Realization of Real-Time Fault-Tolerant Quantum Error Correction
Open accessAbstract
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, midcircuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a 10-qubit quantum charge-coupled device trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]] color code, first proposed by Steane [Phys. Rev. Lett. 77, 793 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.793]. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical state preparation and measurement (SPAM) error of 1.7(2)×10^{-3}, compared to the average physical SPAM error 2.4(4)×10^{-3} of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a T[over ¯]|+⟩_{L} magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudothreshold.
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No quantum processor has definitively shown below-threshold performance.
“Although many platforms have demonstrated different features of quantum error correction14–20, no quantum processor has definitively shown below-threshold performance.”
From Quantum error correction below the surface code threshold (2025) · cites this paper as [14]
One primary path to scaling trapped-ion quantum computers is the quantum charge-coupled device (QCCD) architecture, which involves arrays of trapping zones between which ions are shuttled during algorithms.
“One of the primary paths to scaling is the quantum charge-coupled device (QCCD) architecture, which involves arrays of trapping zones between which ions are shuttled during algorithms13–16.”
From Penning micro-trap for quantum computing (2024) · cites this paper as [16]
The time to rearrange ions for arbitrary circuit connectivity scales poorly for the linear trap design as the number of qubits increases, proportional to N.
“The time to rearrange ions for arbitrary circuit connectivity scales poorly for the linear trap design as the number of qubits increases (proportional to N [16]).”
From A Race-Track Trapped-Ion Quantum Processor (2023) · cites this paper as [16]
Error suppression by increasing the colour code size had not been demonstrated on any experimental platform.
“As a result, error suppression by increasing the colour code size has not yet been demonstrated, to our knowledge, on any experimental platform.”
From Scaling and logic in the colour code on a superconducting quantum processor (2025) · cites this paper as [12]
The Steane code has been implemented with trapped ions and neutral atoms, demonstrating logical operations and logical circuits.
“The smallest instance of the colour code, the Steane code37, has been implemented with trapped ions11,12 and neutral atoms18, demonstrating logical operations and logical circuits13–19.”
From Scaling and logic in the colour code on a superconducting quantum processor (2025) · cites this paper as [12]
The trapped-ion platform has the capability for quantum-information processing with dozens of addressed qubits in a single trap and fidelities sufficient for fault-tolerant gate operations and error correction.
“a particular strength of the trapped-ion platform is the capability for quantuminformation processing with dozens of addressed qubits in a single trap [22,23] and fidelities sufficient for faulttolerant gate operations and error correction [62,63].”
From Entanglement of Trapped-Ion Qubits Separated by 230 Meters (2023) · cites this paper as [63]
For most near-term circuit architectures, including H1-1, single-qubit gates are relatively less noisy than two-qubit entangling gates.
“For most near-term circuit architectures, including H1-1, the single qubit gates are relatively less noisy than the two-qubit entangling gates46,47.”
From Constrained quantum optimization for extractive summarization on a trapped-ion quantum computer (2022) · cites this paper as [47]
Cites
<i>Quantum Computation and Quantum Information</i>
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895 · see citing works
Quantum computation and quantum information
Law, J. (2001) · ACM SIGSOFT Software Engineering Notes · doi:10.1145/505482.505499 · Cited by 20,449 · see citing works
The Jackknife, the Bootstrap and Other Resampling Plans
Efron, B. (1982) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611970319 · Cited by 8,120 · see citing works
Scheme for reducing decoherence in quantum computer memory
Shor, P. W. (1995) · Physical Review A · doi:10.1103/physreva.52.r2493 · Cited by 4,479 · see citing works
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
Shor, P. W. (1999) · SIAM Review · doi:10.1137/s0036144598347011 · Cited by 3,904 · see citing works
Surface codes: Towards practical large-scale quantum computation
Fowler, A. G., Mariantoni, M., Martinis, J. M., & Cleland, A. N. (2012) · Physical Review A · doi:10.1103/physreva.86.032324 · Cited by 3,059 · see citing works
