Abstract
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 )) 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.
Highlights
The learning problem of quantum states, i.e. positive-definite trace class operators of unit trace, is central in modern quantum theory and commonly called quantum state tomography
We develop the quantum transport natural gradient methods and apply them to the quantum statistical learning problems
We consider the optimal control problem of quantum transport natural gradient flows, which leads to the derivation of quantum Schrödinger bridge problem
Summary
The learning problem of quantum states, i.e. positive-definite trace class operators of unit trace, is central in modern quantum theory and commonly called quantum state tomography. The problem of quantum state estimation is ubiquitous in quantum mechanics and has a wide range of applications: This includes the analysis of optical devices [16] as well as the reliable estimation of qubit states in quantum computing [6,24]. Until this day, there have been many recent computationally efficient approaches towards the quantum state estimation problem.
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