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

## TL;DR

This paper introduces a polar decomposition for quantum channels, using the leading Kraus approximation to better understand and bound error propagation in quantum circuits, with implications for fidelity and unitarity assessments.

## Contribution

It proposes a novel unitary-decoherent polar factorization for quantum channels and applies it to bound error growth in quantum circuit performance metrics.

## Key findings

- Bound the evolution of process fidelity and unitarity in quantum circuits.
- Leeway in process fidelity behavior is mainly due to physical unitary operations.
- Introduces the leading Kraus approximation as a simplification tool.

## Abstract

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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Source: https://tomesphere.com/paper/1904.08897