boulderopal.ions.ms_optimize(drives, duration, lamb_dicke_parameters, relative_detunings, target_phases, sample_count=128, robust=False, **optimization_kwargs)

Find optimal pulses to perform a target Mølmer–Sørensen-type operation on a system composed of ions.

This function builds a graph describing the Mølmer–Sørensen operation and calls boulderopal.run_optimization to minimize the target cost.


  • drives (list [OptimizableDrive ]) – A list of optimizable drives addressing the ions. Each ion can only be addressed by a single drive, but there may be ions not addressed by any drive.
  • duration (float) – The duration, in seconds, of the dynamics to be optimized, TT
  • lamb_dicke_parameters (np.ndarray) – The laser-ion coupling strength, {ηjkl}\{\eta_{jkl}\}(3, N, N), where the dimensions indicate, respectively, axis, collective mode, and ion.
  • relative_detunings (np.ndarray) – The difference {δjk=νjkδ}\{\delta_{jk} = \nu_{jk} - \delta\} (in Hz) between each motional mode frequency and the laser detuning from the qubit transition frequency ω0\omega_0(3, N), where the dimensions indicate, respectively, axis and collective mode.
  • target_phases (np.ndarray or None , optional) – The target total relative phases between ion pairs, {Ψln}\{\Psi_{ln}\}(N, N). Ψln\Psi_{ln} with l>nl > n indicates the relative phase between ions ll and nn, while Ψln=0\Psi_{ln} = 0 for lnl \leq n
  • sample_count (int , optional) – The number of times TT
  • robust (bool , optional) – If set to False, the cost corresponds to the infidelity at the end of the gate. If set to True, the cost is the final infidelity plus a dephasing-robust cost term. Defaults to False.
  • \*\*optimization_kwargs – Additional parameters to pass to boulderopal.run_optimization.


The result of the run_optimization call. Its output item is a dictionary containing information about the optimized drive and the evolution of the system, with the following keys:

optimized drives : The piecewise-constant optimized drives implementing the gate. The keys are the names of the drives provided to the function.

sample_times : The times at which the evolution is sampled, as an array of shape (T,).

phases : Acquired phases $\{\Phi_{ln}(t_i) = \phi_{ln}(t_i) + \phi_{nl}(t_i)\}$ for each sample time and for all ion pairs, as a strictly lower triangular matrix of shape (T, N, N). $\Phi_{ln}(t_i)$ with $l > n$ indicates the relative phase between ions $l$ and $n$, while $\Phi_{ln}(t_i) = 0$ for $l \leq n$.

displacements : Displacements $\{\eta_{jkl}\alpha_{jkl}(t_i)\}$ for all mode-ion combinations, as an array of shape (T, 3, N, N), where the dimensions indicate, respectively, time, axis, collective mode, and ion.

infidelities : A 1D array of length T with the operational infidelities of the Mølmer–Sørensen gate at each sample time, $\mathcal{I}(t_i)$.

Return type



boulderopal.ions.ComplexOptimizableDrive : Class describing a piecewise-constant complex-valued optimizable drive.

boulderopal.ions.RealOptimizableDrive : Class describing a piecewise-constant real-valued optimizable drive.

boulderopal.ions.ms_simulate : Simulate a Mølmer–Sørensen-type operation on a trapped ions system.

boulderopal.ions.obtain_ion_chain_properties : Calculate the properties of an ion chain.


See the notes of boulderopal.ions.ms_simulate for the main equations and definitions.

You can use the robust flag to construct a Mølmer–Sørensen gate that is robust against dephasing noise. This imposes a symmetry 1 in the optimizable ion drives and aims to minimize the time-averaged positions of the phase-space trajectories,

αjkl=1tgate0tgateαjkl(t)dt. \langle \alpha_{jkl} \rangle = \frac{1}{t_\text{gate}} \int_0^{t_\text{gate}} \alpha_{jkl}(t) \mathrm{d} t .

This is achieved by adding an additional term to the cost function, consisting of the sum of the square moduli of the time-averaged positions multiplied by the corresponding Lamb–Dicke parameters. That is to say,

Crobust=I+j=13k=1Nl=1Nηjklαjkl2. C_\text{robust} = \mathcal{I} + \sum_{j=1}^{3} \sum_{k=1}^{N} \sum_{l=1}^{N} \left| \eta_{jkl} \langle \alpha_{jkl} \rangle \right|^2 .


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

Was this useful?