To address the challenge posed by noise in real quantum devices, quantum error mitigation techniques play a crucial role. These techniques are resource-efficient, making them suitable for implementation in noisy intermediate-scale quantum devices, unlike the more resource-intensive quantum error correction codes. A notable example of such a technique is Clifford Data Regression, which employs a supervised learning approach. This work investigates two variants of this technique, both of which introduce a non-trivial set of gates into the original circuit. The first variant uses multiple copies of the original circuit, while the second adds a layer of single-qubit rotations. Different characteristics of these methods are analyzed theoretically, such as their complexity, or the scaling of the error with various parameters. Additionally, the performance of these methods is evaluated through numerical experiments, demonstrating a reduction in root mean square error.To address the challenge posed by noise in real quantum devices, quantum error mitigation techniques play a crucial role. These techniques are resource-efficient, making them suitable for implementation in noisy intermediate-scale quantum devices, unlike the more resource-intensive quantum error correction codes. A notable example of such a technique is Clifford Data Regression, which employs a supervised learning approach. This work investigates two variants of this technique, both of which introduce a non-trivial set of gates into the original circuit. The first variant uses multiple copies of the original circuit, while the second adds a layer of single-qubit rotations. Different characteristics of these methods are analyzed theoretically, such as their complexity, or the scaling of the error with various parameters. Additionally, the performance of these methods is evaluated through numerical experiments, demonstrating a reduction in root mean square error.

IEEE Transactions on Quantum EngineeringJan 1, 20260 citations

Extension of Clifford Data Regression Methods for Quantum Error Mitigation

Jordi Pérez-Guijarro, Alba Pagés-Zamora, J.R. Fonollosa

In addressing the challenge posed by noise in actual quantum devices, the application of quantum error mitigation techniques becomes essential. These techniques are resource-efficient, making them viable for implementation in noisy intermediate-scale quantum devices, unlike the resource-intensive quantum error correction codes. A prominent example among these techniques is Clifford Data Regression, which employs a supervised learning approach. This work explores two variants of this technique, both of which add a non-trivial set of gates to the original circuit. The first variant leverages copies of the original circuit, whereas the second approach adds a layer of 1-qubit rotations.In addressing the challenge posed by noise in actual quantum devices, the application of quantum error mitigation techniques becomes essential. These techniques are resource-efficient, making them viable for implementation in noisy intermediate-scale quantum devices, unlike the resource-intensive quantum error correction codes. A prominent example among these techniques is Clifford Data Regression, which employs a supervised learning approach. This work explores two variants of this technique, both of which add a non-trivial set of gates to the original circuit. The first variant leverages copies of the original circuit, whereas the second approach adds a layer of 1-qubit rotations.

IEEE International Conference on Acoustics Speech and Signal ProcessingMar 18, 20242 citations

Quantum error mitigation using energy sampling and extrapolation enhanced Clifford data regression

Zhongqi Zhao, Erik Rosendahl Kjellgren, Sonia Coriani +3 more

Error mitigation is essential for the practical implementation of quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. This work explores and extends Clifford Data Regression (CDR) to mitigate noise in quantum chemistry simulations using the Variational Quantum Eigensolver (VQE). Using the H $_4$ molecule with the tiled Unitary Product State (tUPS) ansatz, we perform noisy simulations with the ibm torino noise model to investigate in detail the effect of various hyperparameters in CDR on the error mitigation quality. Building on these insights, two improvements to the CDR framework are proposed. The first, Energy Sampling (ES), improves performance by selecting only the lowest-energy training circuits for regression, thereby further biasing the sample energies toward the target state. The second, Non-Clifford Extrapolation (NCE), enhances the regression model by including the number of non-Clifford parameters as an additional input, enabling the model to learn how the noisy-ideal mapping evolves as the circuit approaches the optimal one. Our numerical results demonstrate that both strategies outperform the original CDR.Error mitigation is essential for the practical implementation of quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. This work explores and extends Clifford Data Regression (CDR) to mitigate noise in quantum chemistry simulations using the Variational Quantum Eigensolver (VQE). Using the H$_4$ molecule with the tiled Unitary Product State (tUPS) ansatz, we perform noisy simulations with the ibm torino noise model to investigate in detail the effect of various hyperparameters in CDR on the error mitigation quality. Building on these insights, two improvements to the CDR framework are proposed. The first, Energy Sampling (ES), improves performance by selecting only the lowest-energy training circuits for regression, thereby further biasing the sample energies toward the target state. The second, Non-Clifford Extrapolation (NCE), enhances the regression model by including the number of non-Clifford parameters as an additional input, enabling the model to learn how the noisy-ideal mapping evolves as the circuit approaches the optimal one. Our numerical results demonstrate that both strategies outperform the original CDR.

