Quantum process tomography of a Mølmer-Sørensen interaction

We present a simple tomographic protocol, for two-qubit systems, that relies on a single discriminatory transition and no direct spatially selective imaging. This scheme exploits excess micromotion in the trap to realize all operations required to prepare all input states and analyze all output states. We demonstrate a two-qubit entangling gate with a Bell state production fidelity of 0.981(6), and apply the above protocol to perform the first quantum process tomography of a Mølmer-Sørensen entangling gate. We characterize its $\chi$-process matrix, the simplest for an entanglement gate on a separable-states basis, and observe that our dominant source of error is accurately modeled by a quantum depolarization channel.

Figure 1

Two-qubit quantum process tomography. (a) Energy level diagram of the two ions, with the micromotion (MM) sideband (shown as a dashed black line) for the ion not sitting on the rf null, separated by ${\nu }_{\text{rf}}=21.75$ MHz from the carrier transition. The ground-state levels $|S\rangle$ and $|S^{\prime }\rangle$ are Zeeman split by ${\nu }_Z=12.3$ MHz. (b) Absolute value of the reconstructed $\chi$ matrix of the identity process $|{\chi }_{\exp }|$ and the absolute value of the difference with the ideal identity process $|{\chi }_{\exp }-{\chi }_{\mathrm{id}}|$. The basis of the matrix is ${\sigma }_i\otimes {\sigma }_j$ where, for clarity we use the convention $X={\sigma }_x,Y={\sigma }_y,Z={\sigma }_z$.

Figure 2

A Mølmer-Sørensen entangling gate on a ground-state cooled two-ion chain. (a) Evolution of the populations $P_0$ (blue squares), $P_1$ (black circles), and $P_2$ (red triangles) as a function of the interaction duration. The dashed lines are the analytical solutions of the MS model [13]; the solid lines also take into account depolarization (see text). For clarity, single error bars are shown for $t=200$ $\mu$s. (b) Parity scan $(\text{parity}=P_0+P_2-P_1)$ oscillation, obtained by scanning the phase of a $\pi /2$ pulse after a gate time of $t_g=130$ $\mu$s. The red solid line corresponds to the parameters extracted from a maximum-likelihood analysis of the data (see text). Each data point corresponds to 200 repetitions. All error bars of this panel are the binomial standard errors.

Figure 3

Process tomography of a Mølmer-Sørensen interaction. Upper panel: Real and imaginary parts of the reconstructed process matrix Re[$\chi$] and Im[$\chi$] of five consecutive gates. Lower panel: (a) Average fidelity. In blue points, the values extracted from the experimentally reconstructed $\chi$ matrices with respect to the target gate operation ${\mathcal{E}}_0$. In red points, the mean values are calculated with respect to output density matrices for which a depolarizing operation has been applied. Dashed lines are linear fits. (b) Average state purity. In blue points, calculated from the reconstructed process $\chi$ matrices, and in red from a depolarization channel model with a depolarization rate of $1.9\left(1\right)\times {10}^{-2}$ per gate time. The offset value being due to tomographic error, it is set arbitrarily on the red points, and is not used for the analysis of the gate. Error bars represent the standard deviation with respect to all input states (see text).