Coherent Error Suppression in Multiqubit Entangling Gates

We demonstrate a simple pulse shaping technique designed to improve the fidelity of spin-dependent force operations commonly used to implement entangling gates in trapped ion systems. This extension of the Mølmer-Sørensen gate can theoretically suppress the effects of certain frequency and timing errors to any desired order and is demonstrated through Walsh modulation of a two qubit entangling gate on trapped atomic ions. The technique is applicable to any system of qubits coupled through collective harmonic oscillator modes.

Figure 1

(a) The phase space trajectory of the motional wave packets of a single ion during a $W(1,t/t_g)$ spin-dependent force operation with a small detuning error, $\Delta /\delta \ll 1$. The solid and dashed curves show the two different trajectories taken by the two different wave packets associated with spin up and spin down in the ${\widehat{S}}_1$ basis with only the spin up trajectory being labeled for clarity. After being initialized to a state centered at the origin, the spin up wave packet begins its clockwise motion near the point labeled 1. Halfway through the operation, near the point 2, the phase of the drive field is advanced by $\pi$, changing the direction of the applied force. At the end of the gate, near point 3, the wave packet ends up much closer to the origin than the turning point near point 2. Therefore, the two wave packets have more overlap at the end of the gate which is the key to achieving a higher fidelity operation. (b) The Walsh functions $W(1,x)$, $W(3,x)$ and $W(7,x)$ are shown. Notice that $W(3,x)$ can be constructed as two sequential $W(1,x)$ functions with a phase flip on the second pulse. Likewise, $W(7,x)$ can be constructed as sequential $W(3,x)$ pulses with a phase flip on the second pulse.

Figure 2

A single ion prepared in $|\downarrow ⟩$ and with an average excitation number, $\overline{n}\approx 7$, is subjected to the standard and composite spin-dependent force operations and then measured in the ${\widehat{\sigma }}_z$ basis. The data shown are plotted together with theoretical curves assuming an initial thermal state of motion. (a) The data show the probability of finding the ion in $|\uparrow ⟩$ as a function of the symmetric detuning $\delta$ for $t_0=100\mu \sec$. On resonance, $\delta =0$, the motional wave packets quickly become entangled with the spin state, resulting in a maximally mixed spin state. For finite $\delta$, the wave packets trace out circles in phase space resulting in revivals of the initial spin state when $\delta t_0/2\pi$ is a nonzero integer. (b) The spin-dependent force operation is implemented using $W(1,t/t_1)$ for the phase ${\varphi }_s(t)$ with $t_1=\sqrt{2}{t}_0$. (c) $W(3,t/t_3)$ is used for ${\varphi }_s(t)$ with $t_3=2t_0$. Note, the narrow resonance at $\delta t_3/2\pi =2$ corresponds to a trajectory where the phase flips occur when the motional wave packets are not at the origin.

Figure 3

The state fidelity of a two ion MS gate as a function of the detuning $\delta$ is compared for the first three Walsh functions being used for ${\varphi }_s$. In all three data sets, the ions were sideband cooled to the motional ground state before implementing the gate. In Figs. 3a, 3b, 3c, the measured state fidelity as a function of the detuning $\delta$ is compared with the theoretical curves. (d) shows the ideal fidelity curves for $W(0)(\mathrm{\text{blue}})$, $W(1)(\mathrm{\text{red}})$, $W(3)(\mathrm{\text{green}})$, $W(7)(\mathrm{\text{purple}})$ and $W(15)(\mathrm{\text{cyan}})$. The curves are shifted in frequency in order to facilitate comparison and highlight the increasingly large regions of high fidelity for the higher order sequences. As a guide to the eye, the ideal curves shown in Fig. 3d are fit to the data by varying $\Omega$ and including an overall scale factor to account for additional experimental imperfections. The three different fit values of $\Omega$ agree with each other to within 10%. The estimates for the characteristic widths of the high fidelity regions, $B_k$, are shown to increase rapidly with the higher order sequences.