A polar decomposition for quantum channels (with applications to bounding error propagation in quantum circuits)

Arnaud Carignan-Dugas1, Matthew Alexander1, and Joseph Emerson1,2

1Institute for Quantum Computing and the Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

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Inevitably, assessing the overall performance of a quantum computer must rely on characterizing some of its elementary constituents and, from this information, formulate a broader statement concerning more complex constructions thereof. However, given the vastitude of possible quantum errors as well as their coherent nature, accurately inferring the quality of composite operations is generally difficult. To navigate through this jumble, we introduce a non-physical simplification of quantum maps that we refer to as the leading Kraus (LK) approximation. The uncluttered parameterization of LK approximated maps naturally suggests the introduction of a unitary-decoherent polar factorization for quantum channels in any dimension. We then leverage this structural dichotomy to bound the evolution -- as circuits grow in depth -- of two of the most experimentally relevant figures of merit, namely the average process fidelity and the unitarity. We demonstrate that the leeway in the behavior of the process fidelity is essentially taken into account by physical unitary operations.

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