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 $O(d^3/\hat{I}´)$ upper bound and an $\hat{I}\text{c◯}(d/\hat{I}´)$ lower bound for the query complexity of constructing a quantum channel that is $\hat{I}´$-close in diamond norm to a block-encoding of the 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.

5+14 pages, 6 figures. v2; Due to an erroneous analysis of the 2nd order term, claims have been weakened. The upper-bound and query complexity have been correspondingly fixed