Error Correcting Codes in Quantum Theory
Steane, A. M. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.793 · Cited by 2,677 · see citing works
Good quantum error-correcting codes exist
Calderbank, A. R., & Shor, P. W. (1996) · Physical Review A · doi:10.1103/physreva.54.1098 · Cited by 2,516 · see citing works
Feynman, R. P. (1986) · Foundations of Physics · doi:10.1007/bf01886518 · Cited by 1,770 · see citing works
Dennis, E., Kitaev, A., Landahl, A., & Preskill, J. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1499754 · Cited by 1,768 · see citing works
Architecture for a large-scale ion-trap quantum computer
Kielpinski, D., Monroe, C., & Wineland, D. J. (2002) · Nature · doi:10.1038/nature00784 · Cited by 1,589 · see citing works
Universal quantum computation with ideal Clifford gates and noisy ancillas
Bravyi, S., & Kitaev, A. (2005) · Physical Review A · doi:10.1103/physreva.71.022316 · Cited by 1,525 · see citing works
Improved simulation of stabilizer circuits
Aaronson, S., & Gottesman, D. (2004) · Physical Review A · doi:10.1103/physreva.70.052328 · Cited by 1,515 · see citing works
Dennis, E., Kitaev, A., Landahl, A., & Preskill, J. (2001) · CERN Document Server (European Organization for Nuclear Research) · Cited by 1,485 · see citing works
Simulated Quantum Computation of Molecular Energies
Aspuru-Guzik, A., Dutoi, A. D., Love, P. J., & Head-Gordon, M. (2005) · Science · doi:10.1126/science.1113479 · Cited by 1,418 · see citing works
Error Mitigation for Short-Depth Quantum Circuits
Temme, K., Bravyi, S., & Gambetta, J. M. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.180509 · Cited by 1,379 · see citing works
Quantum computing with realistically noisy devices
Knill, E. (2005) · Nature · doi:10.1038/nature03350 · Cited by 1,184 · see citing works
Proceedings 35th Annual Symposium on Foundations of Computer Science
(1994) · doi:10.1109/sfcs.1994 · Cited by 1,133 · see citing works
Preskill, J. (1998) · Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences · doi:10.1098/rspa.1998.0167 · Cited by 994 · see citing works
Deterministic quantum teleportation of atomic qubits
Barrett, M. D., Chiaverini, J., Schaetz, T., Britton, J., Itano, W. M., Jost, J. D., Knill, E., Langer, C., Leibfried, D., Ozeri, R., et al. (2004) · Nature · doi:10.1038/nature02608 · Cited by 975 · see citing works
Error mitigation extends the computational reach of a noisy quantum processor
Kandala, A., Temme, K., Corcoles, A. D., Mezzacapo, A., Chow, J. M., & Gambetta, J. (2019) · Nature · doi:10.1038/s41586-019-1040-7 · Cited by 947 · see citing works
Theory of fault-tolerant quantum computation
Gottesman, D. (1998) · Physical Review A · doi:10.1103/physreva.57.127 · Cited by 945 · see citing works
Extending the lifetime of a quantum bit with error correction in superconducting circuits
Ofek, N., Petrenko, A., Heeres, R., Reinhold, P., Leghtas, Z., Vlastakis, B., Liu, Y., Frunzio, L., Girvin, S. M., Jiang, L., et al. (2016) · Nature · doi:10.1038/nature18949 · Cited by 941 · see citing works
State preservation by repetitive error detection in a superconducting quantum circuit
Kelly, J., Barends, R., Fowler, A. G., Megrant, A., Jeffrey, E. R., White, T., Sank, D., Mutus, J., Campbell, B., Chen, Y., et al. (2015) · Nature · doi:10.1038/nature14270 · Cited by 897 · see citing works
Scalable and robust randomized benchmarking of quantum processes.
Magesan, E., Gambetta, J., & Emerson, J. (2010) · Physical Review Letters · doi:10.1103/physrevlett.106.180504 · Cited by 799 · see citing works
Simulation of Many-Body Fermi Systems on a Universal Quantum Computer
Abrams, D. S., & Lloyd, S. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.2586 · Cited by 745 · see citing works
Quantum computing for finance: Overview and prospects
Orús, R., Mugel, S., & Lizaso, E. (2019) · Reviews in Physics · doi:10.1016/j.revip.2019.100028 · Cited by 714 · see citing works
Topological Quantum Distillation
Bombin, H., & Martin-Delgado, M. A. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.180501 · Cited by 700 · see citing works
Efficient Variational Quantum Simulator Incorporating Active Error Minimization
Li, Y., & Benjamin, S. C. (2017) · Physical Review X · doi:10.1103/physrevx.7.021050 · Cited by 686 · see citing works
Experimental Quantum Error Correction
Cory, D. G., Price, M. D., Maas, W., Knill, E., Laflamme, R., Zurek, W. H., Havel, T. F., & Somaroo, S. S. (1998) · Physical Review Letters · doi:10.1103/physrevlett.81.2152 · Cited by 581 · see citing works
Realization of three-qubit quantum error correction with superconducting circuits
