Eliminating Leakage Errors in Hyperfine Qubits

Population leakage outside the qubit subspace presents a particularly harmful source of error that cannot be handled by standard error correction methods. Using a trapped ${}^{171}{\mathrm{Yb}}^{+}$ ion, we demonstrate an optical pumping scheme to suppress leakage errors in atomic hyperfine qubits. The selection rules and narrow linewidth of a quadrupole transition are used to selectively pump population out of leakage states and back into the qubit subspace. Each pumping cycle reduces the leakage population by a factor of $\sim 3$, allowing for an exponential suppression in the number of cycles. We use interleaved randomized benchmarking on the qubit subspace to show that this pumping procedure has negligible side effects on the qubit subspace, bounding the induced qubit memory error by $\le 2.0(8)\times {10}^{-5}\mathrm{per}$ cycle, and qubit population decay to $\le 1.4(3)\times {10}^{-7}\mathrm{per}$ cycle. These results clear a major obstacle for implementations of quantum error correction and error mitigation protocols.

Figure 1

A level diagram illustrating the leakage repump scheme. The solid blue lines show the 435 nm quadrupole transition driving leaked population from $|L_{\pm }⟩$ to ${{}^{2}D}_{3/2}$. The second step, illustrated by solid red lines, is a dipole transition at 935 nm that drives population to ${}^3[3/2{]}_{1/2}$. As illustrated by the dashed lines, ${}^3[3/2{]}_{1/2}$ states decay to ${{}^{2}S}_{1/2}$ with high probability, thus completing the repump cycle.

Figure 2

Pumping a leaked state $|L_{-}⟩$ into the qubit subspace. Qubit populations, $P_0$ and $P_1$, are shown in blue circles and green diamonds, respectively, and leakage populations, $P_{L_{-}}$ and $P_{L_{+}}$, are shown in red squares and black triangles, respectively. Each point is the average of 1000 shots with a standard error on the 1% level. The solid lines show fitted curves described in the main text and the dashed line shows the ideal decay of $1/3^n$. The pulse sequence is also shown, and we have omitted the final 935 nm pulse with 860 MHz sidebands mentioned in the text.

Figure 3

Benchmarking the identity operation with and without the leakage repump protocol being implemented. The survival frequency of randomized benchmarking quantifies the probability of a random sequence generating a target state and decays exponentially as errors accumulate. This decay is fitted to a function that is related to the average fidelity, which is used to bound the induced error from the leakage repump protocol. We observe no significant increase in the error rate when the leakage repump sequence is implemented during the benchmarked identity operation, meaning that the measurement is limited by memory error. We, therefore, upper bound the induced error at $\le 2.0(8)\times {10}^{-5}\mathrm{per}$ cycle.

Figure 4

Population decay within the qubit subspace during the leakage repump protocol. The measurement is made by preparing $|1⟩$, applying the leakage repump cycle $n$ times and then measuring the $|1⟩$ population. Fitting a decaying exponential to the data results in a decay per cycle of $1.4(3)\times {10}^{-7}$, which is depicted by the black dashed line.