Stability thresholds and calculation techniques for fast entangling gates on trapped ions

Fast entangling gates have been proposed for trapped ions that are orders of magnitude faster than current implementations. We present here a detailed analysis of the challenges involved in performing a successful fast gate. We show that the rotating wave approximation is stable with respect to pulse numbers: the time scale on which we can neglect terms rotating at the atomic frequency is negligibly affected by the number of pulses in the fast gate. In contrast, we show that the laser pulse instability does give rise to a pulse-number-dependent effect; the fast gate infidelity is compounded with the number of applied imperfect pulses. Using the dimensional reduction method presented here, we find bounds on the pulse stability required to achieve two-qubit gate fidelity thresholds.

Figure 1

(h) Center-of-mass (c.m.) phase-space trajectory for the center of a coherent state during the gate operation. The sides of the trajectory correspond to momentum kicks. The angle of each vertex corresponds to free evolution between kicks and marks the gate evolution point for the other figures, from (a) to (g). (a–g) Population occupying c.m. mode number states, for both ions in the excited state, at different points during the GZC gate operation, with $n=1$ (14 total pulse pairs). Blue circles represent an ideal gate, satisfying the rotating wave approximation (RWA) and with no pulse area imperfections given by $(1-\xi )$. A gate with systematic pulse imperfections $\left(\xi =0.95\right)$ is also considered (yellow squares), as well as a gate where the RWA is invalid (green diamonds) such that the pulse duration and laser phase must be defined $(\tau =5$ fs, $\varphi =3\pi /5)$. After the nonideal gates, population is lost to other internal states, and some of the population is imperfectly restored to the initial c.m. state, ${|2\rangle }_c$, the second excited number state. Values were chosen to illustrate the effects of these errors. The point in the gate operation described by (a)–(g) corresponds to a–g in (h).

Figure 2

Infidelity following (a) GZC and (b) FRAG gate operations with different numbers of pulses, governed by $n$. The effect of changing the duration of the pulses composing the gate is shown. The initial motional state is ${|2\rangle }_c{|2\rangle }_r$, the second excited number state for each mode. We determine the mean and the standard deviation (error bars) by varying the phase $\varphi$ for a given pulse duration $\tau$.

Figure 3

A GZC gate with $n=1$ is applied with varying rotation infidelity $\left(1-\mathrm{fidelity}\right)$ for individual pulses and different initial motional occupation. (a) The mean and standard deviation (error bars) of the occupation of motional number states following the gate. (b) The gate fidelity is calculated as a function of the pulse rotation fidelity for initial number states (0, 1, and 2) using the fidelity lower-bound method. The exact fidelity is calculated for both the ground and the first excited number states (0 and 1, exact) and a thermal state with mean phonon number $\overline{n}=0.5$ for comparison, using a representative internal state.

Figure 4

(a, b) GZC and (c, d) FRAG fast gates are applied to ${|ee\rangle |1\rangle }_c{|1\rangle }_r$ with varying $n$ and pulse rotation error. (a, c) The mean and standard deviation (error bars) of the occupation of the c.m. mode are shown following the gate applied to the $|ee\rangle$ internal state. (b,d) The gate fidelity is shown as a function of the pulse rotation infidelity.

Figure 5

Population in the (a) $|ee\rangle$, (b) $|eg\rangle$, and (c) $|gg\rangle$ states for different c.m. number states after a GZC gate applied to $|ee\rangle \otimes {|2\rangle }_c$ with $n=1$. The fraction $\xi$ of a perfect square $\pi$ pulse performed determines the restoration of the internal state and c.m. motional mode to the initial state.

Figure 6

Fast gate fidelity distribution for 300 runs of a GZC gate $\left(n=1\right)$ with random pulse-area fluctuations according to a Gaussian distribution with standard deviation 0.02. The ions are initially in the first excited state of each motional mode.

Figure 7

(a) GZC gate fidelities for varying $n$ with Gaussian fluctuations in the pulse area with varying standard deviation in $\xi$ $\left({\sigma }_{\xi }\right)$. The initial motional states are in the first excited number state for each mode. Error bars correspond to the standard deviation of the gate fidelity distribution. (b) The relationship between ${\sigma }_{\xi }$ and the mean rotational infidelity are shown for comparison with systematic error results.