Singular value transformation for unknown quantum channels

TL;DR

This paper introduces a quantum algorithm that transforms the singular values of unknown quantum channels using a novel block-encoding scheme, enabling advanced spectral analysis and testing properties like entanglement breaking.

Contribution

It develops an approximate block-encoding method for the Hermitized Liouville representation of quantum channels, facilitating polynomial transformations of their singular values with quantum singular value transformation.

Findings

Established upper and lower bounds for constructing approximate channel encodings.

Applied the method to learn singular value moments for quantum channels.

Implications for testing if a quantum channel is entanglement breaking.

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 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.