Error Mitigation for Universal Gates on Encoded Qubits
Open accessAbstract
The Eastin-Knill theorem states that no quantum error-correcting code can have a universal set of transversal gates. For Calderbank-Shor-Steane codes that can implement Clifford gates transversally, it suffices to provide one additional non-Clifford gate, such as the T gate, to achieve universality. Common methods to implement fault-tolerant T gates, e.g., magic state distillation, generate a significant hardware overhead that will likely prevent their practical usage in the near-term future. Recently, methods have been developed to mitigate the effect of noise in shallow quantum circuits that are not protected by error correction. Error mitigation methods require no additional hardware resources but suffer from a bad asymptotic scaling and apply only to a restricted class of quantum algorithms. In this Letter, we combine both approaches and show how to implement encoded Clifford+T circuits where Clifford gates are protected from noise by error correction while errors introduced by noisy encoded T gates are mitigated using the quasiprobability method. As a result, Clifford+T circuits with a number of T gates inversely proportional to the physical noise rate can be implemented on small error-corrected devices without magic state distillation. We argue that such circuits can be out of reach for state-of-the-art classical simulation algorithms.
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Piveteau et al. mitigated the error for universal gates on programmed qubits.
“Similarly, Piveteau etal. mitigated the error for universal gates on programmed qubits [55].”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [55]
The paper states on p. 1 that it mitigates errors introduced by noisy encoded T-gates in encoded Clifford+T circuits, enabling a universal gate set. It discusses encoded qubits, not “programmed qubits,” and does not claim to mitigate errors in every gate of the set. (LLM, conf=0.86)
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“[41] uses quantum error mitigation for implementing T -gate without magic-state distillation and shows efficient characterization methods for the errors of T -gate under the assumption that logical Clifford operations are perfect.”
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Optimal Wire Cutting With Classical Communication
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“Additionally, mitigating and understanding sources and types of runtime or algorithmic errors may allow for reducing the threshold for which fault-tolerant quantum algorithms can be applied [62, 63] by lowering the target total fidelity at the cost of additional sampling overhead or runtime overhead.”
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“Recently, some proposals for integrating error correction and error mitigation techniques, thus moving towards fault tolerant quantum architectures, have also been put forward [11, 12, 26].”
BackgroundAutomated quantum error mitigation based on probabilistic error reduction
McDonough, B., Mari, A., Shammah, N., et al. (2022) · IEEE/ACM International Workshop on Quantum Computing Software (QCS) · doi:10.1109/qcs56647.2022.00015 · Cited by 14 · see citing works
Zero Noise Extrapolation on Logical Qubits by Scaling the Error Correction Code Distance
Wahl, M., Mari, A., Shammah, N., Zeng, W. J., & Ravi, G. S. (2023) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce57702.2023.00103 · Cited by 14 · see citing works
“Recently, PEC has been theoretically applied on logical qubits to reduce the noise of logical gates and effectively increase the code distance [31], [40].”
Virtual quantum resource distillation: General framework and applications
Takagi, R., Yuan, X., Regula, B., & Gu, M. (2024) · Physical Review A · doi:10.1103/physreva.109.022403 · Cited by 13 · see citing works
Error Suppression for Arbitrary-Size Black Box Quantum Operations
Lee, G., Hann, C. T., Puri, S., Girvin, S. M., & Jiang, L. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.190601 · Cited by 13 · see citing works
Liu, Z., Zhang, X., Fei, Y.-Y., & Cai, Z. (2025) · PRX Quantum · doi:10.1103/prxquantum.6.020325 · Cited by 13 · see citing works
“This provides one of the first frameworks that seamlessly combines QEM and QEC beyond the concatenation of QEC and QEM [20–22].”
InfluentialCompilation of a simple chemistry application to quantum error correction primitives
Blunt, N. S., Gehér, G. P., & Moylett, A. E. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013325 · Cited by 11 · see citing works
“…include implementing basic logical gates: implementing Pauli gates through transversal operations, non-Pauli Clifford gates through lattice surgery techniques [9–15], and non-Clifford gates initially through error mitigation techniques [16] and later through magic state distillation [17–19].”
