ms_infidelity

Graph.ms_infidelity(phases, displacements, target_phases, mean_phonon_numbers=None, *, name=None)

Calculate the final operational infidelity of the Mølmer–Sørensen gate.

This function calculates the operational infidelity with respect to the target phases that you specify in the target_phases array. It can use the tensors returned from ms_phases() and ms_displacements() to calculate the infidelity tensor.

Parameters:

  • phases (np.ndarray(real) or Tensor(real)) – Acquired phases \(\{\Phi_{kl}\}\) for all ion pairs with shape (N, N) without time samples or (T, N, N), where T is the number of samples and N is the number of ions. For each sample the phases array must be a strictly lower triangular matrix.

  • displacements (np.ndarray(complex) or Tensor(complex)) – Motional displacements \(\{\eta_{jkl} \alpha_{jkl}\}\) in phase-space with shape (3, N, N) without time samples or (T, 3, N, N), where T is the number of samples, 3 is the number of spatial axes, and N is the number of ions that is equal to the number of modes along an axis. The first dimension \(j\) indicates the axis, the second dimension \(k\) indicates the mode number along the axis, and the third dimension \(l\) indicates the ion.

  • target_phases (np.ndarray) – 2D array containing target relative phases \(\{\psi_{kl}\}\) between ion pairs. For ions \(k\) and \(l\), with \(k > l\), the total relative phase target is the \((k, l)\)-th element. The target_phases must be a strictly lower triangular matrix.

  • mean_phonon_numbers (np.ndarray or None, optional) – 2D array with shape (3, N) of positive real numbers for each motional mode which corresponds to the mean phonon occupation \(\{\bar{n}_{jk}\}\) of the given mode, where 3 is the number of spatial axes and N is the number of ions. If not provided, \(\bar{n}_{jk} = 0\), meaning no occupation of each mode.

  • name (str or None, optional) – The name of the node.

Returns:

A scalar or 1D tensor of infidelities with shape (T,) where T is the number of samples and one infidelity value per sample.

Return type:

Tensor(real)

See also

ms_dephasing_robust_cost

Cost for robust optimization of a Mølmer–Sørensen gate.

ms_displacements

Displacements for each mode/ion combination.

ms_phases

Relative phases for all pairs of ions.

Notes

The infidelity is calculated according to [1]

\[\begin{split}1 - \mathcal{F}_\mathrm{av} = 1 - \left| \left( \prod_{\substack{k=1 \\ l<k}}^N \cos ( \Phi_{kl} - \psi_{kl}) \right) \left( 1 - \sum_{j=1}^3 \sum_{k,l=1}^N \left[ |\eta_{jkl}|^2 |\alpha_{jkl}|^2 \left(\bar{n}_{jk}+\frac{1}{2} \right) \right] \right) \right|^2 ,\end{split}\]

which assumes that the displacements \(\alpha_{jkl}\) are small and eliminates terms of the fourth or higher order in them.

References

Examples

Refer to the How to optimize error-robust Mølmer–Sørensen gates for trapped ions user guide to find how to use this and related nodes.