Singular value transformation for unknown quantum channels

Abstract:Given the ability to apply an unknown quantum channel acting on a $d$-dimensional system, we develop a quantum algorithm for transforming its singular values. The spectrum of a quantum channel as a superoperator is naturally tied to its Liouville representation, which is in general non-Hermitian. Our key contribution is an approximate block-encoding scheme for this representation in a Hermitized form, given only black-box access to the channel; this immediately allows us to apply polynomial transformations to the channel's singular values by quantum singular value transformation (QSVT). We then demonstrate an $O(d^3/\delta)$ upper bound and an $\Omega(d/\delta)$ lower bound for the query complexity of constructing a quantum channel that is $\delta$-close in diamond norm to a block-encoding of the unnormalized Hermitized Liouville representation. We show our method applies practically to the problem of learning the $q$-th singular value moments of unknown quantum channels for arbitrary $q>2, q\in \mathbb{R}$, which has implications for testing if a quantum channel is entanglement breaking. Our results establish a general framework for manipulating black-box quantum channels with QSVT and related methods, enabling direct evaluation of the channel's spectral properties without tomography or significant classical post-processing.

5+15 pages, 7 figures. v2: corrected the upper bound. v3: replaced with published version

arXiv:2506.24112 [quant-ph]

(or arXiv:2506.24112v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2506.24112 Focus to learn more arXiv-issued DOI via DataCite

Phys. Rev. A 114, L030401 (2026)

https://doi.org/10.1103/321q-k5dh Focus to learn more DOI(s) linking to related resources

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