BackgroundMethodInfluentialAkahoshi, Y., Toshio, R., Fujisaki, J., Oshima, H., Sato, S., & Fujii, K. (2025) · PRX Quantum · doi:10.1103/93zr-1ykb · Cited by 12 · see citing works
Symmetric Clifford twirling for cost-optimal quantum error mitigation in early FTQC regime
Tsubouchi, K., Mitsuhashi, Y., Sharma, K., & Yoshioka, N. (2025) · npj Quantum Information · doi:10.1038/s41534-025-01050-9 · Cited by 11 · see citing works
“This implies that a cost-optimal QEM method, if available, could permit a logical error rate up to twice as large, allowing for a smaller code distance in logical qubits.”
Position Papers for the ASCR Workshop on Basic Research Needs in Quantum Computing and Networking
Lougovski, P., Parekh, O., Broz, J., Byrd, M., Chembo, Y. K., de Jong, W. A., Figueroa, E. V., Humble, T. S., Larson, J. M., Quiroz, G., et al. (2023) · doi:10.2172/2283320 · Cited by 10 · see citing works
Quantum error mitigation via quantum-noise-effect circuit groups
Hama, Y., & Nishi, H. (2024) · Scientific Reports · doi:10.1038/s41598-024-52485-7 · Cited by 10 · see citing works
“First, we make a comparison between our method and PEC 26,27,50,53,57,60,65,68–71, 73, 74, 76 .”
Circuit knitting facing exponential sampling-overhead scaling bounded by entanglement cost
Jing, M., Zhu, C., & Wang, X. (2025) · Physical Review A · doi:10.1103/physreva.111.012433 · Cited by 10 · see citing works
Demonstrating quantum error mitigation on logical qubits
Zhang, A., Xie, H., Gao, Y., Yang, J., Bao, Z., Zhu, Z., Chen, J., Wang, N., Zhang, C., Zhong, J., et al. (2025) · Nature Communications · doi:10.1038/s41467-025-67768-4 · Cited by 11 · see citing works
“Speci fi cally, we apply ZNE to reduce post-correction logical errors by amplifying noise on physical qubits.”
Unlocking early fault-tolerant quantum computing with mitigated magic dilution
Luthra, S., Moylett, A. E., Browne, D. E., & Campbell, E. T. (2025) · Quantum Science and Technology · doi:10.1088/2058-9565/ae0aef · Cited by 7 · see citing works
“The quantity λ is defined as the l 1 -norm of the coefficients in this decomposition, that is such that λ 2 is the sampling overhead of each decomposition from Hoeffding’s inequality [25, 27].”
InfluentialVirtual quantum error detection
Tsubouchi, K., Suzuki, Y., Tokunaga, Y., Yoshioka, N., & Endo, S. (2023) · Physical Review A · doi:10.1103/physreva.108.042426 · Cited by 9 · see citing works
Mitigating quantum errors via truncated Neumann series
Wang, K., Chen, Y.-A., & Wang, X. (2023) · Science China Information Sciences · doi:10.1007/s11432-023-3786-1 · Cited by 8 · see citing works
“For example, there is emerging interest in syncretizing error mitigation and correction techniques [74, 75].”
BackgroundSurf-Deformer: Mitigating Dynamic Defects on Surface Code via Adaptive Deformation
Yin, K., Fang, X., Humble, T. S., Li, A., Shi, Y., & Ding, Y. (2024) · IEEE/ACM International Symposium on Microarchitecture (MICRO) · doi:10.1109/micro61859.2024.00061 · Cited by 8 · see citing works
“Various error mitigation approaches [79]–[81] based on statistical methods are also proposed.”
