Show simple item record

dc.contributor.authorBecker, S
dc.contributor.authorLi, W
dc.date.accessioned2021-01-24T16:16:26Z
dc.date.available2021-01-24T16:16:26Z
dc.date.issued2021
dc.date.submitted2020-09-15
dc.identifier.issn0022-4715
dc.identifier.others10955-020-02682-1
dc.identifier.other2682
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/316643
dc.description.abstract<jats:title>Abstract</jats:title><jats:p>In this article, we introduce a new approach towards the statistical learning problem <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>argmin</mml:mi> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> to approximate a target quantum state <jats:inline-formula><jats:alternatives><jats:tex-math>$$\rho _{\star }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> by a set of parametrized quantum states <jats:inline-formula><jats:alternatives><jats:tex-math>$$\rho (\theta )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in a quantum <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^*$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>
dc.languageen
dc.publisherSpringer Science and Business Media LLC
dc.rightsAttribution 4.0 International (CC BY 4.0)
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.subjectArticle
dc.subjectQuantum transport information geometry
dc.subjectQuantum state estimation
dc.subjectQuantum Wasserstein information matrix
dc.subjectQuantum Wasserstein natural gradient
dc.subjectQuantum Schrödinger bridge problem
dc.titleQuantum Statistical Learning via Quantum Wasserstein Natural Gradient
dc.typeArticle
dc.date.updated2021-01-24T16:16:26Z
prism.issueIdentifier1
prism.publicationNameJournal of Statistical Physics
prism.volume182
dc.identifier.doi10.17863/CAM.63755
dcterms.dateAccepted2020-12-07
rioxxterms.versionofrecord10.1007/s10955-020-02682-1
rioxxterms.versionVoR
rioxxterms.licenseref.urihttp://creativecommons.org/licenses/by/4.0/
dc.identifier.eissn1572-9613
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/L016516/1)
cam.issuedOnline2021-01-07


Files in this item

Thumbnail
Thumbnail

This item appears in the following Collection(s)

Show simple item record

Attribution 4.0 International (CC BY 4.0)
Except where otherwise noted, this item's licence is described as Attribution 4.0 International (CC BY 4.0)