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 (single qubit) and numerically (two qubit) 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.

Figure 1

Visualization of von Mises–Fisher distributions. Top: 50 000 random states selected according to von Mises–Fisher distribution around the $|0\rangle$ state (red point) with (left) $\kappa =10$ and (right) $\kappa =100$. Bottom: Fidelity between the $|0\rangle$ state and states $|\psi \rangle$ that are drawn from von Mises–Fisher distribution (i.e., between the “red point” state and states represented by “blue points”) expressed in the form of histograms (orange: $\kappa =10$; red (and inset): $\kappa =100$).

Figure 2

Analytical augmented fidelities for processes that share the same ${\chi }_{00}=0.985$ element (depolarizing rate) corresponding to 99% fidelity. Red (dashed) and blue (dotted) curves bound the region between the minimal and maximal fidelities of a noisy process. Green region (inside) corresponds to diagonal $\chi$ matrices (i.e., Pauli channels). Top: augmented fidelity over polar cap distribution, Eq. (6), parametrized by polar angle $\mathrm{\Theta }$. Bottom: augmented fidelity, Eq. (12), with respect to von Mises–Fisher distribution (normal distribution in directional statistics) as a function of $\kappa$ parameters (corresponding to the inverse of variance in standard statistics for normal distribution). Error bars indicate standard deviation of fidelities (color coded) for minimal and maximal values. Black (dashed) line represents the depolarizing channel that would be identified through RB.

Figure 3

Augmented fidelity of Pauli channels. Different channels (line styles) characterized by values in Table 1 with polar cap (top panel) and von Mises–Fisher (bottom) distributions. Black dashed line corresponds to the depolarizing channel with fidelity 99%. Color coded are distributions centered around different initial states: (red) $|+\rangle$ state, (blue) $|y_{+}\rangle$ (eigenvector of Pauli $Y$ to eigenvalue $+1$), and (green) $|0\rangle$.

Figure 4

Heatmaps of two-qubit augmented fidelities for local polar cap distribution (top panel) and von Mises–Fisher distribution (bottom). The left (right) panel(s) show the minimum (maximum) augmented fidelity envelope with all process matrices considered. Note, that in the limit ${\mathrm{\Theta }}_{1,2}\to \pi$ and ${\kappa }_{1,2}\to 0$, the augmented fidelity fails to reach 99% fidelity, which is related to Eq. (17) and the additional contribution of other diagonal elements.