Reed, M. D., DiCarlo, L., Nigg, S. E., Sun, L., Frunzio, L., Girvin, S. M., & Schoelkopf, R. J. (2012) · Nature · doi:10.1038/nature10786 · Cited by 577 · see citing works
Practical Quantum Error Mitigation for Near-Future Applications
Endo, S., Benjamin, S. C., & Li, Y. (2018) · Physical Review X · doi:10.1103/physrevx.8.031027 · Cited by 562 · see citing works
Demonstration of the trapped-ion quantum CCD computer architecture
Pino, J. M., Dreiling, J. M., Figgatt, C., Gaebler, J. P., Moses, S. A., Allman, M. S., Baldwin, C. H., Foss-Feig, M., Hayes, D., Mayer, K., et al. (2021) · Nature · doi:10.1038/s41586-021-03318-4 · Cited by 547 · see citing works
Self-citationNoise tailoring for scalable quantum computation via randomized compiling
Wallman, J. J., & Emerson, J. (2016) · Physical Review A · doi:10.1103/physreva.94.052325 · Cited by 535 · see citing works
Realization of quantum error correction
Chiaverini, J., Leibfried, D., Schaetz, T., Barrett, M. D., Blakestad, R. B., Britton, J., Itano, W. M., Jost, J. D., Knill, E., Langer, C., et al. (2004) · Nature · doi:10.1038/nature03074 · Cited by 517 · see citing works
Fault-Tolerant Quantum Computation with Constant Error Rate
Aharonov, D., & Ben-Or, M. (2008) · SIAM Journal on Computing · doi:10.1137/s0097539799359385 · Cited by 513 · see citing works
Corcoles, A. D., Magesan, E., Srinivasan, S. J., Cross, A. W., Steffen, M., Gambetta, J., & Chow, J. M. (2015) · Nature Communications · doi:10.1038/ncomms7979 · Cited by 477 · see citing works
Olmschenk, S., Younge, K. C., Moehring, D. L., Matsukevich, D. N., Maunz, P., & Monroe, C. (2007) · Physical Review A · doi:10.1103/physreva.76.052314 · Cited by 464 · see citing works
Quantum error correction of a qubit encoded in grid states of an oscillator
Campagne-Ibarcq, P., Eickbusch, A., Touzard, S., Zalys-Geller, E., Frattini, N. E., Sivak, V. V., Reinhold, P., Puri, S., Shankar, S., Schoelkopf, R. J., et al. (2020) · Nature · doi:10.1038/s41586-020-2603-3 · Cited by 459 · see citing works
ProjectQ: An open source software framework for quantum computing
Steiger, D. S., Häner, T., & Troyer, M. (2018) · Quantum · doi:10.22331/q-2018-01-31-49 · Cited by 428 · see citing works
Quantum computations on a topologically encoded qubit
Nigg, D., Müller, M., Martinez, E. A., Schindler, P., Hennrich, M., Monz, T., Martin-Delgado, M. A., & Blatt, R. (2014) · Science · doi:10.1126/science.1253742 · Cited by 400 · see citing works
Encoding a qubit in a trapped-ion mechanical oscillator
Flühmann, C., Nguyen, T. L., Marinelli, M., Negnevitsky, V., Mehta, K., & Home, J. P. (2019) · Nature · doi:10.1038/s41586-019-0960-6 · Cited by 394 · see citing works
Exponential suppression of bit or phase errors with cyclic error correction
Google Quantum, A. I., Chen, Z., Satzinger, K. J., Atalaya, J., Korotkov, A. N., Dunsworth, A., Sank, D., Quintana, C., McEwen, M., Barends, R., et al. (2021) · Nature · doi:10.1038/s41586-021-03588-y · Cited by 393 · see citing works
Methodology for quantum logic gate construction
Zhou, X., Leung, D. W., & Chuang, I. L. (2000) · Physical Review A · doi:10.1103/physreva.62.052316 · Cited by 355 · see citing works
Fault-tolerant control of an error-corrected qubit
Egan, L., Debroy, D. M., Noel, C., Risinger, A., Zhu, D., Biswas, D., Newman, M., Li, M., Brown, K. R., Cetina, M., et al. (2021) · Nature · doi:10.1038/s41586-021-03928-y · Cited by 348 · see citing works
Experimental Repetitive Quantum Error Correction
Schindler, P., Barreiro, J. T., Monz, T., Nebendahl, V., Nigg, D., Chwalla, M., Hennrich, M., & Blatt, R. (2011) · Science · doi:10.1126/science.1203329 · Cited by 343 · see citing works
Repeated quantum error detection in a surface code
Andersen, C. K., Remm, A., Lazar, S., Krinner, S., Lacroix, N., Norris, G. J., Gabureac, M. S., Eichler, C., & Wallraff, A. J. (2020) · Nature Physics · doi:10.1038/s41567-020-0920-y · Cited by 329 · see citing works
Single ion qubit with estimated coherence time exceeding one hour
Wang, P., Luan, C.-Y., Qiao, M., Um, M., Zhang, J., Wang, Y., Yuan, X., Gu, M., Zhang, J., & Kim, K. (2021) · Nature Communications · doi:10.1038/s41467-020-20330-w · Cited by 321 · see citing works
Implementing a universal gate set on a logical qubit encoded in an oscillator
Heeres, R., Reinhold, P., Ofek, N., Frunzio, L., Jiang, L., Devoret, M., & Schoelkopf, R. J. (2017) · Nature Communications · doi:10.1038/s41467-017-00045-1 · Cited by 309 · see citing works
Quantum error correction and universal gate set operation on a binomial bosonic logical qubit
Hu, L., Ma, Y., Cai, W., Mu, X., Xu, Y., Wang, W., Wu, Y., Wang, H., Song, Y. P., Zou, C.-L., et al. (2019) · Nature Physics · doi:10.1038/s41567-018-0414-3 · Cited by 303 · see citing works
Proceedings of Symposia in Applied Mathematics.
