Removing leakage-induced correlated errors in superconducting quantum error correction

Abstract

Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing.

Fig. 1. Removing leakage with reset.

a Schematic of the multi-level reset protocol. The qubit starts with a population in its first three excited states (closed circles), with the readout resonator in the ground state (open circle). (i) The qubit is swept adiabatically past the resonator to swap excitations. (ii) Resonator occupation decays to the environment while the qubit holds. (iii) After the resonator is sufficiently depleted, the qubit returns diabatically to its operating frequency. The total duration of the reset protocol is about 250 ns. b Circuit for the bit-flip stabilizer code including reset (R). Measure qubits (Q_M) cyclically apply parity measurements to neighboring data qubits (Q_D) using Hadamard (H) and CZ gates. We add X gates to data qubits to depolarize energy relaxation error. When introducing reset, leakage errors (stars) may be removed from both measure and data qubits, either directly or via transport through the CZ gates (red lines).

Fig. 2. Reset gate benchmarking.

a The qubit frequency trajectory for implementing reset consists of three stages. We plot the ground state infidelity when resetting the first three excited states of the qubit versus swap (b) and vs hold times (c). We include experimental data (points) and theory prediction (solid lines). Reset error versus swap and hold for the experiment (d) and theory (e) show a wide range of optimal parameters. Dashed white lines indicate linecuts for (b) and (c). White circle indicates the point of operation.

Fig. 3. Leakage during the bit-flip code.

The growth in $∣2⟩$ population vs. stabilizer code length. The circuit is run for a number of rounds and terminated with a readout sensitive to $∣2⟩$ population. The experimental data are averaged over measure or data qubits and fitted to an exponential (dashed lines) to extract rates. Further data are included in Supplementary Note 3. The inset shows the 21 qubit chain as implemented on the Sycamore device.

Fig. 4. Injection of leakage.

Detection event fraction when a full $∣1⟩\to ∣2⟩$ rotation is inserted in round 10 after the first Hadamards a on measure qubit 5 and b on the data qubit between measure qubits 4 (circles) and 5 (triangles). Insets show the event fraction across all measure qubits, indicating the traces plotted in the main figure (dashed lines). See Fig. 3 inset for qubit locations.

Fig. 5. Correlations caused by leakage.

p_ij matrices show the strength of non-local correlations in the detected errors. These undesired correlations are significantly reduced with the addition of reset. a The error graph for the bit-flip code, highlighting examples of non-local correlations on both space and time, indicating their corresponding p_ij elements below (boxes). b, c Time-correlations on measure qubit 6, with and without reset. d, e Cross-correlations between measure qubits 5 and 6, with and without reset.

Fig. 6. Logical code performance.

a The logical error rate for 30 rounds vs system size. The error suppression factor Λ_bit is fitted to the data from nine qubits up. b Λ_bit versus code depth, showing that with reset logical error suppression is improved consistently. The error bars indicate the standard deviation error in the fit of error rate versus number of qubits. The threshold for the bit-flip code (unity) is shown as a dashed line. The arrow indicates the data in (a).