An Efficient Parameterized Algorithm for Computing Quantum Channel Fidelity via Symmetries Exploitation

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programmi...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 71; no. 9; pp. 7003 - 7015
Main Authors: Chee, Yeow Meng, Ta, Hoang, Vu, Van Khu
Format: Journal Article
Language:English
Published: IEEE 01.09.2025
Subjects:
bilinear optimization
Data mining
de Finetti theorems
Hilbert space
Matrix converters
Noise measurement
Optimization
Polynomials
Quantum channels
Quantum entanglement
Quantum error correction
Quantum mechanics
semidefinite programming
Symmetric matrices
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">\mathrm {poly}(1/\epsilon, \text {input dimension}) </tex-math></inline-formula> time, compared to the <inline-formula> <tex-math notation="LaTeX">\exp (1/\epsilon, \text {input dimension}) </tex-math></inline-formula> running time required for direct computation.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2025.3589660