The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices

Palma, Giacomo De; Trevisan, Dario

doi:10.1007/s00023-023-01340-y

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2024

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Cited by 3 publications

References 75 publications

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Quantum Optimal Transport: Quantum Channels and Qubits

De Palma,

Trevisan

2024

Bolyai Society Mathematical Studies

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Quantum Optimal Transport: Quantum Channels and Qubits

De Palma,

Trevisan

2024

Bolyai Society Mathematical Studies

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Quantum optimal transport: an invitation

Trevisan

2024

Boll Unione Mat Ital

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The optimal mass transport problem was formulated centuries ago, but only recently there has been a surge in its applications, particularly in functional inequalities, geometry, stochastic analysis, and numerical solutions for partial differential equations. Quantum optimal transport aims to extend this success story to non-commutative systems, where density operators replace probability measures. This brief review paper aims to describe the latest approaches, highlighting their advantages, disadvantages, and open mathematical problems relevant to applications.

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Classical shadows meet quantum optimal mass transport

De Palma,

Klein,

Pastorello

2024

Journal of Mathematical Physics

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Classical shadows constitute a protocol to estimate the expectation values of a collection of M observables acting on O(1) qubits of an unknown n-qubit state with a number of measurements that is independent of n and that grows only logarithmically with M. We propose a local variant of the quantum Wasserstein distance of order 1 of De Palma et al. [IEEE Trans. Inf. Theory 67, 6627–6643 (2021)] and prove that the classical shadow obtained measuring O(log n) copies of the state to be learned constitutes an accurate estimate with respect to the proposed distance. We apply the results to quantum generative adversarial networks, showing that quantum access to the state to be learned can be useful only when some prior information on such state is available.

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The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices

Cited by 3 publications

References 75 publications

Quantum Optimal Transport: Quantum Channels and Qubits

Quantum Optimal Transport: Quantum Channels and Qubits

Quantum optimal transport: an invitation

Classical shadows meet quantum optimal mass transport

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