# ms_dephasing_robust_cost

`ions.ms_dephasing_robust_cost(drives, lamb_dicke_parameters, relative_detunings, *, name=None)`

Calculate the cost for robust optimization of a Mølmer–Sørensen gate.

Add the tensor that this function returns to the infidelity of your target operation to obtain a cost that you can use to create a Mølmer–Sørensen gate that is robust against dephasing noise. You can further multiply the robust cost by a scaling factor to weigh how much importance you give to the robustness compared to the original cost.

### Parameters

**drives**(*list**[**Pwc**or**None**]*) – The piecewise-constant drives, $\{\gamma_j\}$, one for each of the $N$**lamb_dicke_parameters**(*np.ndarray*) – The laser-ion coupling strength, $\{\eta_{jkl}\}$`(3, N, N)`

, where the dimensions indicate, respectively, axis, collective mode, and ion.**relative_detunings**(*np.ndarray*) – The difference $\{\delta_{jk} = \nu_{jk} - \delta\}$ (in Hz) between each motional mode frequency and the laser detuning from the qubit transition frequency $\omega_0$`(3, N)`

, where the dimensions indicate, respectively, axis and collective mode.**name**(*str**or**None**,**optional*) – The name of the node.

### Returns

The cost term that you can use to optimize a Mølmer–Sørensen gate that is robust against dephasing noise. The cost is the sum of the square moduli of the time-averaged positions of the phase-space trajectories, weighted by the corresponding Lamb–Dicke parameters.

### Return type

Tensor(scalar, real)

### SEE ALSO

`Graph.ions.ms_infidelity`

: Final operational infidelity of a Mølmer–Sørensen gate.

`boulderopal.ions.obtain_ion_chain_properties`

: Function to calculate the properties of an ion chain.

## Notes

You can construct a Mølmer–Sørensen gate that is robust against
dephasing noise by a combination of minimizing the time-averaged
positions of the phase-space trajectories and imposing a symmetry in
each ion’s drive ^{1}.

The displacement of the $l$-th ion in the $k$-th mode of
oscillation in dimension $j$^{2}

For a gate of duration $t_\text{gate}$

$\langle \alpha_{jkl} \rangle = \frac{1}{t_\text{gate}} \int_0^{t_\text{gate}} \alpha_{jkl}(t) \mathrm{d} t .$This function returns the sum of the square moduli of the time-averaged
positions multiplied by the corresponding Lamb–Dicke parameters. These
parameters weight the time-averaged positions in the same way that the
$\alpha_{jkl}(t)$`Graph.ions.ms_infidelity`

).

In other words, the robust cost that this function returns is

$C_\text{robust} = \sum_{j=1}^{3} \sum_{k=1}^{N} \sum_{l=1}^{N} \left| \eta_{jkl} \langle \alpha_{jkl} \rangle \right|^2 .$You can add this to the infidelity with the respect to the target gate to create the cost function that optimizes a gate that is also robust against dephasing. You can further multiply $C_\text{robust}$

## 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.