quant-pharXiv:2511.03556Nov 6, 20250 citations

Clifford Perturbation Approximation for Quantum Error Mitigation

Zhang, Ruiqi, Yuguo Shao, Fuchuan Wei +3 more

Quantum error mitigation (QEM) is critical for harnessing the potential of near-term quantum devices. Particularly, QEM protocols can be designed based on machine learning, where the mapping between noisy computational outputs and ideal ones can be learned on a training set consisting of Clifford circuits or near-Clifford circuits that contain only a limited number of non-Clifford gates. This learned mapping is then applied to noisy target circuits to estimate the ideal computational output. In this work, we propose a learning-based error mitigation framework called Clifford Perturbation Data Regression (CPDR), which constructs training sets by Clifford circuits with small perturbations. Specifically, these circuits are parameterized quantum circuits, where the rotation angles of the gates are restricted to a narrow range, ensuring that the gates remain close to Clifford gates. This design enables the efficient simulation of the training circuits using the Sparse Pauli Dynamics method. As a result, CPDR is able to utilize training sets with a better diversity to train the model, compared with previous learning-based QEM models that construct training sets with only Clifford or near-Clifford circuits. Numerical simulations on small-scale Ising model circuits demonstrate that the performance of CPDR dramatically outperforms that of existing methods such as Zero-Noise Extrapolation and learning-based Probabilistic Error Cancellation. Furthermore, using the experimental data from IBM's 127-qubit Eagle processor, our findings suggest that CPDR demonstrates improved accuracy compared to the original mitigation results reported in [Nature 618, 500 (2023)].Quantum error mitigation (QEM) is critical for harnessing the potential of near-term quantum devices. Particularly, QEM protocols can be designed based on machine learning, where the mapping between noisy computational outputs and ideal ones can be learned on a training set consisting of Clifford circuits or near-Clifford circuits that contain only a limited number of non-Clifford gates. This learned mapping is then applied to noisy target circuits to estimate the ideal computational output. In this work, we propose a learning-based error mitigation framework called Clifford Perturbation Data Regression (CPDR), which constructs training sets by Clifford circuits with small perturbations. Specifically, these circuits are parameterized quantum circuits, where the rotation angles of the gates are restricted to a narrow range, ensuring that the gates remain close to Clifford gates. This design enables the efficient simulation of the training circuits using the Sparse Pauli Dynamics method. As a result, CPDR is able to utilize training sets with a better diversity to train the model, compared with previous learning-based QEM models that construct training sets with only Clifford or near-Clifford circuits. Numerical simulations on small-scale Ising model circuits demonstrate that the performance of CPDR dramatically outperforms that of existing methods such as Zero-Noise Extrapolation and learning-based Probabilistic Error Cancellation. Furthermore, using the experimental data from IBM's 127-qubit Eagle processor, our findings suggest that CPDR demonstrates improved accuracy compared to the original mitigation results reported in [Nature 618, 500 (2023)].

arXiv:2412.09518Dec 12, 20240 citations

Learning-Based Quantum Error Mitigation

Armands Strikis, Dayue Qin, Yanzhu Chen +2 more

If noisy-intermediate-scale-quantum-era quantum computers are to perform useful tasks, they will need to employ powerful error mitigation techniques.Quasiprobability methods can permit perfect error compensation at the cost of additional circuit executions, provided that the nature of the error model is fully understood and sufficiently local both spatially and temporally.Unfortunately, these conditions are challenging to satisfy.Here we present a method by which the proper compensation strategy can instead be learned ab initio.Our training process uses multiple variants of the primary circuit where all non-Clifford gates are substituted with gates that are efficient to simulate classically.The process yields a configuration that is near optimal versus noise in the real system with its non-Clifford gate set.Having presented a range of learning strategies, we demonstrate the power of the technique both with real quantum hardware (IBM devices) and exactly emulated imperfect quantum computers.The systems suffer a range of noise severities and types, including spatially and temporally correlated variants.In all cases the protocol successfully adapts to the noise and mitigates it to a high degree.If noisy-intermediate-scale-quantum-era quantum computers are to perform useful tasks, they will need to employ powerful error mitigation techniques.Quasiprobability methods can permit perfect error compensation at the cost of additional circuit executions, provided that the nature of the error model is fully understood and sufficiently local both spatially and temporally.Unfortunately, these conditions are challenging to satisfy.Here we present a method by which the proper compensation strategy can instead be learned ab initio.Our training process uses multiple variants of the primary circuit where all non-Clifford gates are substituted with gates that are efficient to simulate classically.The process yields a configuration that is near optimal versus noise in the real system with its non-Clifford gate set.Having presented a range of learning strategies, we demonstrate the power of the technique both with real quantum hardware (IBM devices) and exactly emulated imperfect quantum computers.The systems suffer a range of noise severities and types, including spatially and temporally correlated variants.In all cases the protocol successfully adapts to the noise and mitigates it to a high degree.

arXiv:2005.07601PRX QuantumNov 10, 2021237 citations

Unified approach to data-driven quantum error mitigation

Angus Lowe, Max Hunter Gordon, Piotr Czarnik +3 more

Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise extrapolation (ZNE) and Clifford data regression (CDR). Here, we propose a scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford circuits (which are classically simulable) and second by varying the noise levels in these circuits. We employ a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8-qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors of 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise extrapolation (ZNE) and Clifford data regression (CDR). Here, we propose a scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford circuits (which are classically simulable) and second by varying the noise levels in these circuits. We employ a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8-qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors of 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.

arXiv:2011.01157Physical Review ResearchJul 28, 202189 citations

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molecule with the tiled Unitary Product State (tUPS) ansatz, we perform noisy simulations with the ibm torino noise model to investigate in detail the effect of various hyperparameters in CDR on the error mitigation quality. Building on these insights, two improvements to the CDR framework are proposed. The first, Energy Sampling (ES), improves performance by selecting only the lowest-energy training circuits for regression, thereby further biasing the sample energies toward the target state. The second, Non-Clifford Extrapolation (NCE), enhances the regression model by including the number of non-Clifford parameters as an additional input, enabling the model to learn how the noisy-ideal mapping evolves as the circuit approaches the optimal one. Our numerical results demonstrate that both strategies outperform the original CDR.