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, 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 .
The displacement of the \(j\)-th ion in the \(p\)-th mode of
oscillation is the following :
\[\alpha_{pj}(t) = \int_0^t d\tau \frac{\gamma_j(\tau)}{2}
e^{i \delta_p \tau} .\]
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.