ms_infidelity

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

Calculates 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, 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, 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 function is defined as 1:

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

References

1

C. D. B. Bentley, H. Ball, M. J. Biercuk, A. R. R. Carvalho, M. R. Hush, and H. J. Slatyer, Advanced Quantum Technologies 3, 2000044 (2020).