Experimental Bayesian estimation of quantum state preparation, measurement, and gate errors in multiqubit devices

We introduce a Bayesian method for the estimation of single-qubit errors in quantum devices, and use it to characterize these errors on three superconducting 27-qubit devices. We self-consistently estimate up to seven parameters of each qubit's state preparation, readout, and gate errors, analyze the stability of these errors as a function of time, and demonstrate easily implemented approaches for mitigating different errors before a quantum computation experiment. On the investigated devices we find non-negligible qubit reset errors that cannot be parametrized as a diagonal mixed state, but manifest as a coherent phase in a superposition with a small contribution from the qubit's excited state. We are able to mitigate such errors by applying prerotations on the initialized qubits, which we demonstrate with multiqubit entangled states. Our results show that Bayesian estimation can resolve small parameters—including those pertaining to quantum gate errors—with a high relative accuracy, at a lower measurement cost as compared with standard characterization approaches.

Figure 1

Two components of the quantum state preparation (initialization) errors; (a) $x_0$ and (b) $y_0$, as measured using three parallelization modes in one single job on the IBM Quantum device ibmq_paris, accessed via the cloud. The estimation method was Bayesian as described in the text (and in Sec. 2d in detail), with five state preparation and measurement parameters [Eq. (32)] estimated self-consistently. As indicated in the legend, the parameters are estimated either for all qubits in parallel, in two parallel experiments of physically disconnected qubits, or completely sequentially (where only one qubit is being operated on during a given experiment). The data is consistent with no clear correlations in the measurement results between the qubits. The error bars are calculated within the Bayesian approach as in Eq. (31). Qubit 5 was removed from the analysis because of excessive errors (possibly due to interaction with a transient two-level system in the environment).

Figure 2

(a) The initial state component $z_0$ and (b) one POVM parameter corresponding to classical measurement errors ${\pi }_z$ as estimated in the same experiment of Fig. 1. For an ideal readout without errors, ${\pi }_z=0.5$. Here ${\pi }_z<0.5$ for all qubits, which indicates that measuring 0 when the qubit is in $|1\rangle$ is slightly more probable than the reverse error, consistent with $T_1$ decay events during measurement. See the text for a discussion of the correlation in the error bars seen for some of the qubits, between the classical state preparation and readout error parameters.

Figure 3

(a) The transverse $[{\rho }_0$ of Eq. (34)] and (b) the initial longitudinal ($z_0$) components of each qubit in the 27-qubit IBM Quantum device ibmq_paris, accessed via the cloud. Seven identical experiments were run at different times within a window of 8 hours. The plot depicts with horizontal bars the minimal, mean and maximal values estimated for each qubit, while the “violin” shape indicates the averaged probability shape (being narrow near less frequently observed values). Typical quantum initialization errors (${\rho }_0\sim 0.05$) are larger than the classical errors (given by $1-z_0\sim 0.01$), and the initialization errors are mostly relatively stable for time scales of hours. Qubit 5 was removed from the analysis because of excessive measurement errors.

Figure 4

(a) The plane polar angle (${\phi }_0$), and (b) the initial transverse component (${\rho }_0$), both defined in Eq. (34), of the first four qubits (chosen arbitrarily), plotted vs time for the same experiments as in Fig. 3. Seven identical experiments were run at different times within a window of 8 hours. The initialization errors can be seen to be relatively stable for time scales of hours.

Figure 5

Two components of the quantum state preparation errors, (a) $x_0$ and (b) $y_0$, for each qubit in the 27-qubit IBM Quantum device ibmq_toronto, in an experiment of estimation and mitigation run in two steps. The first estimation starting (arbitrarily) at $t=0$, was used to calculate the parameters needed for mitigation of the state preparation errors. The second experiment included two estimations of $x_0$ and $y_0$ parameters, with and without their mitigation by rotation gates. As can be seen, the mitigation reduced the initialization error in the device. Here, qubit 18 was removed from the analysis because of excessive errors. See the text for a detailed description of the protocol employed.

Figure 6

Comparison of expectation values obtained using different possible steps of mitigation, in an experiment performed on the 27-qubit IBM Quantum device ibmq_sydney, accessed via the cloud. In each experiment the $N$-qubit cat state of Eq. (36) was prepared (with some inevitable noise), and then the expectation value of the $N$-qubit operator $\otimes X^N$ was measured, while $N$ was increased from step to step. Each expectation value was calculated four times; without any mitigation, with prerotation of the participating qubits in order to mitigate the estimated state preparation errors, with mitigation of the readout errors using the estimated readout error probabilities, and with both state preparation and readout mitigation. The ideal expectation value is shown with a dashed line. These results show that the estimated SPAM parameters of the device were accurate enough to enable non-negligible improvements of the results via error mitigation.

Figure 7

(a) The gate depolarization error parameter $[\varepsilon$ of Eq. (18)], and (b) gate rotation error parameter $[\theta$ of Eq. (16)], of each qubit in the 27-qubit IBM Quantum device ibmq_sydney. As indicated in the legend, two experiments are shown, with the first experiment starting (arbitrarily) at $t=0$. For the second experiment, a small heuristic correction to the gate pulse amplitude was applied based on the first experiment's estimated rotation error (see the text for more details). As can be seen, this ad hoc correction has significantly reduced the overrotation error for most qubits. Here, the Bayesian estimation method described in the text, with seven state preparation, measurement and gate error parameters were self-consistently estimated using eight gate sequences [Eq. (37)]. Here, qubits 3 and 18 were removed from the analysis because of excessive measurement errors.

Figure 8

The two single-qubit gate error parameters of each qubit in the 27-qubit IBM Quantum device ibmq_sydney, as a function of the powers that the $X_{90}$ gate is taken to, with $n$ being the exponent in ${\left(X_{90}\right)}^{4n}$ and ${\left(X_{90}\right)}^{4n+1}$. It can be seen that for these parameters, the estimated $\theta$ values typically converge for $n\gtrsim 6$, and the same holds for $\varepsilon$, with a few exceptions.

Figure 9

Bottom: The ping-pong comparison experiment for all qubits of the 27-qubit IBM Quantum device ibmq_sydney. Top: The depolarization parameter estimated using the Bayesian method in the same experiment, shown for reference. For most qubits, despite a few notable exceptions, the estimation of $\theta$ using the Bayesian approach is close (within statistical fluctuations) to the estimation using an extended ping-pong method with variation of the gate amplitude. This serves as an independent validation of the Bayesian estimation parametrization and experiments.

Figure 10

The extended ping-pong experiment with a variation of the pulse amplitude, shown corresponding to a single qubit (number 8), for the same experimental run as Fig. 10. (a) The experiment points and fit curve of Eq. (C2) for $A=1.0157$, and (b) the experiment points and fit curve of Eq. (C3).