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)

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

\alpha_{jkl}(t) = \int_0^t d\tau \frac{\gamma_l(\tau)}{2} \exp(i 2 \pi \delta_{jk} \tau) .

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

[1] A. R. Milne, C. L. Edmunds, C. Hempel, F. Roy, S. Mavadia, and M. J. Biercuk, Phys. Rev. Appl. 13, 024022 (2020).

[2] C. D. B. Bentley, H. Ball, M. J. Biercuk, A. R. R. Carvalho, M. R. Hush, and H. J. Slatyer, Adv. Quantum Technol. 3, 2000044 (2020).

Examples