Quantum statistical learning via Quantum Wasserstein natural gradient
- URL: http://arxiv.org/abs/2008.11135v1
- Date: Tue, 25 Aug 2020 16:06:43 GMT
- Title: Quantum statistical learning via Quantum Wasserstein natural gradient
- Authors: Simon Becker and Wuchen Li
- Abstract summary: We introduce a new approach towards the statistical learning problem $operatornameargmin_rho(theta).
We approximate a target quantum state $rho_star$ by a set of parametrized quantum states $rho(theta)$ in a quantum $L2$-Wasserstein metric.
- Score: 5.426691979520477
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: In this article, we introduce a new approach towards the statistical learning
problem $\operatorname{argmin}_{\rho(\theta) \in \mathcal P_{\theta}} W_{Q}^2
(\rho_{\star},\rho(\theta))$ to approximate a target quantum state
$\rho_{\star}$ by a set of parametrized quantum states $\rho(\theta)$ 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.
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