The Future Era of Quantum Computing
Kahanda, G., Patel, V., Parikh, M., Ippolito, M. J., Solanki, M., & Ahmed, S. M. M. (2023) · Advanced sciences and technologies for security applications · doi:10.1007/978-3-031-20160-8_27 · Cited by 8 · see citing works
Quantum error mitigation for rotation-symmetric bosonic codes with symmetry expansion
Endo, S., Suzuki, Y., Tsubouchi, K., Asaoka, R., Yamamoto, K., Matsuzaki, Y., & Tokunaga, Y. (2025) · Physical Review A · doi:10.1103/physreva.111.062402 · Cited by 9 · see citing works
“An example of other methods than SE is probabilistic error cancellation (PEC) [37, 39] on the code space to effectively increase the code distances [60, 68], compensate for gate decomposition errors such as approximation errors in Solovay-Kitaev algorithms [60] and T gate errors [60, 69, 70].”
Majumder, S., Yale, C. G., Morris, T., Lobser, D., Burch, A. D., Chow, M. N. H., Revelle, M., Clark, S., & Pooser, R. C. (2023) · Quantum · doi:10.22331/q-2023-05-15-1006 · Cited by 7 · see citing works
Error Mitigation and Circuit Division for Early Fault-Tolerant Quantum Phase Estimation
Dutkiewicz, A., Polla, S., Scheurer, M., Gogolin, C., Huggins, W. J., & O’Brien, T. E. (2025) · PRX Quantum · doi:10.1103/mlmy-yskj · Cited by 7 · see citing works
“This can be achieved either by developing algorithms that have some natural robustness to error [15, 25, 102], or by augmenting error correction with quantum error mitigation (EM) [26, 35, 36, 103].”
InfluentialError crafting in mixed quantum gate synthesis
Yoshioka, N., Akibue, S., Morisaki, H., Tsubouchi, K., & Suzuki, Y. (2025) · npj Quantum Information · doi:10.1038/s41534-025-01032-x · Cited by 6 · see citing works
“Published in partnership with The University of New South Wales https://doi.org/10.1038/s41534-025-01032-x”
InfluentialShadow Simulation of Quantum Processes
Zhao, X., Wang, X., & Chiribella, G. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.120804 · Cited by 5 · see citing works
Power of quantum measurement in simulating unphysical operations
Zhao, X., Zhang, L., Zhao, B., & Wang, X. (2025) · Physical Review Research · doi:10.1103/physrevresearch.7.013334 · Cited by 7 · see citing works
“Potential use cases include circuit knitting [43–45], Hamiltonian simulation [46–48], and quantum error correction [49–51].”
A Practical Framework for Quantum Error MitigationPreprint
Cai, Z. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2110.05389 · Cited by 5 · see citing works
“…many experiments [3–7], and is expected to be a key component for practical quantum applications from the noisy intermediate-scale quantum (NISQ) era that we are entering now, all the way up to the moment right before we have full QEC fault-tolerance [8, 9] in which error is no longer a bottleneck.”
BackgroundQuantum information processing with superconducting circuits: A perspective
Wendin, G. P. (2023) · Elsevier eBooks · doi:10.1016/b978-0-323-90800-9.00226-2 · Cited by 11 · see citing works
Quantum Computing for Photosensitizer Design in Photodynamic Therapy
Zehr, H., Baiardi, A., Tacchino, F., Gandon, A., Fischer, L. E., Xu, Y., DiFilippo, F. P., Guidoni, L., Haase, P. A. B., Talarico, N. W., et al. (2025) · Annual Review of Biomedical Data Science · doi:10.1146/annurev-biodatasci-103123-095644 · Cited by 6 · see citing works
Degeneration of the Grover search algorithm with depolarization in the oracle-box wires
Rastegin, A. E., & Shemet, A. M. (2023) · Modern Physics Letters A · doi:10.1142/s021773232350030x · Cited by 3 · see citing works
Quantum Gate Decomposition: A Study of Compilation Time vs. Execution Time Trade-offs
Rosa, E. C. R., Marchi, J., Duzzioni, E. I., & de Santiago, R. (2025) · doi:10.5753/sblp.2025.10459 · Cited by 3 · see citing works
Compiling a Simple Chemistry Application to Quantum Error Correction Primitives
Blunt, N. S., Gehér, G. P., & Moylett, A. E. (2023) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce57702.2023.10247 · Cited by 2 · see citing works
“We find that even this application requires hundreds of physical qubits and thousands of QEC rounds, suggesting that in early fault-tolerance further techniques will be required [8].”
