ms_dephasing_robust_cost

Graph.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(1D, complex)]) – The list of piecewise-constant drives \(\{\gamma_j\}\). The number of drives must be the same as the number of ions \(N\). Drive values must be one-dimensional arrays and in rad/s. Drive durations must be in seconds. All drives must have the same total duration, but they can have different numbers of segments.

  • lamb_dicke_parameters (np.ndarray) – A (3, N, N) \(\{ \eta_{jkn} \}\) array of parameters specifying the laser-ion coupling strength, where \(N\) equals the number of ions. The first dimension \(j\) indicates the axis, the second dimension \(k\) indicates the collective mode number, and the third dimension \(n\) indicates the ion.

  • relative_detunings (np.ndarray) – The 2D array \(\{\delta_{jk} = \nu_{jk} - \delta\}\) specifying the difference (in Hz) between each motional mode frequency \(\nu_{jk}\) and the laser detuning \(\delta\) (the detuning from the qubit transition frequency). Its shape must be (3, N) where the first dimension \(j\) indicates the axis and the second dimension \(k\) indicates the collective mode number.

  • 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

calculate_ion_chain_properties()

Function to calculate the properties of an ion chain.

ms_infidelity

Final operational infidelity of a Mølmer–Sørensen gate.

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 \(j\)-th ion in the \(p\)-th mode of oscillation is the following [2]:

\[\alpha_{pj}(t) = \int_0^t d\tau \frac{\gamma_j(\tau)}{2} e^{i \delta_p \tau} .\]

where the axis dimension and the collective mode dimension are combined into a single index \(p\) for simplicity. For a gate of duration \(t_\text{gate}\), the time-averaged position is:

\[\langle \alpha_{pj} \rangle = \frac{1}{t_\text{gate}} \int_0^{t_\text{gate}} \alpha_{pj}(t) \mathrm{d} t .\]

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 \(\alpha_{pj}(t)\) are weighted in the formula for the infidelity of a Mølmer–Sørensen gate.

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

\[C_\text{robust} = \sum_{p,j} \left| \eta_{pj} \langle \alpha_{pj} \rangle \right|^2.\]

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 \(C_\text{robust}\) by a scaling factor to weigh how much importance you give to robustness.

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.