# ms_dephasing_robust_cost¶

static OperationNamespace.ms_dephasing_robust_cost(drives, lamb_dicke_parameters, relative_detunings, *, name=None)

Calculates 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[TensorPwc(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 in rad/s and 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, 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)

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

1

A. R. Milne, C. L. Edmunds, C. Hempel, F. Roy, S. Mavadia, and M. J. Biercuk, Physical Review Applied 13, 024022 (2020).

2

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