量子态的普适归纳推理
Universal Inductive Inference of Quantum States
- Hon Hai (Foxconn) Research Institute（鸿海（富士康）研究院）
- Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出普适量子归纳推理框架，将Solomonoff归纳推广至量子源，给出信息论推理算法及其匹配下界，并构建非i.i.d.态层析算法，探讨计算难度。
AI中文摘要：
Solomonoff的普适归纳推理[Inf. Control. 1964]是用于从序列数据学习和预测未来观测的最一般框架之一。它为任何可计算的随机源提供预测保证。我们询问是否可以为量子系统发展类似的普适归纳理论。为此，我们引入普适量子归纳推理，这是一个用于学习和预测可能表现出跨时间任意相关性和纠缠的量子源的框架。给定过去测量的结果和相应的测量后量子系统，学习器产生一个联合态，该联合态预测下一个测量结果和测量后系统，同时保持与过去系统的相关性。我们将量子源建模为一个多部分量子态，其描述由未知的$s$位程序在给定时间界限内生成。学习器需要产生一个在迹距离上与目标$\u03b5$-接近且概率至少为$1-\u03b4$的状态。我们建立了一个信息论推理算法，其轮复杂度为$O(s\u03b5^{-2}\u03b4^{-1})$。我们进一步证明了在相关参数范围内匹配的下界，即使对于经典源也是如此。作为第二个结果，我们构建了一个非独立同分布态层析成像算法，该算法输出给定过去测量结果的下一个系统的条件态的经典描述。我们的信息论层析成像算法实现了轮复杂度$O(s\uff1cmin\uff1e{2^s,2^n}\u03b5^{-2}\u03b4^{-1})$，其中$n$是每轮接收的量子比特数。最后，我们研究了在量子密码假设下普适量子归纳推理的计算难度。
英文摘要：
Solomonoff's universal inductive inference [Inf. Control. 1964] is one of the most general frameworks for learning from sequential data and predicting future observations. It provides prediction guarantees for any computable stochastic source. We ask whether an analogous universal theory of induction can be developed for quantum systems. To this end, we introduce universal quantum inductive inference, a framework for learning and predicting quantum sources that may exhibit arbitrary correlations and entanglement across time. Given the outcomes of past measurements and the corresponding post-measurement quantum systems, the learner produces a joint state that predicts the next measurement outcome and post-measurement system while preserving correlations with the past systems. We model a quantum source as a multipartite quantum state whose description is generated by an unknown $s$-bit program within a given time bound. The learner is required to produce a state that is $ε$-close in trace distance to the target with probability at least $1-δ$. We establish an information-theoretic inference algorithm with round complexity $O(sε^{-2}δ^{-1})$. We further prove a matching lower bound in the relevant parameter regime, even for classical sources. As a second result, we construct a non-i.i.d. state tomography algorithm that outputs a classical description of the conditional state of the next system given past measurement outcomes. Our information-theoretic tomography algorithm achieves round complexity $O(s\min\{2^s,2^n\}ε^{-2}δ^{-1})$, where $n$ is the number of qubits received in each round. Finally, we investigate the computational hardness of universal quantum inductive inference under quantum cryptographic assumptions.
