Augmented fidelities for single qubit gates

An average gate fidelity is a standard performance metric to quantify deviation between an ideal unitary gate transformation and its realistic experimental implementation. The average is taken with respect to states uniformly distributed over the full Hilbert space. We analytically show how this average changes if the uniform distribution condition is relaxed, replaced by parametrized distributions -- polar cap and von Mises-Fisher distributions -- and how the resulting fidelities can differentiate certain noise models. In particular, we demonstrate that Pauli channels with different noise rates along the three axes can be faithfully distinguished using these augmented fidelities.