BackgroundMethodPerspectives on Utilization of Measurements in Quantum Algorithms
Uotila, V., Salmenperä, I., Becker, L., de Griend, A. M.-V., Shinde, A. R., & Nurminen, J. K. (2025) · IEEE International Conference on Quantum Software (QSW) · doi:10.1109/qsw67625.2025.00016 · Cited by 2 · see citing works
Space-time trade-off in networked virtual distillation
Araki, T., Goodwin, J. F., & Koczor, B. (2025) · Physical Review A · doi:10.1103/t828-wvgl · Cited by 2 · see citing works
“Solving practical problems beyond sys-tem sizes that are classically tractable before achieving full fault-tolerance requires mitigating errors either in physical gates or logical gates in early fault-tolerant architectures [4–6].”
Layout-Aware Quantum Circuitry and Algorithmic Extensions to Grover's Algorithm
Al-Bayaty, A. (2025) · doi:10.15760/etd.3924 · Cited by 2 · see citing works
Chen, Y.-A., Zhu, C., He, K., Jing, M., & Wang, X. (2025) · IEEE Transactions on Information Theory · doi:10.1109/tit.2025.3568128 · Cited by 1 · see citing works
“…significant interest across various domains, including quantum error mitigation [21], [22], [23], [24], distributed quantum computing [25], [26], [27], correlation functions [28], entanglement detection [29], [30], [31], quantum broadcasting [32], [33], and fault-tolerant quantum computing [34].”
Ma, S., Wu, Y., Xu, S., et al. (2025) · Physical review. B./Physical review. B · doi:10.1103/4dd2-xbcl · Cited by 1 · see citing works
State-adaptive quantum error correction and fault-tolerant quantum computing
Wang, D. (2026) · Physical Review A · doi:10.1103/3z8s-jh28 · Cited by 1 · see citing works
Symmetric Channel Verification for Purifying Noisy Quantum Channels
Tsubouchi, K., Mitsuhashi, Y., Takagi, R., & Yoshioka, N. (2025) · PRX Quantum · doi:10.1103/jcd6-lft3 · Cited by 1 · see citing works
“13114v1 [ qu a n t - ph ] 17 M a r 2025”
Noise-Agnostic Unbiased Quantum Error Mitigation for Logical Qubits
Xie, H., Yoshioka, N., Tsubouchi, K., & Li, Y. (2025) · Physical Review Letters · doi:10.1103/2n23-4qg8 · Cited by 2 · see citing works
“Note that Pauli twirling is applied to the encoding and decoding operations, such that the associated errors are effectively Pauli, i.e., commutative. on logical qubits typically exhibit negligible noise [7, 13], allowing them to serve as ideal tools for twirling stabilizer operations.”
Enhancing the yield of bucket brigade quantum random access memory using redundancy repair
Kim, D., Heng, S., Lee, S., & Han, Y. (2025) · Scientific Reports · doi:10.1038/s41598-025-32955-2 · Cited by 4 · see citing works
“If fabrication defects occur in the topology of the surface code lattice, the distances of the code and the quality of the encoded logical qubits are seriously reduced 46,51,53,54 .”
Layered KIK quantum error mitigation for dynamic circuits
Bar, B., Santos, J. P., & Uzdin, R. (2026) · npj Quantum Information · doi:10.1038/s41534-026-01207-0 · Cited by 1 · see citing works
Dual-map framework for noise characterization of quantum computers
Sud, J., Marshall, J., Wang, Z., Rieffel, E. G., & Wudarski, F. (2022) · Physical Review A · doi:10.1103/physreva.106.012606 · Cited by 1 · see citing works
“[1] Christophe Piveteau, David Sutter, Sergey Bravyi, Jay M.”
