ms_simulate
The Boulder Opal Toolkits are currently in beta phase of development. Breaking changes may be introduced.
- ms_simulate(drives, lamb_dicke_parameters, relative_detunings, duration, target_phases=None, sample_count=128)
Simulate a Mølmer–Sørensen-type operation on a system composed of ions.
- Parameters
drives (list[Drive]) – A list of drives addressing the ions. Each ion can only be addressed by a single drive, but there may be ions not addressed by any drive.
lamb_dicke_parameters (np.ndarray) – A 3D array of shape
(3, N, N), where \(N\) is the number of ions in the system, specifying the laser-ion coupling strength, \(\{\eta_{jkl}\}\). The three dimensions indicate, respectively, the axis, the collective mode number, and the ion.relative_detunings (np.ndarray) – A 2D array of shape
(3, N)specifying the difference, in Hz, between each motional mode frequency and the laser detuning (with respect to the qubit transition frequency \(\omega_0\)), \(\{\delta_{jk} = \nu_{jk} - \delta\}\). The two dimensions indicate, respectively, the axis and the collective mode number.duration (float) – The duration, in seconds, of the dynamics to be simulated, \(T\). It must be greater than zero.
target_phases (np.ndarray, optional) – A 2D array of shape
(N, N)with the target relative phases between ion pairs, \(\{\Psi_{kl}\}\), as a strictly lower triangular matrix. Its \((k, l)\)-th element indicates the total relative phase target for ions \(k\) and \(l\), with \(k > l\). If not provided, the function does not return the operational infidelities.sample_count (int, optional) – The number of times \(T\) between 0 and duration (included) at which the evolution is sampled. Defaults to 128.
- Returns
A dictionary containing information about the evolution of the system. The dictionary keys are:
sample_timesThe times at which the evolution is sampled, as an array of shape
(T,).phasesAcquired phases \(\{\Phi_{jk}(t_i) = \phi_{jk}(t_i) + \phi_{kj}(t_i)\}\) for each sample time and for all ion pairs, as a strictly lower triangular matrix of shape
(T, N, N). \(\Phi_{jk}\) records the relative phase between ions \(j\) and \(k\); matrix elements where \(j \leq k\) are zero.displacementsDisplacements \(\{\eta_{pj}\alpha_{pj}(t_i)\}\) for all mode-ion combinations, as an array of shape
(T, 3, N, N). The four dimensions indicate the sample time, the axis, the collective mode number, and the ion.infidelitiesA 1D array of length
Trepresenting the operational infidelities of the Mølmer–Sørensen gate at each sample time, \(\mathcal{I}(t_i)\). Only returned if target relative phases are provided.
- Return type
dict
See also
ions.DriveClass describing non-optimizable drives.
ions.ms_optimize()Find optimal pulses to perform Mølmer–Sørensen-type operations on trapped ions systems.
ions.obtain_ion_chain_properties()Calculate the properties of an ion chain.
Notes
The internal and motional Hamiltonian of \(N\) ions is
\[H_0 = \sum_{p = 1}^{3N} \hbar\nu_p \left(a_p^\dagger a_p + \frac{1}{2}\right) + \sum_{j = 1}^N \frac{\hbar \omega_0}{2} \sigma_{z,j} ,\]where the axis dimension and collective mode dimension are combined into a single index \(p\) for simplicity, \(a_p\) is the annihilation operator for the mode \(p\), and \(\sigma_{z,j}\) is the Pauli \(Z\) operator for the ion \(j\). The interaction Hamiltonian for Mølmer–Sørensen-type operations in the rotating frame with respect to \(H_0\) is:
\[H_I(t) = i\hbar\sum_{j = 1}^N \sigma_{x, j} \sum_{p = 1}^{3N} (-\beta_{pj}^*(t)a_p + \beta_{pj}(t) a_p^\dagger) ,\]where \(\sigma_{x, j}\) is the Pauli \(X\) operator for the ion \(j\) and \(\beta_{pj}(t) = \eta_{pj} \frac{\gamma_j(t)}{2} e^{i\delta_p t}\), indicating the coupling of the ion \(j\) to the motional mode \(p\), where \(\{\gamma_j\}\) is the total drive acting on ion \(j\).
The corresponding unitary operation is given by 1
\[U(t) = \exp\left[ \sum_{j=1}^N \sigma_{x, j} B_j(t) + i\sum_{j=1}^N\sum_{k=1}^{j - 1} (\phi_{jk}(t) + \phi_{kj}(t)) \sigma_{x, j} \sigma_{x, k} \right] ,\]where
\[ \begin{align}\begin{aligned}B_j(t) &\equiv \sum_{p = 1}^{3N} \left(\eta_{pj}\alpha_{pj}(t)a_p^\dagger - \eta_{pj}^{\ast}\alpha_{pj}^\ast(t)a_p \right) ,\\\phi_{jk}(t) &\equiv \mathrm{Im} \left[ \sum_{p=1}^{3N} \int_{0}^{t} d \tau_1 \int_{0}^{\tau_1} d \tau_2 \beta_{pj}(\tau_1)\beta_{pk}^{\ast}(\tau_2) \right] ,\end{aligned}\end{align} \]and
\[\alpha_{pj}(t) = \int_0^t d\tau \frac{\gamma_j(\tau)}{2} e^{i \delta_p \tau} .\]The operational infidelity of the Mølmer–Sørensen gate is defined as 1:
\[\begin{split}\mathcal{I} = 1 - \left| \left( \prod_{\substack{k=1 \\ l<k}}^N \cos ( \phi_{kl} - \psi_{kl}) \right) \left( 1 - \sum_{j=1}^3 \sum_{k,l=1}^N \left[ |\eta_{jkl}|^2 |\alpha_{jkl}|^2 \left(\bar{n}_{jk}+\frac{1}{2} \right) \right] \right) \right|^2 .\end{split}\]References