Hardware-Efficient Leakage-Reduction Scheme for Quantum Error Correction with Superconducting Transmon Qubits

Leakage outside of the qubit computational subspace poses a threatening challenge to quantum error correction (QEC). We propose a scheme using two leakage-reduction units (LRUs) that mitigate these issues for a transmon-based surface code, without requiring an overhead in terms of hardware or QEC-cycle time as in previous proposals. For data qubits, we consider a microwave drive to transfer leakage to the readout resonator, where it quickly decays, ensuring that this negligibly disturbs the computational states for realistic system parameters. For ancilla qubits, we apply a $|1⟩\leftrightarrow |2⟩\pi$ pulse conditioned on the measurement outcome. Using density-matrix simulations of the distance-3 surface code, we show that the average leakage lifetime is reduced to almost one QEC cycle, even when the LRUs are implemented with limited fidelity. Furthermore, we show that this leads to a significant reduction of the logical error rate. This LRU scheme opens the prospect for near-term scalable QEC demonstrations.

Popular Summary

The development of quantum computers might allow one to solve computational problems otherwise impossible to solve on a classical computer. However, quantum computers are subject to a variety of errors that hinder the accuracy of their output. Quantum error correction codes have been tailored to protect against regular errors within the computational subspace of the qubits. Leakage outside of the computational subspace, present in leading platforms such as superconducting qubits, poses a threat as quantum error correction codes do not provide an inherent protection against this kind of error. Leakage-reduction units have been proposed to bring a leaked state back into the computational subspace. However, the leakage-reduction units proposed so far have a high cost in terms of qubit count, required gates, and/or time.

In this work we introduce two leakage-reduction units that use hardware already present on chip, without requiring additional qubits, two-qubit gates, or time. In particular, for data qubits, we use a microwave drive to transfer leakage to their dedicated readout resonators, while for ancilla qubits, we use a rotation conditioned on the measurement outcome. Using simulations of a small-distance transmon-based surface code, we show that the average leakage lifetime is reduced to almost its minimum and that the logical error rate of the code is significantly lowered, even when these operations are implemented with limited fidelity.

Our leakage-reduction units open the prospect for near-term scalable demonstrations of quantum error correction and are potentially applicable to superconducting qubits other than transmons.

Figure 1

Concept of the readout-resonator LRU. (a) The state $|20⟩$ (with the notation $|\mathrm{transmon,resonator}⟩$) is connected to $|01⟩$ by two main paths via either $|11⟩$ or $|10⟩$, due to the capacitive coupling $g$ or the transmon-drive amplitude $\mathrm{\Omega }$, respectively. This generates an effective coupling $\tilde{g}$ that can be used to swap $|20⟩\leftrightarrow |01⟩$. The latter quickly decays to $|00⟩$ due to the typically high coupling $\kappa$ of the readout resonator to the transmission-line environment, overall removing leakage from a leaked transmon. (b) In the rotating frame of the drive, $|20⟩$ and $|01⟩$ show an avoided crossing as a function of the drive frequency ${\omega }_d$, centered at ${\omega }_d^{\ast }$. The effective coupling $\tilde{g}({\omega }_d^{\ast })$ is equal to half the energy separation at that point. (c),(e) The values $\mathrm{\Delta }{\omega }_d^{\ast }:={\omega }_d^{\ast }-(2{\omega }_q+\alpha -{\omega }_r)$ and $\tilde{g}({\omega }_d^{\ast })$ are respectively evaluated either exactly by full numerical diagonalization of $H$ in Eq. (1), or by approximate analytical formulas (see Sec. 1a and Appendix pp1) for the parameters in Table 1. The absolute errors with respect to the exact curves are shown in (d) and (f), respectively.