Tompkins, C. B., Bellman, R., & Hall, M. (1961) · Mathematics of Computation · doi:10.2307/2002914 · Cited by 297 · see citing works
Quantum computing enhanced computational catalysis
von Burg, V., Low, G. H., Häner, T., Steiger, D. S., Reiher, M., Roetteler, M., & Troyer, M. (2021) · Physical Review Research · doi:10.1103/physrevresearch.3.033055 · Cited by 259 · see citing works
Detecting bit-flip errors in a logical qubit using stabilizer measurements
Ristè, D., Poletto, S., Huang, M.-Z., Bruno, A., Vesterinen, V., Saira, O.-P., & DiCarlo, L. (2015) · Nature Communications · doi:10.1038/ncomms7983 · Cited by 254 · see citing works
Repeated quantum error correction on a continuously encoded qubit by real-time feedback
Cramer, J., Kalb, N., Rol, M. A., Hensen, B., Blok, M., Markham, M., Twitchen, D. J., Hanson, R., & Taminiau, T. H. (2016) · Nature Communications · doi:10.1038/ncomms11526 · Cited by 242 · see citing works
Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits
Takita, M., Cross, A. W., Córcoles, A. D., Chow, J. M., & Gambetta, J. M. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.180501 · Cited by 228 · see citing works
Quantum Error Correction with Only Two Extra Qubits
Chao, R., & Reichardt, B. W. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.050502 · Cited by 221 · see citing works
Ultrahigh Error Threshold for Surface Codes with Biased Noise
Tuckett, D. K., Bartlett, S. D., & Flammia, S. T. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.050505 · Cited by 216 · see citing works
Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code
Knill, E. H., Laflamme, R., Martínez, R., & Negrevergne, C. (2001) · Physical Review Letters · doi:10.1103/physrevlett.86.5811 · Cited by 215 · see citing works
Quantum Communication, Computing, and Measurement
Hirota, O., Holevo, A., & Caves, C. (1997) · Springer eBooks · doi:10.1007/978-1-4615-5923-8 · Cited by 208 · see citing works
Experimental demonstration of topological error correction
Yao, X.-C., Wang, T., Chen, H.-Z., Gao, W., Fowler, A. G., Raussendorf, R., Chen, Z., Liu, N.-L., Lu, C., Deng, Y., et al. (2012) · Nature · doi:10.1038/nature10770 · Cited by 204 · see citing works
Strong resilience of topological codes to depolarization
Bombín, H., Andrist, R. S., Ohzeki, M., Katzgraber, H., & Martin-Delgado, M. (2012) · Physical Review X · doi:10.1103/physrevx.2.021004 · Cited by 182 · see citing works
Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor
Hashim, A., Naik, R., Morvan, A., Ville, J.-L., Mitchell, B., Kreikebaum, J. M., Davis, M., Smith, E., Iancu, C., O’Brien, K. P., et al. (2021) · Physical Review X · doi:10.1103/physrevx.11.041039 · Cited by 165 · see citing works
Fault-tolerant quantum error detection
Linke, N. M., Gutiérrez, M., Landsman, K. A., Figgatt, C., Debnath, S., Brown, K. R., & Monroe, C. R. (2017) · Science Advances · doi:10.1126/sciadv.1701074 · Cited by 163 · see citing works
Errors in trapped-ion quantum gates due to spontaneous photon scattering
Ozeri, R., Itano, W. M., Blakestad, R. B., Britton, J., Chiaverini, J., Jost, J. D., Langer, C., Leibfried, D. G., Reichle, R., Seidelin, S. L., et al. (2007) · Physical Review A · doi:10.1103/physreva.75.042329 · Cited by 162 · see citing works
Quantum error correction beyond qubits
Aoki, T., Takahashi, G., Kajiya, T., Yoshikawa, J.-I., Braunstein, S. L., van Loock, P., & Furusawa, A. (2009) · Nature Physics · doi:10.1038/nphys1309 · Cited by 144 · see citing works
Quantum Universality from Magic States Distillation Applied to CSS Codes
Reichardt, B. W. (2005) · Quantum Information Processing · doi:10.1007/s11128-005-7654-8 · Cited by 143 · see citing works
Fault-Tolerant Logical Gates in the IBM Quantum Experience
Harper, R., & Flammia, S. T. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.080504 · Cited by 141 · see citing works
