Figure 1

Qubit encoding in the symmetric states of an ensemble of $(2N+1)$-level systems. $|s⟩$ denotes the reservoir state. The figure depicts the state $|10\dots 01⟩$.Reuse & Permissions

Figure 2

Schematic illustration of our procedure to remove single atom occupancy of register states, already occupied by the ensemble. The second and third lines of the figure depict superposition states. We use the shorthand notation $|\eta ⟩=c_0|0⟩+c_1|1⟩$, $|r_{\eta }⟩=c_0|r_0⟩+c_1|r_1⟩$, $|ϵ⟩=\alpha |\overline{0}{⟩}_{K-1}+\beta |\overline{1}{⟩}_{K-1}$, and $|r_{ϵ}⟩=\alpha |{\overline{r}}_0{⟩}_{K-1}+\beta |{\overline{r}}_1{⟩}_{K-1}$. Our error detection sequence proceeds as follows: (i) $0\leftrightarrow r_0$ and $1\leftrightarrow r_1$ resonant composite pulses are applied. (ii) After detection of either $|r_0⟩$ or $|r_1⟩$, either a $s\leftrightarrow r_0$ or a $s\leftrightarrow r_1$ resonant pulse is applied. (iii) Depending on the result of the measurement in (ii) either $|r_0⟩$ or $|r_1⟩$ is ionized. If an ion signal is observed, the final state is $|\overline{ϵ}{⟩}_{K-1}$; otherwise $|\eta ⟩$ is ionized, yielding the state $|S{⟩}_{K-1}$, subsequently transferred into the final state $|\overline{0}{⟩}_{K-1}$ by a $s\leftrightarrow 0$ resonant pulse.Reuse & Permissions