BackgroundT-Count Optimizing Genetic Algorithm for Quantum State Preparation
Wright, A. L., Lewis, M., Zuliani, P., et al. (2024) · IEEE International Conference on Quantum Software (QSW) · doi:10.1109/qsw62656.2024.00020 · Cited by 1 · see citing works
“Research into different methods for building these non-Clifford gates such as the T gate has shown reduced error rates during the subsequent computation [23].”
MethodJi, N., 宋, P.-L. 平. S., Wang, W., 孙, H.-Q. 汇. S., & Ma, H. (2025) · Chinese Physics B · doi:10.1088/1674-1056/ae1818
Prospects of quantum error mitigation for quantum signal processing
Liaubaitė, U., & Skelton, S. E. (2025) · Physica Scripta · doi:10.1088/1402-4896/ae216a
“This can be understood as assuming access to an oracle with no internal noise, which might apply to a scenario where QEM protocols are used to complement error correction protocols on the oracle, such as the ones explored in [15] and [16].”
Q-Cluster: Quantum Error Mitigation Through Noise-Aware Unsupervised Learning
Patil, H. P., Baron, D., & Zhou, H. (2025) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce65121.2025.00097
Low-overhead tailoring and learning of noise in graph states
Park, G., Sun, J., & Jeong, H. (2026) · Physical Review A · doi:10.1103/rr8y-yz5m
“State or gate-set tomography [22, 40, 41] can, in principle, re-construct the full noise structure, but its sampling and time complexities scale exponentially with the number of qubits [41].”
Mitigating errors in state preparation and measurement with noncomputational states
Haupt, C. J., Vazquez, A. C., Fischer, L. E., Woerner, S., & Egger, D. J. (2025) · Physical Review Applied · doi:10.1103/jv9h-spzw
“In the fault-tolerant regime error mitigation will remain relevant by enhancing performance and reducing residual logical errors beyond the capabilities of error correction alone [5, 6].”
Logical Accreditation: A Framework for Efficient Certification of Fault-Tolerant Computations
Mills, J. K., Sireesh, A., Leichtle, D., Roffe, J., & Kashefi, E. (2026) · PRX Quantum · doi:10.1103/7pj4-gcbq
“Error mitigation techniques have been proposed as a means of extending the computational performance of logical computations with unpurified magic states [15], and where code distances are too low to suppress noise sufficiently to achieve desired logical error rates [5].”
InfluentialRuntime reduction in lattice surgery utilizing timelike soft information
Akahoshi, Y., Toshio, R., Fujisaki, J., Oshima, H., Sato, S., & Fujii, K. (2026) · Physical Review Research · doi:10.1103/wz3p-sgdq · Cited by 1 · see citing works
Wu, Y., Ma, S., Chen, R., Xu, S., Zhang, Y., Liu, J., Li, C., Lefkidis, G., Hübner, W., & Jin, W. (2026) · The Journal of Chemical Physics · doi:10.1063/5.0344787
Dataset for "Unlocking early fault-tolerant quantum computing with mitigated magic dilution"
Luthra, S., Moylett, A. E., Browne, D. E., & Campbell, E. T. (2025) · Zenodo (CERN European Organization for Nuclear Research) · doi:10.5281/zenodo.15427988 · Cited by 1 · see citing works
- Authors
- Piveteau, C., Sutter, D., Bravyi, S., Gambetta, J. M., & Temme, K.
- Affiliations
- ETH Zurich (CH)
- IBM Research - Zurich (CH)
- IBM (United States) (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2021-11-12
- Volume
- 127
- Issue
- 20
- Pages
- 200505-200505
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W3135181313
- Cited by
- 81
- References
- 37
- Citations by year
- 2026: 6 · 2025: 27 · 2024: 16 · 2023: 22 · 2022: 7 · 2021: 3
- FWCI
- 4.24
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
- National Center of Competence in Research Quantum Science and Technology
- IBM Research Frontiers Institute
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- error mitigation, encoded qubits, Calderbank-Shor-Steane codes, Clifford+T circuits, transversal gates, T-gate, fault-tolerant quantum computing, shallow quantum circuits, magic state distillation, quantum error correction, classical simulation of quantum circuits
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