Entangling logical qubits with lattice surgery
Erhard, A., Nautrup, H. P., Meth, M., Postler, L., Stricker, R., Stadler, M., Negnevitsky, V., Ringbauer, M., Schindler, P., Briegel, H. J., et al. (2021) · Nature · doi:10.1038/s41586-020-03079-6 · Cited by 141 · see citing works
Fault-tolerant quantum computation with few qubits
Chao, R., & Reichardt, B. W. (2018) · npj Quantum Information · doi:10.1038/s41534-018-0085-z · Cited by 137 · see citing works
Repeated multi-qubit readout and feedback with a mixed-species trapped-ion register
Negnevitsky, V., Marinelli, M., Mehta, K. K., Lo, H.-Y., Flühmann, C., & Home, J. (2018) · Nature · doi:10.1038/s41586-018-0668-z · Cited by 128 · see citing works
Strong Resilience of Topological Codes to Depolarization
Bombín, H., Andrist, R. S., Ohzeki, M., Katzgraber, H. G., & Martin-Delgado, M. A. (2012) · Institutional Repositories DataBase (IRDB) · doi:10.3929/ethz-b-000058359 · Cited by 126 · see citing works
Entangling logical qubits with lattice surgery
Erhard, A., Nautrup, H. P., Meth, M., Postler, L., Stricker, R., Stadler, M., Negnevitsky, V., Ringbauer, M., Schindler, P., Briegel, H. J., et al. (2021) · RePEc: Research Papers in Economics · Cited by 103 · see citing works
Reliable transport through a microfabricated <i>X</i> -junction surface-electrode ion trap
Wright, K., Amini, J. M., Faircloth, D. L., Volin, C., Charles Doret, S., Hayden, H., Pai, C.-S., Landgren, D. W., Denison, D., Killian, T., et al. (2013) · New Journal of Physics · doi:10.1088/1367-2630/15/3/033004 · Cited by 87 · see citing works
Quantum Information Science and Its Contributions to Mathematics
Lomonaco, S. J. (2010) · Proceedings of symposia in applied mathematics · doi:10.1090/psapm/068 · Cited by 83 · see citing works
Experimental quantum coding against qubit loss error
Lu, C., Gao, W., Zhang, J., Zhou, X., Yang, T., & Pan, J.-W. (2008) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0800740105 · Cited by 75 · see citing works
Reichardt, B. W. (2020) · Quantum Science and Technology · doi:10.1088/2058-9565/abc6f4 · Cited by 70 · see citing works
Near Ground-State Cooling of Two-Dimensional Trapped-Ion Crystals with More than 100 Ions
Jordan, E., Gilmore, K., Shankar, A., Safavi-Naini, A., Bohnet, J., Holland, M., & Bollinger, J. J. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.053603 · Cited by 70 · see citing works
Self-citationExperimental exploration of five-qubit quantum error-correcting code with superconducting qubits
Gong, M., Yuan, X., Wang, S., Wu, Y., Zhao, Y., Zha, C., Li, S., Zhang, Z., Zhao, Q., Liu, Y., et al. (2021) · National Science Review · doi:10.1093/nsr/nwab011 · Cited by 66 · see citing works
Fault-Tolerant Operation of a Quantum Error-Correction Code
Egan, L., Debroy, D. M., Noel, C., Risinger, A., Zhu, D., Biswas, D., Newman, M., Li, M., Brown, K. R., Cetina, M., et al. (2020) · arXiv · doi:10.48550/arxiv.2009.11482 · Cited by 65 · see citing works
Crosstalk Suppression for Fault-tolerant Quantum Error Correction with Trapped Ions
Parrado-Rodríguez, P., Ryan-Anderson, C., Bermudez, A., & Müller, M. (2021) · Quantum · doi:10.22331/q-2021-06-29-487 · Cited by 63 · see citing works
Self-citationIntegrated optical addressing of a trapped ytterbium ion
Ivory, M., Setzer, W. J., Karl, N., McGuinness, H., DeRose, C., Blain, M., Stick, D., Gehl, M., & Parazzoli, L. P. (2021) · Physical Review X · doi:10.1103/physrevx.11.041033 · Cited by 62 · see citing works
Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code
Goto, H. (2016) · Scientific Reports · doi:10.1038/srep19578 · Cited by 61 · see citing works
Coherence in logical quantum channels
Iverson, J. K., & Preskill, J. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab8e5c · Cited by 61 · see citing works
