In situ calibration of unitary operations during quantum error correction

Abstract

Quantum error correction uses the measurement of syndromes and classical decoding algorithms to estimate the location and type of errors while protecting the encoded quantum bits. Here we consider how prior information and Bayesian updates can play a critical role in improving the performance of QEC in the scenario of a particularly noisy qubit. This allows for leveraging even distance codes, which typically are less valued in QEC, to handle the noisy qubit, changing the power-law scaling of the logical error rate with the baseline physical error rate. A crucial component of this is updating the prior by real time feeding of decoder outputs into a approximate Kalman filter. Thus our approach provides a bootstrap to the actual error rates. We show this via simulation of the full closed-loop system: starting from uniform priors, the update procedure gradually learns site-specific error rates, enabling the decoder to outperform a fixed-prior baseline. In turn, we show that this enables in situ calibration of unitary operations via injection of gate set tomography operations with only moderate overhead in the more typical scenario of low noise qubits.

Figures (4)

• Figure 1: The intuition behind this work, for the case of the toy example: the [[4,2,2]] code. (Top left) The [[4,2,2]] code, composed of four data qubits (blue dots) has a $Z$-stabilizer (central square) which touches all of them. The top right qubit is especially error-prone leading to a bit flip error (lightning), stabilizer measurement registers an error (red). (Bottom left) The control flow diagram of this work. A decoder using prior error rate information can correct the error shown above without inducing a logical error, and the decoding result can then be used to update the prior. (Right) Circuits showing application of prior-assisted decoding to improve error correction performance (top) by preferring corrections where priors are high, and to in situ gate calibration (bottom) by inferring the value of gate parameters from decoding data while protecting logical information.
• Figure 2: (Left) The rotated surface code. Data qubits are vertices of the graph whereas $X$-checks ($Z$-checks) are on the grey (white) faces. (Right) Logical error rate for $d=3$ up to $d=6$ rotated surface codes for each of the three cases from the main text, which we plot on a log-log scale versus $\varepsilon$ to extract the scaling of logical state preservation with system-wide error rate. For $d=3$ and $d=4$, the logical error rate for case 1 (case 2) has the same scaling for both distances, i.e. as $\varepsilon^2$ ($\varepsilon^1$), but case 3, however, $d=4$ scales better than $d=3$, as we expect from Statement 4. Specifically, the case 3 and case 1 lines are nearly indistinguishable for $d=4$ -- the code handles both an unknown and known error with $\varepsilon^2$ logical error. Analogous behavior holds for $d=5$ and $d=6$, but the scaling of all error rates increased by a power of $\varepsilon$.
• Figure 3: Application of the Kalman update on a distance 4 rotated surface code showing the enhanced performance after updating. (Left) The workflow: noisy quantum hardware gives rise to syndrome measurements, then the decoder decodes the syndrome and uses the resulting error estimate to update its priors and, possibly, apply a correction. (Top right) We plot the estimated bit-flip probabilities updated according to Eq. \ref{['eq:Kalman_update']}. We see that the estimated rate for qubit 1 increases as it should, and qubit 2 does as well. (Bottom right) We then compare the performance of the decoder running on the priors at the beginning and end of the protocol and see different scaling with $\varepsilon$.
• Figure 4: Applying error correction with updated priors to calibration in the unrotated surface code with distance 4. As explained in the main text, we switch to the unrotated surface code because it does not suffer from the degeneracy shown in Fig. \ref{['fig:Kalman']}(top right) as it has no weight 2 stabilizers, leaving the rotated surface code to future work. (left top) The applied rotation angle $\theta$. Over the course of the experiment, it increases until reaching $\theta_{\rm target} - \theta_0$ (dashed line), and then oscillates. (left bottom) Estimated bit flip rates and the true bit flip rate for qubit 1 (dashed blue) over the course of the experiment. The actual bit flip rate from the gate $U_\theta$ decreases as a result of calibration feedback, while the prior is also updated, both ultimately stabilize around the target (dashed grey). (right) Decoder performance after calibration, fixing the actual and estimated bit flip rate of qubit 1 and varying the error rate $\varepsilon$ of all other qubits. Data agrees with linear fit in log-log space (dashed orange) corresponding to $\varepsilon^2$ scaling.