In many quantum tasks, there is an unknown quantum object that one wishes to
learn. An online strategy for this task involves adaptively refining a
hypothesis to reproduce such an object or its measurement statistics. A common
evaluation metric for such a strategy is its regret, or roughly the accumulated
errors in hypothesis statistics. We prove a sublinear regret bound for learning
over general subsets of positive semidefinite matrices via the
regularized-follow-the-leader algorithm and apply it to various settings where
one wishes to learn quantum objects. For concrete applications, we present a
sublinear regret bound for learning quantum states, effects, channels,
interactive measurements, strategies, co-strategies, and the collection of
inner products of pure states. Our bound applies to many other quantum objects
with compact, convex representations. In proving our regret bound, we establish
various matrix analysis results useful in quantum information theory. This
includes a generalization of Pinsker's inequality for arbitrary positive
semidefinite operators with possibly different traces, which may be of
independent interest and applicable to more general classes of divergences.

通过应用正则化跟随领导算法，在学习正半定矩阵的通用子集和其他量子物体时，证明了一个次线性的遗憾界，并建立了在量子信息理论中有用的各种矩阵分析结果。

在线学习众多量子对象

Online learning of a panoply of quantum objects

Optimization is one of the keystones of modern science and engineering. Its
applications in quantum technology and machine learning helped nurture
variational quantum algorithms and generative AI respectively. We propose a
general approach to design variational optimization algorithms based on
generative models: the Variational Generative Optimization Network (VGON). To
demonstrate its broad applicability, we apply VGON to three quantum tasks:
finding the best state in an entanglement-detection protocol, finding the
ground state of a 1D quantum spin model with variational quantum circuits, and
generating degenerate ground states of many-body quantum Hamiltonians. For the
first task, VGON greatly reduces the optimization time compared to stochastic
gradient descent while generating nearly optimal quantum states. For the second
task, VGON alleviates the barren plateau problem in variational quantum
circuits. For the final task, VGON can identify the degenerate ground state
spaces after a single stage of training and generate a variety of states
therein.

基于生成模型的变分生成优化网络（VGON）是一个通用方法，用于设计变分优化算法，并在量子任务中得到广泛应用。