Wootton, J. R., & Loss, D. (2018) · Physical Review A · doi:10.1103/physreva.97.052313 · Cited by 60 · see citing works
Crosstalk Suppression for Fault-tolerant Quantum Error Correction with Trapped IonsPreprint
Parrado-Rodríguez, P., Ryan-Anderson, C., Bermúdez, A., & Müller, M. (2020) · RWTH Publications (RWTH Aachen) · Cited by 52 · see citing works
Self-citationDeterministic reordering of40Ca+ions in a linear segmented Paul trap
Splatt, F., Harlander, M., Brownnutt, M., Zähringer, F., Blatt, R., & Hänsel, W. (2009) · New Journal of Physics · doi:10.1088/1367-2630/11/10/103008 · Cited by 51 · see citing works
Gutiérrez, M., Müller, M., & Bermúdez, A. (2019) · Physical Review A · doi:10.1103/physreva.99.022330 · Cited by 48 · see citing works
Quantum teleportation of physical qubits into logical code spaces
Luo, Y.-H., Chen, M.-C., Erhard, M., Zhong, H.-S., Wu, D., Tang, H., Zhao, Q., Wang, X., Fujii, K., Li, L., et al. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2026250118 · Cited by 45 · see citing works
Fault-tolerant protection of near-term trapped-ion topological qubits under realistic noise sources
Bermúdez, A., Xu, X., Gutiérrez, M., Benjamin, S. C., & Müller, M. (2019) · Physical Review A · doi:10.1103/physreva.100.062307 · Cited by 37 · see citing works
Subspace benchmarking high-fidelity entangling operations with trapped ions
Baldwin, C. H., Bjork, B. J., Gaebler, J. P., Hayes, D., & Stack, D. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.013317 · Cited by 34 · see citing works
Self-citationEliminating Leakage Errors in Hyperfine Qubits
Hayes, D., Stack, D. T., Bjork, B., Potter, A. C., Baldwin, C. H., & Stutz, R. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.170501 · Cited by 32 · see citing works
Self-citationStabilizer Slicing: Coherent Error Cancellations in Low-Density Parity-Check Stabilizer Codes
Debroy, D. M., Li, M., Newman, M., & Brown, K. R. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.250502 · Cited by 30 · see citing works
Real-time processing of stabilizer measurements in a bit-flip code
Ristè, D., Govia, L. C. G., Donovan, B., Fallek, S. D., Kalfus, W. D., Brink, M., Bronn, N. T., & Ohki, T. (2020) · npj Quantum Information · doi:10.1038/s41534-020-00304-y · Cited by 29 · see citing works
Coherent rotations of qubits within a surface ion-trap quantum computer
van Mourik, M. W., Martinez, E. A., Gerster, L., Hrmo, P., Monz, T., Schindler, P., & Blatt, R. (2020) · Physical Review A · doi:10.1103/physreva.102.022611 · Cited by 23 · see citing works
Step-by-step magic state encoding for efficient fault-tolerant quantum computation
Goto, H. (2014) · Scientific Reports · doi:10.1038/srep07501 · Cited by 13 · see citing works
The Physical Implementation of Quantum Computation
DiVincenzo, D. P. (2000) · Fortschritte der Physik · doi:10.1002/1521-3978(200009)48:9/11
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Suppressing quantum errors by scaling a surface code logical qubit
Google Quantum, A. I., Acharya, R., Aleiner, I., Allen, R., Andersen, T. I., Ansmann, M., Arute, F., Arya, K., Asfaw, A., Atalaya, J., et al. (2023) · Nature · doi:10.1038/s41586-022-05434-1 · Cited by 1,103 · see citing works
“Several works have reported quantum error correction on small codes able to correct a single error [20–26] and in continuous variable codes [27–30].”
Self-citationQuantum error correction below the surface code threshold
Collaborators, G. Q. A. A., Acharya, R., Abanin, D. A., Aghababaie-Beni, L., Aleiner, I., Andersen, T. I., Ansmann, M., Arute, F., Arya, K., Asfaw, A., et al. (2025) · Nature · doi:10.1038/s41586-024-08449-y · Cited by 778 · see citing works
“We note there are other accurate decoders that would be interesting to test on this data, including beliefmatch-ing [13], BP+OSD [14], and tensor network decoding [1].”
