Scalable in situ qubit calibration during repetitive error detection

We present a method to optimize qubit control parameters during error detection which is compatible with large-scale qubit arrays. We demonstrate our method to optimize single or two-qubit gates in parallel on a nine-qubit system. Additionally, we show how parameter drift can be compensated for during computation by inserting a frequency drift and using our method to remove it. We remove both drift on a single qubit and independent drifts on all qubits simultaneously. We believe this method will be useful in keeping error rates low on all physical qubits throughout the course of a computation. Our method is $O\left(1\right)$ scalable to systems of arbitrary size, providing a path towards controlling the large numbers of qubits needed for a fault-tolerant quantum computer.

Figure 1

Error detection circuit and gate error detection. (a) Linear chain of nine superconducting qubits with nearest-neighbor coupling. (b) Error detection circuit for the bit-flip repetition code, where data qubits (black and gold) hold the quantum state, and measurement qubits [red (dark), blue (dark gray), green (light gray), and orange (gray)] are used to detect errors. Here we perform three experiments i–iii, where rotation angles are tuned from their optimal values and errors are recorded. Gate-induced bit errors on data qubits (i and ii) are copied to measurement qubits through cz gates where they are detected as detection events; see text. Gate errors on measurement qubits (iii) are localized to those qubits. Hadamard gates are physically implemented with variable phase $\frac{\pi }{2}$ rotations. (c) Detection event fraction $\zeta$ vs variable-angle $X$ gates on data (i and ii) and measurement (iii) qubits. Each data point is the average of 6000 instances of eight rounds of detection. Change in gate parameters from their ideal increases $\zeta$. Change in $\zeta$ is localized to measurement qubits near the gate being varied, as $\zeta$ for unaffected qubits is constant.

Figure 2

Parallel gate optimization using independent hardware groupings. (a) Schematic of ADEPT for optimizing gates on measurement qubits. Gate parameters are chosen and error detection is run to determine $\zeta$ that is used as an error metric for the Nelder-Mead optimization algorithm, which chooses new gate parameters. (b) Parallel optimization of measurement qubit gate parameters. Each measurement qubit is optimized independently, and each iteration is 4500 instances of eight rounds of detection. Average $\zeta$ is improved from 0.202 to 0.179. (c) Using ADEPT to optimize cz gates. As cz gates include data qubits, both neighboring measurement qubits will change $\zeta$ with changing gate parameters. To avoid error overlap, we optimize cz that are well separated (see text). (d) Parallel Nelder-Mead optimization of two cz gates. The metric is the average $\zeta$ of the measurement qubits that neighbor the data qubit in the cz. Each iteration is 4500 instances of eight rounds of detection. Average $\zeta$ is improved from 0.350 to 0.172.

Figure 3

Tracking and negating frequency drift with ADEPT. (a) A slow sinusoidally varying frequency drift of up to 10 MHz is inserted onto the orange (gray) measure qubit. (b) Uncorrected, this will cause $\zeta$ of the orange (gray) qubit to double within 20 iterations (with each iteration 12 000 instances of eight rounds of detection), and eventually saturate near the randomization limit of 0.5. (c) The frequency drift can be compensated for by feeding $\zeta$ into a following algorithm. The algorithm samples points above and below the ideal bias value, and fits these points to a parabolic error model (see Appendix pp3). (d) The drift-following algorithm tracks and compensates for the inserted frequency drift through one oscillation. The other algorithms produce no compensation, as no drift is inserted. (e) The measured $\zeta$ remain flat throughout the experiment, in contrast to (b).

Figure 4

Tracking and negating independent frequency drift on all qubits. (a) A different amplitude frequency drift is inserted onto each of nine qubits in the device. (b) ADEPT is used in conjunction with a frequency-following algorithm on all qubits by interleaving the active hardware pattern. In i, $\zeta$ is used to inform the frequency-following algorithm for the measurement qubits. In ii and iii $\zeta$ is used to inform the frequency-following algorithm of neighboring data qubits. (c) Compensation algorithms associated with each hardware pattern i–iii are active one pattern a time in a sequential repeating fashion. (d) Independent frequency drift on each qubit is tracked independently. The traces have been spaced along the $y$ axis for viewing clarity. Each data point is 6000 instances of eight rounds of detection. (e) $\zeta$ is stabilized throughout the course of the experiment, indicating that all qubits have their frequency drift compensated for.

Figure 5

Hardware patterns and local groupings. (a) Error detection circuit, where gate color corresponds to which measurement qubit will detect errors from that gate. (b) First hardware pattern, where each of four groupings contains a measurement qubit and the single-qubit operations for that qubit. Errors from these gates will not propagate back to neighboring qubits. The relative $\zeta$ from the measurement qubit can be used to inform changes in gate parameters. (c), (d) Second and third hardware pattern, where each grouping contains one data qubit and up to two measurement qubits. Groupings contain single-qubit gates on measurement qubits and cz gates. The $\zeta$ of measurement qubits within each grouping can be used to inform gate parameters for gates within the grouping.

Figure 6

Emulation of continuously running error detection. (a) Error detection experiments are composed of four steps: initialization, error detection, ending the code, and system reset. (b) $N$ rounds of continuous error detection would involve initialization, $N$ rounds of error detection, then ending the code and resetting the system. (c) We emulate continuous error detection by running an ensemble of experiments that each contain eight rounds of detection, such that we accumulate $N$ total error detection rounds. (d) Ideally, ADEPT would run without ever needing to end the code of reset the system. Statistics over $N$ rounds of detection would be used to calculate $\zeta$, and then parameters would be updated on-the-fly. (e) We emulate this behavior by running ensembles of experiments to gather $N$ rounds of error detection to compute $\zeta$, and then updating parameters between ensembles. In main text Figs. 3 and 4, the inserted bias is updated between ensembles of experiments.

Figure 7

Parabolic error model. Errors take the form of Eq. (C1). After some time, the ideal parameter $x_0$ at $t=0$ drifts to $x_1$ at $t=1$. Detection fraction $\zeta$ is sampled at $x_0\pm \mathrm{\Delta }x$ defined in Eq. (C3). $\zeta$ is increased by $F{\zeta }_0$ when using this sampling. After fitting to a parabolic error model, a new parameter $x_1$ is determined by adding $\delta x$ as defined in Eq. (C6).