Bayesian mitigation of measurement errors in multiqubit experiments

TL;DR

The paper addresses measurement error mitigation in multiqubit experiments by proposing a Bayesian framework that uses full detector information, including analog IQ data, to infer true qubit state distributions. It develops subspace-reduction heuristics to manage the exponential state space and introduces a two-variable Bayesian update per pair of populations for scalable inference. It demonstrates improvements on superconducting devices and shows that using analog readout data and discretized detector response functions yields noticeable gains, while maintaining manageable classical compute times. Compared to IBU and Mthree, the Bayesian method, especially with analog data, provides higher accuracy and can be effectively integrated on top of existing mitigation schemes.

Abstract

In Phys. Rev. A 108, L060402 (2023), we introduced a Bayesian measurement error mitigation algorithm, which leveraged complete information from the readout signal, and validated the protocol on a quantum device with five superconducting qubits. Here, we present an improved algorithm's implementation, tailored for multiqubit experiments on near-term superconducting qubit quantum devices. In particular, we provide a detailed algorithm workflow, from calibrating the detector response functions to the postprocessing of measurement outcomes, offering a computationally efficient solution for the output size typical of current quantum computing devices. We show how the numerical representation of the noise function affects the performance of the error mitigation algorithm and test the convergence criteria. We benchmark our protocol on actual quantum computers with superconducting qubits, where the readout signal encodes the measurement information as unprocessed analog data before qubit state assignment. Finally, we compare the performance of our algorithm against other measurement error mitigation methods, such as iterative Bayesian unfolding and the Mthree method, and show how our method can be integrated on top of other readout error mitigation protocols.

Figures (7)

• Figure 1: (a) Workflow of standard readout error mitigation. (b) Workflow of Bayesian measurement error mitigation. (c) Workflow of Bayesian measurement error mitigation when integrating other readout error mitigation techniques. (d) Pictorial sketch of the heuristic algorithm to perform Bayesian measurement error mitigation in a multiqubit experiment. In the first step (I), we collect the noisy bitstrings counts and reduce the $2^{\mathrm{N_q}}$ outcome subspace to mitigate to the $M$ measured noisy counts for which we have at least one measurement. In step (II), we select two populations to use as variables while keeping all others constant, and apply Bayesian measurement error mitigation to obtain two new mitigated values. The Bayesian update is performed for all possible pairs. Step (II) is repeated until a specified convergence criterion or exit condition is met, resulting in the full set of mitigated values (III).
• Figure 2: (a): (Top) Measurement clouds in the IQ plane, where each point corresponds to a single shot. The blue cloud represents measurements of the "0" state, i.e., when no operation is performed on the qubit besides the measurement. The orange cloud represents measurements of the "1" state, where a single X-gate is applied to the qubit before measurement. (Bottom) Projected histograms from the IQ plane onto the Q-axis.
• Figure 3: (a): Quantum circuit for the preparation of a random multiqubit string: single qubit X gates are applied on the specific qubits ti flip. (b): Comparison between the mitigated (orange) and unmitigated (gray) success probability of measuring the prepared bitstring as a function of the string length. (c): Comparison between the mitigated (orange) and unmitigated (gray) success probability of measuring a prepared 16 qubits bitstring as a function of the number of shots. The distribution shown in (b) and (c) correspond to 20 different random bitstrings, with the middle bar representing the mean across the different realization, while the upper and lower bars are the highest and lowest success probabilities.
• Figure 4: (a): Comparison of mitigated success probabilities for measuring a prepared bitstring as a function of string length, for different number of partitions of the Q-axis. The results are obtained with $10^3$ shots and averaged over 50 realizations, with error bars corresponding to the standard deviations. (b): Success probability of measuring a single qubit as a function of $\mathrm{N_{\mathrm{bin}}}$ with the inset showing only the average values. (c): Success probability of measuring a 20-qubit string as a function of $\mathrm{N_{\mathrm{bin}}}$. The results shown in (b) and (c) are obtained with $10^3$ shots and show the results for all the 50 realizations. The middle horizontal bar represent the average mitigated success probability, while the upper and lower bars are the maximum and minimum of the measured samples.
• Figure 5: (a): Mitigated success probability as a function of the Hamming distance between population pairs undergoing the Bayesian update cycle. (b): Total variation distance between the mitigated probabilities after each Bayesian iteration as a function of the iteration number for different string lengths. (b): Number of iterations required by our Bayesian mitigation scheme to reach the convergence condition. (c): Time required by our Bayesian mitigation scheme to mitigate the outcomes of the string preparation experiment as a function of the number of qubits measured. The results in (a), (b), and (c) show the results obtained for 50 realizations with the middle horizontal bar line representing the mean of the distributions. In all cases the mitigation algorithm is applied using $\mathrm{N}_{\mathrm{bin}}=20$ as discretization parameter for the detector response function.