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arXiv:2307.08741 · Physical Review Applied

Characterization of coherent errors in gate layers with robustness to Pauli noise

Noah Kaufmann, Ivan Rojkov, Florentin Reiter

Published Mar 7, 2025• 5 citations
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Abstract

Characterization of quantum devices generates insights into their sources of disturbances. State-of-the-art characterization protocols often focus on incoherent noise and eliminate coherent errors when using Pauli or Clifford twirling techniques. This approach biases the structure of the effective noise and adds a circuit and sampling overhead. We motivate the extension of an incoherent local Pauli noise model to coherent errors and present a practical characterization protocol for an arbitrary gate layer. Notably, the coherent noise estimation is robust to Pauli noise. We demonstrate our protocol on a superconducting hardware platform and identify the leading coherent errors. To verify the characterized noise structure, we mitigate its coherent and incoherent components using a gate-level coherent noise mitigation scheme in conjunction with probabilistic error cancelation. The proposed characterization procedure opens up possibilities for device calibration, hardware development, and improvement of error mitigation and correction techniques.

DOI
10.1103/physrevapplied.23.034014

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Characterization of quantum devices generates insights into their sources of disturbances. State-of-the-art characterization protocols often focus on incoherent noise and eliminate coherent errors when using Pauli or Clifford twirling techniques. This approach biases the structure of the effective noise and adds a circuit and sampling overhead. We motivate the extension of an incoherent local Pauli noise model to coherent errors and present a practical characterization protocol for an arbitrary gate layer. Notably, the coherent noise estimation is robust to Pauli noise. We demonstrate our protocol on a superconducting hardware platform and identify the leading coherent errors. To verify the characterized noise structure, we mitigate its coherent and incoherent components using a gate-level coherent noise mitigation scheme in conjunction with probabilistic error cancelation. The proposed characterization procedure opens up possibilities for device calibration, hardware development, and improvement of error mitigation and correction techniques.
Abstract Coherent noise can be much more damaging than incoherent (probabilistic) noise in the context of quantum error correction. One solution is to use twirling to turn coherent noise into incoherent Pauli channels. In this Article, we show that some of the coherence of the noise channel can actually be used to improve its logical fidelity by simply sandwiching the noise with a chosen pair of Pauli gates, which we call Pauli conjugation. Using the optimal Pauli conjugation, we can achieve a higher logical fidelity than using twirling and doing nothing. We devise a way to search for the optimal Pauli conjugation scheme and apply it to Steane code, 9-qubit Shor code and distance-3 surface code under global coherent ZZZ Z noise. The optimal conjugation schemes show improvement in logical fidelity over twirling while the weights of the conjugation gates we need to apply are lower than the average weight of the twirling gates. In our example noise and codes, the concatenated threshold obtained using conjugation is consistently higher than the twirling threshold and can be up to 1.5 times higher than the original threshold where no mitigation is applied. Our simulations show that Pauli conjugation can be robust against gate errors. With the help of logical twirling, the undesirable coherence in the noise channel can be removed and the advantages of conjugation over twirling can persist as we go to multiple rounds of quantum error correction.
Accurate noise characterization is essential for reliable quantum computation. Effective Pauli noise models have emerged as powerful tools, offering a detailed description of the error processes with a manageable number of parameters, which guarantees the scalability of the characterization procedure. However, a fundamental limitation in the learnability of Pauli eigenvalues impedes a full high-accuracy characterization of both general and effective Pauli noise models, thereby restricting, e.g., the performance of noise-aware error mitigation techniques. We introduce multilayer cycle benchmarking (MLCB), an enhanced characterization protocol that improves the learnability associated with effective Pauli noise models by jointly analyzing multiple layers of Clifford gates. We show a simple experimental implementation and demonstrate that, in realistic scenarios, MLCB can reduce unlearnable noise degrees of freedom by up to 75 % , improving the accuracy of sparse Pauli-Lindblad noise models and boosting the performance of error mitigation techniques like probabilistic error cancellation. Our results highlight MLCB as a scalable, practical tool for precise noise characterization and improved quantum computation.