Figure 2

Lindblad simulations of the transmon-resonator system for the readout-resonator LRU. In (a),(b) the initial state is $|2⟩⟨2|\otimes {\sigma }_{\mathrm{th}}$, while in (c),(d) it is $|0⟩⟨0|\otimes {\sigma }_{\mathrm{th}}$, where ${\sigma }_{\mathrm{th}}$ is the resonator thermal state. (a),(c) Transmon leakage population $p^{|2⟩}=⟨2|{\mathrm{Tr}}_r[\rho (T_{\mathrm{slot}})]|2⟩$ at the end of the time slot of $T_{\mathrm{slot}}=440\mathrm{ns}$. For each choice of $(\mathrm{\Omega },{\omega }_d)$, we optimize the total pulse duration $t_p\le T_{\mathrm{slot}}$ to minimize $p^{|2⟩}$ given the initial state $|2⟩⟨2|\otimes {\sigma }_{\mathrm{th}}$ for fixed $t_{\mathrm{rise}}=30\mathrm{ns}$. The white star indicates the chosen operating point ($\mathrm{\Omega }/2\pi \approx 204\mathrm{MHz}$, ${\omega }_d/2\pi \approx 5.2464\mathrm{GHz}$, $t_p=178.6\mathrm{ns}$) with $p_{\mathrm{op.}}^{|2⟩}\approx 0.5\mathrm{\%}$ in (a). The induced leakage in (c) is $p^{|2⟩}\approx 0.48\mathrm{\%}$ at the operating point. The purple line corresponds to the higher-order estimate of the optimal drive frequency ${\omega }_d^{\ast }$ as a function of $\mathrm{\Omega }$ [see Fig. 1]. The heatmaps are sampled using the adaptive package [66]. (b),(d) Time evolution of the populations in a few selected states for the operating point. The vertical dashed line indicates the used $t_p$. The inset in (d) shows a schematic of the pulse $\mathrm{\Omega }(t)$.

Figure 3

(a) Schematic overview of the Surface-17 layout [46, 57]. Pink (red) circles with $D$ labels represent low- (high-)frequency data qubits, while blue (green) circles with $X$ ($Z$) labels represent ancilla qubits, which have an intermediate frequency. Ancilla qubits and high-frequency data qubits are prone to leakage during the $\mathrm{cz}$ gates. (b) The quantum circuit for a single QEC cycle employed in simulation for the unit-cell scheduling defined in Ref. [57], in which we insert the LRUs. The res-LRUs (orange) are applied unconditionally on the high-frequency data qubits after the $\mathrm{cz}$ gates, while the $\pi$-LRUs (teal) are applied on the ancilla qubits depending on the measurement outcome. Gray elements correspond to operations belonging to the previous or the following QEC cycle. The duration of each operation is given in Appendix pp3 2b1. The arrow at the bottom indicates the repetition of QEC cycles.

Figure 4

Average leakage lifetime $l_{\mathrm{avg}}^{\mathcal{L}}$ [(a),(c)] and leakage steady state ${\overline{p}}_{\mathrm{SS}}^{\mathcal{L}}$ [(b),(d)] as a function of the leakage-reduction rate $R$ for data qubits [(a),(b)] and as a function of the readout probability $p_M(2|2)$ for ancilla qubits [(c),(d)]. Here we fix the $\mathrm{cz}$ leakage rate to $L_1=0.5\mathrm{\%}$. The insets in (b) and (d) show that ${\overline{p}}_{\mathrm{SS}}^{\mathcal{L}}$ tends to approximately $N_{\mathrm{flux}}{L}_1$ ($N_{\mathrm{flux}}=4$ for $D_4$, 3 for $D_3,D_5$, 1 for $Z_0,Z_3$, and 2 for the remaining ancilla qubits). The vertical dashed lines correspond to the values used in Sec. 2 2d. These results are extracted from $2\times {10}^4$ runs of 20 QEC cycles each per choice of parameters. Error bars are estimated using bootstrapping and are mostly smaller than the symbol size.

