Entangling-gate error from coherently displaced motional modes of trapped ions

Entangling gates in trapped-ion quantum computers are most often applied to stationary ions with initial motional distributions that are thermal and close to the ground state, while those demonstrations that involve transport generally use sympathetic cooling to reinitialize the motional state prior to applying a gate. Future systems with more ions, however, will face greater nonthermal excitation due to increased amounts of ion transport and exacerbated by longer operational times and variations over the trap array. In addition, pregate sympathetic cooling may be limited due to time costs and laser access constraints. In this paper, we analyze the impact of such coherent motional excitation on entangling-gate error by performing simulations of Mølmer-Sørenson (MS) gates on a pair of trapped-ion qubits with both thermal and coherent excitation present in a shared motional mode at the start of the gate. We quantify how a small amount of coherent displacement erodes gate performance in the presence of experimental noise, and we demonstrate that adjusting the relative phase between the initial coherent displacement and the displacement induced by the gate or using Walsh modulation can suppress this error. We then use experimental data from transported ions to analyze the impact of coherent displacement on MS-gate error under realistic conditions.

Figure 1

Entanglement infidelity ${\epsilon }_e$ and diamond error ${\epsilon }_{\diamond }$ for an MS gate vs motional frequency error $\delta \nu /2\pi$ with ${\overline{n}}_{\text{th}}=0$. From bottom (lightest) to top (darkest) in each plot, the calculations are for ${\left|\alpha \right|}^2=0$ to 2 in steps of 0.4 with $\varphi =0$ (solid lines) and with $\varphi =\pi /2$ (dashed lines). For ${\left|\alpha \right|}^2=0$, the gate error is the same for both values of $\varphi$.

Figure 2

Entanglement infidelity ${\epsilon }_e$ and diamond error ${\epsilon }_{\diamond }$ for an MS gate averaged over a Gaussian distribution of motional frequencies with width $\sigma /2\pi =600$ Hz and centered at ${\nu }_0/2\pi =3$ MHz vs the phase $\varphi$ of the initial coherent state. From bottom (lightest) to top (darkest) in each plot, the calculations are for ${\left|\alpha \right|}^2=0$ to 2 in steps of 0.4 with ${\overline{n}}_{\text{th}}=0$ (a) and for ${\overline{n}}_{\text{th}}=0$ to 2 in steps of 0.4 with ${\left|\alpha \right|}^2=1$ (b).

Figure 3

Entanglement infidelity ${\epsilon }_e$ (upper plots) and diamond error ${\epsilon }_{\diamond }$ (lower plots) for an MS gate averaged over a Gaussian distribution of motional frequencies with width $\sigma /2\pi =600$ Hz and centered at ${\nu }_0/2\pi =3$ MHz vs ${\left|\alpha \right|}^2$ and ${\overline{n}}_{\text{th}}$, for $\varphi =0$ (a) and for $\varphi =\pi /2$ (b). In each plot, the color represents increasing gate error from lightest to darkest, and the contours start at 0.01 and increase in steps of 0.01 from the lower left to the upper right.

Figure 4

Entanglement infidelity ${\epsilon }_e$ (a) and diamond error ${\epsilon }_{\diamond }$ (b) for an MS gate with first-order Walsh modulation averaged over a Gaussian distribution of motional frequencies with width $\sigma /2\pi =600$ Hz and centered at ${\nu }_0/2\pi =3$ MHz vs ${\left|\alpha \right|}^2$ and ${\overline{n}}_{\text{th}}$. There is no dependence on $\varphi$. In each plot, the color represents increasing gate error from lightest to darkest, and the contours start at 0.002 and increase in steps of 0.002 from the lower left to the upper right.

Figure 5

Excited-state probability $P_e$ during blue-sideband Rabi-flopping experiments and the log-likelihood vs ${\left|\alpha \right|}^2$ and ${\overline{n}}_{\text{th}}$ for $E_z=0$ (a) and for $E_z=40$ V/m (b). In the upper plots, the black dots are the average of $M=500$ measurements at each time step. The blue vertical line segments are the corresponding statistical error bars, and the orange line is the best-fit model of $P_e$ based on a maximum likelihood estimation. In the lower plots, the color scales range from the maximum likelihood (lightest) to $e^{-1}$ times the maximum likelihood (darkest), i.e., to one sigma. The black contours occur every five sigma.