Optimal encoding of classical data for statistical inference using quantum
computing is investigated. A universal encoder is sought that is optimal for a
wide array of statistical inference tasks. Accuracy of any statistical
inference is shown to be upper bounded by a term that is proportional to
maximal quantum leakage from the classical data, i.e., the input to the
inference model, through its quantum encoding. This demonstrates that the
maximal quantum leakage is a universal measure of the quality of the encoding
strategy for statistical inference as it only depends on the quantum encoding
of the data and not the inference task itself. The optimal universal encoding
strategy, i.e., the encoding strategy that maximizes the maximal quantum
leakage, is proved to be attained by pure states. When there are enough qubits,
basis encoding is proved to be universally optimal. An iterative method for
numerically computing the optimal universal encoding strategy is presented.

利用量子计算，研究经典数据在统计推断中的最佳编码。寻找一种适用于各种统计推断任务的优化编码器。通过量子编码，实证任何统计推断的准确性都被一个与量子编码中的经典数据最大泄漏成比例的项上界所限制，这表明最大量子泄漏是统计推断编码策略质量的普适度量，仅取决于数据的量子编码而不依赖于推断任务本身。证明了最佳的普适编码策略，即最大化最大量子泄漏的编码策略，是通过纯态实现的。当量子比特足够多时，基底编码被证明是普遍最优的。提出了一种迭代方法，用于数值计算最佳的普适编码策略。

统计推断的最优通用量子编码

Optimal Universal Quantum Encoding for Statistical Inference

The concept of attention, numerical weights that emphasize the importance of
particular data, has proven to be very relevant in artificial intelligence.
Relative entropy (RE, aka Kullback-Leibler divergence) plays a central role in
communication theory. Here we combine these concepts, attention and RE. RE
guides optimal encoding of messages in bandwidth-limited communication as well
as optimal message decoding via the maximum entropy principle (MEP). In the
coding scenario, RE can be derived from four requirements, namely being
analytical, local, proper, and calibrated. Weighted RE, used for attention
steering in communications, turns out to be improper. To see how proper
attention communication can emerge, we analyze a scenario of a message sender
who wants to ensure that the receiver of the message can perform well-informed
actions. If the receiver decodes the message using the MEP, the sender only
needs to know the receiver's utility function to inform optimally, but not the
receiver's initial knowledge state. In case only the curvature of the utility
function maxima are known, it becomes desirable to accurately communicate an
attention function, in this case a by this curvature weighted and re-normalized
probability function. Entropic attention communication is here proposed as the
desired generalization of entropic communication that permits weighting while
being proper, thereby aiding the design of optimal communication protocols in
technical applications and helping to understand human communication. For
example, our analysis shows how to derive the level of cooperation expected
under misaligned interests of otherwise honest communication partners.

提出了一种新型的基于关注度和相对熵的通信模型，旨在设计和优化技术应用中的最佳通信协议，以及理解人类通信的方式。