Q-Cluster: Quantum Error Mitigation Through Noise-Aware Unsupervised Learning

TL;DR

This work addresses the challenge of mitigating quantum noise in the pre-fault-tolerant era by introducing Q-Cluster, an unsupervised-learning-based QEM method that reshapes noisy bit-string distributions under a bit-flip model. The method clusters measured bit-strings using Hamming distance, computes centroids via qubit-wise majority vote, and applies Bayesian redistribution to reverse noise, with an iterative scheme to determine the number of clusters. To tackle real hardware, it integrates noise-tailoring through Dynamical Decoupling and Pauli Twirling and uses an ExtraTrees regressor to estimate an effective bit-flip rate $p_e$ from calibration and circuit features, enabling accurate mitigation on IBM devices. Empirically, Q-Cluster improves fidelity by about 1.46× on low-entropy benchmarks and outperforms state-of-the-art QEM methods M3, HAMMER, and QBEEP by 1.29×, 1.47×, and 2.65× respectively, highlighting the value of combining unsupervised learning, noise-tailoring, and data-driven error-rate estimation for practical quantum advantage.

Abstract

Quantum error mitigation (QEM) is critical in reducing the impact of noise in the pre-fault-tolerant era, and is expected to complement error correction in fault-tolerant quantum computing (FTQC). In this paper, we propose a novel QEM approach, Q-Cluster, that uses unsupervised learning (clustering) to reshape the measured bit-string distribution. Our approach starts with a simplified bit-flip noise model. It first performs clustering on noisy measurement results, i.e., bit-strings, based on the Hamming distance. The centroid of each cluster is calculated using a qubit-wise majority vote. Next, the noisy distribution is adjusted with the clustering outcomes and the bit-flip error rates using Bayesian inference. Our simulation results show that Q-Cluster can mitigate high noise rates (up to 40% per qubit) with the simple bit-flip noise model. However, real quantum computers do not fit such a simple noise model. To address the problem, we (a) apply Pauli twirling to tailor the complex noise channels to Pauli errors, and (b) employ a machine learning model, ExtraTrees regressor, to estimate an effective bit-flip error rate using a feature vector consisting of machine calibration data (gate & measurement error rates), circuit features (number of qubits, numbers of different types of gates, etc.) and the shape of the noisy distribution (entropy). Our experimental results show that our proposed Q-Cluster scheme improves the fidelity by a factor of 1.46x, on average, compared to the unmitigated output distribution, for a set of low-entropy benchmarks on five different IBM quantum machines. Our approach outperforms the state-of-art QEM approaches M3 [24], Hammer [35], and QBEEP [33] by 1.29x, 1.47x, and 2.65x, respectively.

Figures (9)

• Figure 1: Q-Cluster workflow. Stage A: transpilation of the input circuit to the target quantum computer. Stage B: application of noise tailoring techniques, Dynamical Decoupling and Pauli Twirling, to shape quantum noise into a local depolarizing model (Sec. \ref{['sec:NoiseTailor']}). Stage C: execution of the noise-tailored circuits on the target device, and the results, along with noise and timing characteristics, are collected. Stage D: Approximation of the tailored noise effects as a bit-flip noise model with an effective error rate $p_e$ (Sec. \ref{['subsec:pe']}). Stage E: Noise mitigation in the output probability distribution using noise-aware clustering (Sec. \ref{['sec:cluster']}) and noise-aware redistribution (Sec. \ref{['sec:redist']}).
• Figure 2: Illustrative example of Q-Cluster. (a) shows the ideal distribution, while (b) shows the noisy distribution after applying a bit-flip noise with error rate of p = 0.15. The noisy distribution serves as input to Q-Cluster. (c) shows the result after the first iteration of Q-Cluster, where the number of clusters (k) is 1. (d) and (e) show the resulting distributions of Q-Cluster for k = 2 and k = 3, respectively. (Bit-strings are sorted according to Hamming distance from '000000')
• Figure 3: Q-Cluster fidelity improvement with different parameters (Higher Improvement; better mitigation). Each point on the plot shows the average result over 10 randomly generated probability distributions, where the number of qubits is 14 and $d$ is the number of dominant states in the ideal distribution. (a) Effect of different bit-flip error rates. (b) Effect of mis-estimated bit-flip error rate $p$ (actual p = 0.2). (c) Effect of different stopping thresholds..
• Figure 4: Performance of Q-Cluster when the user provides the expected number of clusters (Exact Clusters). The experimental set up is the same as used in Fig. \ref{['fig:Sensitivity Analysis']} with $p = 0.15$. We also test the effects under-estimating the number of clusters by 25% and 50% (Clusters -25% and Clusters -50%) or overestimating the number of clusters by same amount (Clusters +25% and Clusters +50%). We contrast this with Q-Cluster's iterative Cluster finding approach.
• Figure 5: The feature importance scores of the trained ExtraTreesRegressor model, showing the relative significance of each feature in estimating the effective error rate $p_e$. Among them, ESP has the highest importance, followed by the number of measurements, the number of qubits, and entropy.