论文标题
量子统计学习通过量子剂量恒星天然梯度
Quantum statistical learning via Quantum Wasserstein natural gradient
论文作者
论文摘要
在本文中，我们介绍了一种针对统计学习问题的新方法$ \ operatorName {argmin} _ {ρ（θ）\ in \ Mathcalp_θ} w_ {q}^2（ρ_{\ star}，ρ（par star}，ρ（θ），ρ（θ），ρ（θ）$，以近似目标量子$ partim $ partim $ρ_ partaum parte $ρ（θ）$中的量子$ l^2 $ - wasserstein衡量标准。我们通过考虑有限维$ C^*$代数的密度运算符的Wasserstein天然梯度流来解决此估计问题。对于密度运算符的连续参数模型，我们撤回量子剂量恒星度量，以使参数空间与量子Wasserstein Information矩阵成为Riemannian歧管。使用Benamou-Brenier公式的量子类似物，我们在参数空间上得出了自然梯度流。我们还通过研究相关的Wigner概率分布的运输来讨论某些连续变化的量子状态。
In this article, we introduce a new approach towards the statistical learning problem $\operatorname{argmin}_{ρ(θ) \in \mathcal P_θ} W_{Q}^2 (ρ_{\star},ρ(θ))$ to approximate a target quantum state $ρ_{\star}$ by a set of parametrized quantum states $ρ(θ)$ in a quantum $L^2$-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional $C^*$ algebras. For continuous parametric models of density operators, we pull back the quantum Wasserstein metric such that the parameter space becomes a Riemannian manifold with quantum Wasserstein information matrix. Using a quantum analogue of the Benamou-Brenier formula, we derive a natural gradient flow on the parameter space. We also discuss certain continuous-variable quantum states by studying the transport of the associated Wigner probability distributions.