Quantum noise is a central challenge in quantum computing across many applications. Extensive work has examined how qubits couple to their environment, leading to decoherence and irreversible relaxation. This work studies a continuous coherent noise model for quantum circuits and compares it with a discrete Pauli model. The focus is on small, coherent gate errors that accumulate across circuit depth. These errors are modeled as random rotations on the Bloch sphere using a von Mises-Fisher distribution. In the small-angle limit, the model reduces to an isotropic Gaussian distribution. We test the model on quantum error-correction circuits based on the [[5,1,3]] and [[7,1,3]] codes. A variant of Grover's search circuit with different qubit counts is also examined. To enable fair comparison, we introduce a model-independent matching scheme. Pauli and continuous noise channels are aligned using the binary entropy at readout. This isolates the effect of noise structure at fixed uncertainty. An approximate analytical method for the propagation of coherent errors is also developed. The method tracks error distributions both on Clifford and non-Clifford circuits without full Monte Carlo sampling. It reduces simulation cost while preserving accuracy for circuit-level error estimates. The approximation is validated against brute-force simulations, identifying its regime of validity with Clifford and non-Clifford circuits under error correction. Our results show that continuous coherent noise and Pauli noise lead to comparable logical-error trends in the stabilizer-code circuits studied here. This behavior is consistent with syndrome extraction projecting a general single-qubit error onto Pauli-error subspaces before recovery. They also show where simplified propagation models work well and where their accuracy is reduced.
Using near-term quantum computers to achieve a quantum advantage requires efficient strategies to improve the performance of the noisy quantum devices presently available. We develop and experimentally validate two efficient error mitigation protocols named ``Noiseless Output Extrapolation" and ``Pauli Error Cancellation" that can drastically enhance the performance of quantum circuits composed of noisy cycles of gates. By combining popular mitigation strategies such as probabilistic error cancellation and noise amplification with efficient noise reconstruction methods, our protocols can mitigate a wide range of noise processes that do not satisfy the assumptions underlying existing mitigation protocols, including non-local and gate-dependent processes. We test our protocols on a four-qubit superconducting processor at the Advanced Quantum Testbed. We observe significant improvements in the performance of both structured and random circuits, with up to 86% improvement in variation distance over the unmitigated outputs. Our experiments demonstrate the effectiveness of our protocols, as well as their practicality for current hardware platforms.
Noise characterization of multi-qubit entangling Clifford operations is a key practical bottleneck for quantum error mitigation and for the calibration, validation, and optimization of quantum error-correction protocols, especially in the presence of state preparation and measurement (SPAM) errors. Although cycle benchmarking can isolate some Pauli error components, it cannot resolve the problem of coupled error parameters, which leads to unlearnable degrees of freedom even in simple noisy gates, not to mention general nnn-qubit Clifford gates. Here we introduce mid-circuit-measurement-based generalized cycle benchmarking, a framework that makes otherwise unidentifiable Pauli fidelities and non-Markovian noise learnable via repeated measurements and classical post-processing. Applying the deferred feed-forward principle to generalized cycle benchmarking, we show that an insertion of mid-circuit measurements can reverse Pauli cycles induced by a general Clifford gate. This fact enables us to reveal a Pauli-noise learnability condition for Clifford gates. Assuming sufficient state preparation quality, we numerically demonstrate the feasibility of characterizing the previously unlearnable noise components. We implement the protocol on superconducting quantum processing units and validate its effectiveness in disambiguating the coupled noise components, benchmarked against conventional tomography. Finally, we observe consistent measurement-induced bit-flip bias and non-Markovian correlations, which define a range of applicability for the Pauli noise model and the proposed noise-characterization protocol.
Complex quantum systems and their various applications are susceptible to noise of coherent and incoherent nature. Characterization of noise and its sources is an open, key challenge in quantum technology applications, especially in terms of distinguishing between inherent incoherent noise and systematic coherent errors. In this paper, we study a scheme of repeated sequential measurements that enables the characterization of incoherent errors by reducing the effects of coherent errors. We demonstrate this approach using a coherently controlled nitrogen vacancy in diamond, coupled to both a natural nuclear spin bath (non-Markovian) and to experimentally controlled relaxation through an optical pumping process (nearly Markovian). Our results show mitigation of coherent errors both for Markovian and non-Markovian incoherent noise profiles. We apply this scheme to the estimation of the dephasing time (T2*) due to incoherent noise. We observe an improved robustness against coherent errors in the estimation of dephasing time (T2*) compared to the standard (Ramsey) measurement.