Self-citationRealizing repeated quantum error correction in a distance-three surface code
Krinner, S., Lacroix, N., Remm, A., Di Paolo, A., Genois, E., Leroux, C., Hellings, C., Lazar, S., Swiadek, F., Herrmann, J., et al. (2022) · Nature · doi:10.1038/s41586-022-04566-8 · Cited by 686 · see citing works
Real-time quantum error correction beyond break-even
Sivak, V. V., Eickbusch, A., Royer, B., Singh, S., Tsioutsios, I., Ganjam, S., Miano, A., Brock, B. L., Ding, A. Z., Frunzio, L., et al. (2023) · Nature · doi:10.1038/s41586-023-05782-6 · Cited by 407 · see citing works
A Race-Track Trapped-Ion Quantum Processor
Moses, S. A., Baldwin, C. H., Allman, M. S., Ancona, R., Ascarrunz, L., Barnes, C., Bartolotta, J., Bjork, B., Blanchard, P., Bohn, M., et al. (2023) · Physical Review X · doi:10.1103/physrevx.13.041052 · Cited by 391 · see citing works
Self-citationDemonstration of fault-tolerant universal quantum gate operations
Postler, L., Heuβen, S., Pogorelov, I., Rispler, M., Feldker, T., Meth, M., Marciniak, C. D., Stricker, R., Ringbauer, M., Blatt, R., et al. (2022) · Nature · doi:10.1038/s41586-022-04721-1 · Cited by 287 · see citing works
Entanglement of Trapped-Ion Qubits Separated by 230 Meters
Krutyanskiy, V., Galli, M., Krcmarsky, V., Baier, S., Fioretto, D. A., Pu, Y., Mazloom, A., Sekatski, P., Canteri, M., Teller, M., et al. (2023) · Physical Review Letters · doi:10.1103/physrevlett.130.050803 · Cited by 217 · see citing works
Beating the break-even point with a discrete-variable-encoded logical qubit
Ni, Z., Li, S., Deng, X., Cai, Y., Zhang, L., Wang, W., Yang, Z.-B., Yu, H., Yan, F., Liu, S., et al. (2023) · Nature · doi:10.1038/s41586-023-05784-4 · Cited by 176 · see citing works
Mid-circuit correction of correlated phase errors using an array of spectator qubits
Singh, K., Bradley, C. E., Anand, S., Ramesh, V., White, R., & Bernien, H. (2023) · Science · doi:10.1126/science.ade5337 · Cited by 119 · see citing works
Measurement as a Shortcut to Long-Range Entangled Quantum Matter
Lu, T.-C., Lessa, L. A., Kim, I. H., & Hsieh, T. H. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.040337 · Cited by 115 · see citing works
Encoding a magic state with beyond break-even fidelity
Gupta, R. S., Sundaresan, N., Alexander, T., Wood, C. J., Merkel, S. T., Healy, M. B., Hillenbrand, M., Jochym-O’Connor, T., Wootton, J. R., Yoder, T. J., et al. (2024) · Nature · doi:10.1038/s41586-023-06846-3 · Cited by 115 · see citing works
Sundaresan, N., Yoder, T. J., Kim, Y., Li, M., Chen, E. H., Harper, G., Thorbeck, T., Cross, A. W., Córcoles, A. D., & Takita, M. (2023) · Nature Communications · doi:10.1038/s41467-023-38247-5 · Cited by 108 · see citing works
Scalable Measures of Magic Resource for Quantum Computers
Haug, T., & Kim, M. S. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.010301 · Cited by 104 · see citing works
Learning high-accuracy error decoding for quantum processors
Bausch, J., Senior, A. W., Heras, F. J. H., Edlich, T., Davies, A., Newman, M., Jones, C., Satzinger, K., Niu, M. Y., Blackwell, S., et al. (2024) · Nature · doi:10.1038/s41586-024-08148-8 · Cited by 106 · see citing works
“As physical error rates in quantum hardware have been brought down, researchers have started to conduct error-suppression experiments on real quantum devices 4,17,18,20,21,41 .”
Midcircuit Measurements on a Single-Species Neutral Alkali Atom Quantum Processor
Graham, T. M., Phuttitarn, L., Chinnarasu, R., Song, Y., Poole, C., Jooya, K., Scott, J., Scott, A., Eichler, P., & Saffman, M. (2023) · Physical Review X · doi:10.1103/physrevx.13.041051 · Cited by 81 · see citing works
Demonstration of Fault-Tolerant Steane Quantum Error Correction
Postler, L., Butt, F., Pogorelov, I., Marciniak, C. D., Heußen, S., Blatt, R., Schindler, P., Rispler, M., Müller, M., & Monz, T. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.030326 · Cited by 66 · see citing works
An, F. A., Ransford, A., Schaffer, A., Sletten, L. R., Gaebler, J., Hostetter, J., & Vittorini, G. (2022) · Physical Review Letters · doi:10.1103/physrevlett.129.130501 · Cited by 59 · see citing works
Self-citationConstrained quantum optimization for extractive summarization on a trapped-ion quantum computer
Niroula, P., Shaydulin, R., Yalovetzky, R., Minssen, P., Herman, D., Hu, S., & Pistoia, M. (2022) · Scientific Reports · doi:10.1038/s41598-022-20853-w · Cited by 59 · see citing works
