In this article, we introduce a new approach towards the statistical learning problem $$\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$ argmin ρ ( θ ) ∈ P θ W Q 2 ( ρ ⋆ , ρ ( θ ) ) to approximate a target quantum state $$\rho _{\star }$$ ρ ⋆ by a set of parametrized quantum states $$\rho (\theta )$$ ρ ( θ ) in a quantum $$L^2$$ L 2 -Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional $$C^*$$ 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.

中文翻译：

---

通过 Quantum Wasserstein Natural Gradient 进行量子统计学习

在本文中，我们介绍了一种解决统计学习问题的新方法 $$\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta))$$ argmin ρ ( θ ) ∈ P θ WQ 2 ( ρ ⋆ , ρ ( θ ) ) 来近似目标量子态 $$\rho _{\ star }$$ ρ ⋆ 通过一组参数化的量子态 $$\rho (\theta )$$ ρ ( θ ) 在量子 $$L^2$$ L 2 -Wasserstein 度量中。我们通过在有限维 $$C^*$$C* 代数上考虑密度算子的 Wasserstein 自然梯度流来解决这个估计问题。对于密度算子的连续参数模型，我们撤回了量子 Wasserstein 度量，使得参数空间成为具有量子 Wasserstein 信息矩阵的黎曼流形。使用 Benamou-Brenier 公式的量子模拟，我们在参数空间上推导出自然梯度流。我们还通过研究相关的 Wigner 概率分布的传输来讨论某些连续可变的量子态。