Leakage mitigation for quantum error correction using a mixed qubit scheme

Leakage errors take qubits out of the computational subspace and will accumulate if not addressed. A leaked qubit will reduce the effectiveness of quantum error correction protocols due to the cost of implementing leakage reduction circuits and the harm caused by interacting leaked states with qubit states. Ion trap qubits driven by Raman gates have a natural choice between qubits encoded in magnetically insensitive hyperfine states that can leak and qubits encoded in magnetically sensitive Zeeman states of the electron spin that cannot leak. In our previous work, we compared these two qubits in the context of the toric code with a depolarizing leakage error model and found that for magnetic-field noise with a standard deviation less than 32 $\mu \mathrm{G}$ that the ${}^{174}\mathrm{Yb}^{+}$ Zeeman qubit outperforms the ${}^{171}\mathrm{Yb}^{+}$ hyperfine qubit. Here we examine a physically motivated leakage error model based on ions interacting via the Mølmer-Sørenson gate. We find that this greatly improves the performance of hyperfine qubits but the Zeeman qubits are more effective for magnetic-field noise with a standard deviation less than 10 $\mu \mathrm{G}$. At these low magnetic fields, we find that the best choice is a mixed qubit scheme where the hyperfine qubits are the syndrome qubits and the leakage is handled without the need of an additional leakage reduction circuit.

Figure 1

Mixed species surface code layout. Hyperfine (${}^{171}\mathrm{Yb}^{+})$ ions are defined as syndrome qubits (white) and Zeeman (${}^{174}\mathrm{Yb}^{+}$) ions are defined as data qubits (black). The two tones of diamonds represents $X$ and $Z$ stabilizer measurements.

Figure 2

Top circuit is the standard syndrome extraction circuit for the surface code. The bottom circuit is the standard circuit with a SWAP-LRC implemented at the end. Both the homogenous Zeeman system and the mixed species system utilize the top circuit. The homogenous hyperfine system requires the LRC to handle leakage errors.

Figure 3

Comparison of logical error rates for the DP leakage model at distances 3 (black), 5 (light gray), and 7 (dark gray). The solid and dashed colored lines with the squares points represent the Zeeman and mixed species systems, respectively, stabilized to $10\mu \mathrm{G}$ standard deviation from the mean magnetic field per two qubit gate. The solid black line with the right triangle points represents the hyperfine system with the SWAP-LRC implemented. The logical error rate ($p_L$) is proportional to $p_s^{\lceil \frac{d}{2}\rceil }$ for the Zeeman system and $p_s^{\lceil \frac{d}{4}\rceil }$ for mixed species, and hyperfine systems.

Figure 4

Comparison of logical error rates for the MS leakage model at distances 3 (black), 5 (light gray), and 7 (dark gray). The solid and dashed lines with the square points represent the Zeeman and mixed species systems, respectively, stabilized to $10\mu \mathrm{G}$ standard deviation from the mean magnetic field per two qubit gate. The solid black line with the right triangle points represents the hyperfine system with the SWAP-LRC implemented. The logical error rate ($p_L$) is proportional to $p_s^{\lceil \frac{d}{2}\rceil }$ for the Zeeman, mixed species, and hyperfine systems.

Figure 5

Comparison of the different schemes for a distance-3 surface code using the depolarizing leakage model. The solid and dashed colored (gray) lines represent the Zeeman and mixed species systems, respectively. The solid black line with right triangle shows the performance of the hyperfine system with the SWAP-LRC implemented. The different symbols of the lines indicates the standard deviation from the mean magnetic field per two qubit gate: $100\mu \mathrm{G}$ (red triangle), $32\mu \mathrm{G}$ (green circle), $10\mu \mathrm{G}$ (blue square), and $1\mu \mathrm{G}$ (purple diamond).

Figure 6

Comparison of the different schemes for a distance-3 surface code using the MS leakage model. The solid and dashed colored (gray) lines represent the Zeeman and mixed species systems, respectively. The solid black line with the right triangle shows the performance of the hyperfine system with the SWAP-LRC. The different symbols of the lines indicates the standard deviation from the mean magnetic field per two qubit gate: $100\mu \mathrm{G}$ (red triangle), $32\mu \mathrm{G}$ (green circle), $10\mu \mathrm{G}$ (blue square), and $1\mu \mathrm{G}$ (purple diamond).