A high-fidelity quantum matter-link between ion-trap microchip modules
Akhtar, M., Bonus, F., Lebrun-Gallagher, F. R., Johnson, N. I., Siegele-Brown, M., Hong, S., Hile, S. J., Kulmiya, S. A., Weidt, S., & Hensinger, W. K. (2023) · Nature Communications · doi:10.1038/s41467-022-35285-3 · Cited by 59 · see citing works
Baldwin, C. H., Mayer, K., Brown, N. C., Ryan-Anderson, C., & Hayes, D. (2022) · Quantum · doi:10.22331/q-2022-05-09-707 · Cited by 54 · see citing works
Self-citationScaling and logic in the colour code on a superconducting quantum processor
Lacroix, N., Bourassa, A., Heras, F. J. H., Zhang, L. M., Bausch, J., Senior, A. W., Edlich, T., Shutty, N., Sivak, V., Bengtsson, A., et al. (2025) · Nature · doi:10.1038/s41586-025-09061-4 · Cited by 51 · see citing works
Self-citationHigh-fidelity teleportation of a logical qubit using transversal gates and lattice surgery
Ryan-Anderson, C., Brown, N. C., Baldwin, C. H., Dreiling, J. M., Foltz, C., Gaebler, J. P., Gatterman, T. M., Hewitt, N., Holliman, C., Horst, C. V., et al. (2024) · Science · doi:10.1126/science.adp6016 · Cited by 52 · see citing works
Self-citationPenning micro-trap for quantum computing
Jain, S., Sägesser, T., Hrmo, P., Torkzaban, C., Stadler, M., Oswald, R., Axline, C., Bautista-Salvador, A., Ospelkaus, C., Kienzler, D., et al. (2024) · Nature · doi:10.1038/s41586-024-07111-x · Cited by 33 · see citing works
“We demonstrate full quantum control of an ion in this setting, as well as the ability to transport the ion arbitrarily in the trapping plane above the chip.”
Protecting expressive circuits with a quantum error detection code
Self, C. N., Benedetti, M., & Amaro, D. (2024) · Nature Physics · doi:10.1038/s41567-023-02282-2 · Cited by 31 · see citing works
Quantum circuit engineering for correcting coherent noise
Ahsan, M., Naqvi, S. A. Z., & Anwer, H. (2022) · Physical Review A · doi:10.1103/physreva.105.022428 · Cited by 22 · see citing works
A 98-qubit trapped-ion quantum computer with all-to-all connectivity
Ransford, A., Allman, M. S., Arkinstall, J., Campora, J. P., Cooper, S. F., Delaney, R. D., Dreiling, J. M., Estey, B., Figgatt, C., Hall, A., et al. (2026) · Nature · doi:10.1038/s41586-026-10676-4 · Cited by 16 · see citing works
Self-citationA quantum processor based on coherent transport of entangled atom arrays
Bluvstein, D., Levine, H., Semeghini, G., Wang, T. T., Ebadi, S., Kalinowski, M., Keesling, A., Maskara, N., Pichler, H., Greiner, M., et al. (2022) · Nature · doi:10.1038/s41586-022-04592-6 · Cited by 14 · see citing works
Experimental demonstration of the advantage of adaptive quantum circuits
Foss-Feig, M., Tikku, A., Lu, T.-C., Mayer, K., Iqbal, M., Gatterman, T. M., Gerber, J. A., Gilmore, K., Gresh, D., Hankin, A., et al. (2023) · arXiv · doi:10.48550/arxiv.2302.03029 · Cited by 11 · see citing works
Self-citationOptimization and implementation of a surface-electrode ion trap junction
Zhang, C., Mehta, K. K., & Home, J. P. (2022) · New Journal of Physics · doi:10.1088/1367-2630/ac7db6 · Cited by 10 · see citing works
Improving Threshold for Fault-Tolerant Color-Code Quantum Computing by Flagged Weight Optimization
Takada, Y., & Fujii, K. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.030352 · Cited by 10 · see citing works
- Authors
- Ryan-Anderson, C., Bohnet, J. G., Lee, K., Gresh, D., Hankin, A., Gaebler, J. P., Francois, D., Chernoguzov, A., Lucchetti, D., Brown, N. C., Gatterman, T. M., Halit, S. K., Gilmore, K., Gerber, J. A., Neyenhuis, B., Hayes, D., & Stutz, R. P.
- Affiliations
- Quantinuum (United States) (US)
- ORCID
- Venue
- Physical Review X
- Publisher
- American Physical Society (APS)
- Published
- 2021-12-23
- Volume
- 11
- Issue
- 4
- ISSN
- 2160-3308 (ISSN-L)
- OpenAlex
- W3181660950
- Cited by
- 306
- References
- 96
- Citations by year
- 2026: 37 · 2025: 79 · 2024: 77 · 2023: 60 · 2022: 51 · 2021: 2
- FWCI
- 16.72
- Type
- journal article
- Language
- en
- Update notices
- correction (2022-01-11)
- Open access
- gold
- doi.org
- Licence
- Locations
- Physical Review X (publishedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- 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
- quantum error correction, real-time decoding, fault tolerance, color codes, trapped-ion quantum computer, quantum charge-coupled device, logical qubits, syndrome measurement, non-Clifford gates, magic state distillation
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