斯布鲁克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究如何通过随机过程解释量子学习模型，开发概率框架将其表示为配置空间上随机过程，利用正算子值测量诱导转移核，探讨了马尔可夫性、负性等特性及与经典模型的关系，还提及与射影模拟的联系及有限阶核的近似作用。
AI 中文摘要
量子机器学习模型通过相干量子演化和测量定义概率性输入-输出映射。虽然这类模型有计算优势，但其内部运作和决策通常难以用通过中间配置的随机轨迹来解释。与经典（马尔可夫）随机过程不同，量子动力学通常违反查普曼-柯尔莫哥洛夫可分性条件，无法分解为概率上有意义的中间转变。我们开发了一个概率框架，将量子学习模型表示为配置空间上的随机过程，其中动力学被建模为概率分布上的线性映射。从固定的正算子值测量出发，任意量子通道在相关概率表示上诱导转移核。对于信息完备的正算子值测量（特别是对称信息完备正算子值测量），这些核是马尔可夫的但通常是准随机的，非经典性表现为负性。相比之下，射影空间允许正随机核，但由于查普曼-柯尔莫哥洛夫可分性的失效通常需要非马尔可夫动力学。这在负性和对过去配置的依赖性之间产生了权衡，即量子动力学可以由马尔可夫准随机映射或具有更高马尔可夫阶的正随机过程表示。我们讨论了这种量子动力学表示如何能被解释为通过记忆空间的随机游走，类似于射影模拟，一种学习和智能模型，其中决策源于在情节记忆网络上的随机游走。我们进一步概述了有限阶随机核如何能近似这种量子审议过程，并展示了在什么情况下可以恢复经典机器学习模型。
英文摘要
Quantum machine learning models define probabilistic input--output maps through coherent quantum evolution and measurement. While such models can exhibit computational advantages, their internal functioning and decision making generally resists interpretation in terms of stochastic trajectories through intermediate configurations. In contrast to classical (Markovian) stochastic processes, quantum dynamics generically violates the Chapman--Kolmogorov divisibility condition, preventing a decomposition into probabilistically meaningful intermediate transitions. We develop a probabilistic framework for representing quantum learning models as stochastic processes over configuration spaces where the dynamics are modeled as linear maps on probability distributions. Starting from a fixed POVM, arbitrary quantum channels induce transition kernels on the associated probability representation. For informationally complete POVMs, and in particular SIC-POVMs, these kernels are Markovian but generally quasi-stochastic, with non-classicality appearing as negativity. By contrast, projective spaces admit positive stochastic kernels but generally require non-Markovian dynamics due to the failure of Chapman--Kolmogorov divisibility. This yields a trade-off between negativity and dependence on past configurations, i.e. quantum dynamics can be represented either by Markovian quasi-stochastic maps or by positive stochastic processes with higher Markov order. We discuss how such representations of quantum dynamics can be interpreted as stochastic walks through a memory space in the spirit of Projective Simulation, a model of learning and agency in which decisions arise from random walks over an episodic memory network. We further outline how finite-order stochastic kernels can approximate such quantum deliberation processes and show in what regimes the classical machine learning model is recovered.
Comments13 pages, 2 tables, reference added