Figure 5

Logical error rate ${\epsilon }_L$ per QEC cycle for the upper bound (red) and minimum-weight perfect-matching (green) decoders versus the $\mathrm{cz}$ leakage rate $L_1$, in the cases with no LRUs, only res-LRU, only $\pi$-LRU, and both LRUs (the point without leakage at $L_1=0$ is always without LRUs as well). These results are extracted from $2\times {10}^4$ runs of 20 QEC cycles each per choice of parameters. Error bars are estimated using bootstrapping and are smaller than the symbol size.

Figure 6

Effective $T_1$ (a) and $T_2$ (b) that account for the extra decoherence caused by the drive during the time slot $T_{\mathrm{slot}}=440\mathrm{ns}$. We can see that the variation is small as a function of the drive amplitude compared to the values at $\mathrm{\Omega }=0$. The white star indicates the chosen operating point ($\mathrm{\Omega }/2\pi \approx 204\mathrm{MHz}$, ${\omega }_d/2\pi \approx 5.2464\mathrm{GHz}$, $t_p=178.6\mathrm{ns}$; see Sec. 1b). The purple line corresponds to the higher-order estimate of the optimal drive frequency ${\omega }_d^{\ast }$ as a function of $\mathrm{\Omega }$ [see Fig. 1]. The heatmaps are sampled using the adaptive package [66].

Figure 7

Time evolution from the initial state $|2⟩⟨2|\otimes {\sigma }_{\mathrm{th}}$ for $t_{\mathrm{rise}}=30\mathrm{ns}$ and for an otherwise always-on drive during $T_{\mathrm{slot}}$. This is simulated with the same $\mathrm{\Omega }/2\pi \approx 204\mathrm{MHz}$ and ${\omega }_d/2\pi \approx 5.2464\mathrm{GHz}$ as at the operating point in Fig. 2.

Figure 8

Sensitivity of the leakage-reduction rate $R$ of the readout-resonator LRU as a function of the overall residual $ZZ$ coupling $\zeta$. (a) Underdamped regime, specifically at the operating point ($\mathrm{\Omega }/2\pi \approx 204\mathrm{MHz}$, ${\omega }_d/2\pi \approx 5.2464\mathrm{GHz}$, $t_p=178.6\mathrm{ns}$; see Sec. 1b). (b) Critical regime ($\mathrm{\Omega }/2\pi \approx 143\mathrm{MHz}$, ${\omega }_d/2\pi \approx 5.252\mathrm{GHz}$, $t_p=440\mathrm{ns}$).

Figure 9

Logical error rate ${\epsilon }_L$ per QEC cycle as a function of various LRU parameters. In (a) and (b) we use only the res-LRU, while in (c) and (d) we use the $\pi$-LRU. We fix $L_1=0.5\mathrm{\%}$ for all. Vertical dashed lines indicate the values considered in Sec. 2 2d. These results are extracted from $2\times {10}^4$ runs of 20 QEC cycles each per choice of parameters. Error bars are estimated using bootstrapping and are smaller than the symbol size.

Figure 10

Variation of the logical error rate ${\epsilon }_L$ for different choices of leakage conditional phases ${\varphi }^{\mathcal{L}}$. (a) The logical error rate ${\epsilon }_L$ per QEC cycle for the UB (shades of red) and MWPM (shades of green) decoders versus $L_1$, in the cases with no LRUs and both LRUs, each for all ${\varphi }^{\mathcal{L}}$ set to 0, $\pi /2$, or uniformly random in $[0,\pi ]$. These results are extracted from $2\times {10}^4$ runs of 20 QEC cycles each per choice of parameters. Error bars are estimated using bootstrapping and are mostly smaller than the symbol size. (b) The random values for ${\varphi }^{\mathcal{L}}$ used across this work. These values are extracted from a uniform distribution in $[0,\pi ]$. We have excluded negative values as $\pm {\varphi }^{\mathcal{L}}$ corresponds to the same chance of spreading a $Z$ error under the twirling action of the parity-check measurements.