Crosstalk Suppression for Fault-tolerant Quantum Error Correction with Trapped Ions

Pedro Parrado-Rodríguez1, Ciarán Ryan-Anderson1,2, Alejandro Bermudez3, and Markus Müller4,5

1Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
2Honeywell Quantum Solutions, 303 S. Technology Ct., Broomfield, Colorado 80021, USA
3Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain.
4Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, 52428 Jülich, Germany
5JARA-Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany

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Physical qubits in experimental quantum information processors are inevitably exposed to different sources of noise and imperfections, which lead to errors that typically accumulate hindering our ability to perform long computations reliably. Progress towards scalable and robust quantum computation relies on exploiting quantum error correction (QEC) to actively battle these undesired effects. In this work, we present a comprehensive study of crosstalk errors in a quantum-computing architecture based on a single string of ions confined by a radio-frequency trap, and manipulated by individually-addressed laser beams. This type of errors affects spectator qubits that, ideally, should remain unaltered during the application of single- and two-qubit quantum gates addressed at a different set of active qubits. We microscopically model crosstalk errors from first principles and present a detailed study showing the importance of using a coherent vs incoherent error modelling and, moreover, discuss strategies to actively suppress this crosstalk at the gate level. Finally, we study the impact of residual crosstalk errors on the performance of fault-tolerant QEC numerically, identifying the experimental target values that need to be achieved in near-term trapped-ion experiments to reach the break-even point for beneficial QEC with low-distance topological codes.

Ion traps are one of the leading platforms for building scalable quantum computers, offering all-to-all connectivity between qubits encoded in ions that belong to same Coulomb crystal and having demonstrated high fidelity gate operations. These quantum processors, however, are susceptible to multiple sources of noise that can corrupt the results of the computations. The development and implementation of quantum error correction (QEC) techniques is thus crucial to allow for reliable and scalable quantum computing. In ion traps, one major source of errors can be crosstalk, a process that can happen when gates applied to a given set of qubits unwantedly affect neighbouring qubits. But how much does this error affect the computations, and how can it be handled?
In this work, we evaluate the effects of crosstalk on QEC protocols implemented on a state-of-the-art ion trap platform. Using realistic error models derived from first principles, and through extensive numerical simulations, we demonstrate the feasibility of beneficial QEC and the efficiency of techniques like refocussing pulse sequences that can greatly suppress the detrimental effects of crosstalk.
The proposed techniques can be extended to other QEC codes, and our work can guide the path towards experimental realizations of error-corrected trapped-ion quantum processors.

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► References

[1] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. 10.1017/​CBO9780511976667.

[2] John Preskill. Fault-Tolerant Quantum Computation, page 213. 1996. 10.1142/​9789812385253_0008.

[3] M. H. Devoret and R. J. Schoelkopf. Superconducting circuits for quantum information: an outlook. Science, 339 (6124): 1169, 2013. 10.1126/​science.1231930.

[4] Colin D. Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M. Sage. Trapped-ion quantum computing: Progress and challenges. Applied Physics Reviews, 6, 2019. 10.1063/​1.5088164.

[5] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O'Brien. Quantum computers. Nature, 464 (7285): 45, 2010. 10.1038/​nature08812.

[6] D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt. Quantum computations on a topologically encoded qubit. Science, 345: 302, 2014. 10.1126/​science.1253742.

[7] M. Müller, A. Rivas, E. A. Martínez, D. Nigg, P. Schindler, T. Monz, R. Blatt, and M. A. Martin-Delgado. Iterative phase optimization of elementary quantum error correcting codes. Phys. Rev. X, 6: 031030, Aug 2016. 10.1103/​PhysRevX.6.031030.

[8] A. Bermudez, X. Xu, R. Nigmatullin, J. O'Gorman, V. Negnevitsky, P. Schindler, T. Monz, U. G. Poschinger, C. Hempel, J. Home, F. Schmidt-Kaler, M. Biercuk, R. Blatt, S. Benjamin, and M. Müller. Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation. Phys. Rev. X, 7: 041061, Dec 2017. 10.1103/​PhysRevX.7.041061.

[9] Philipp Schindler, Julio T. Barreiro, Thomas Monz, Volckmar Nebendahl, Daniel Nigg, Michael Chwalla, Markus Hennrich, and Rainer Blatt. Experimental repetitive quantum error correction. Science, 332 (6033): 1059, 2011. 10.1126/​science.1203329.

[10] David Kielpinski, Amit Ben-Kish, Joe Britton, Volker Meyer, Mary Rowe, Wayne Itano, David Wineland, Charles Sackett, and Christopher Monroe. Recent results in trapped-ion quantum computing at nist. Quantum Info. Comput., 1: 113, 12 2001. 10.26421/​qic1.s-12.

[11] J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, and D. J. Wineland. Realization of quantum error correction. Nature, 432: 602, 2004. 10.1038/​nature03074.

[12] Tobias Olsacher, Lukas Postler, Philipp Schindler, Thomas Monz, Peter Zoller, and Lukas M. Sieberer. Scalable and parallel tweezer gates for quantum computing with long ion strings. PRX Quantum, 1: 020316, Dec 2020. 10.1103/​PRXQuantum.1.020316.

[13] V. Negnevitsky, M. Marinelli, K. K. Mehta, H. Y. Lo, C. Flühmann, and J. P. Home. Repeated multi-qubit readout and feedback with a mixed-species trapped-ion register. Nature, 563: 527, 2018. 10.1038/​s41586-018-0668-z.

[14] Laird Egan, Dripto M. Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R. Brown, Marko Cetina, and Christopher Monroe. Fault-tolerant operation of a quantum error-correction code, 2020. URL https:/​/​​abs/​2009.11482.

[15] D. Zhu, N. M. Linke, M. Benedetti, K. A. Landsman, N. H. Nguyen, C. H. Alderete, A. Perdomo-Ortiz, N. Korda, A. Garfoot, C. Brecque, L. Egan, O. Perdomo, and C. Monroe. Training of quantum circuits on a hybrid quantum computer. Science Advances, 5 (10), 2019. 10.1126/​sciadv.aaw9918.

[16] Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I.-J. Wang, Simon Gustavsson, and William D. Oliver. Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics, 11 (1): 369, Mar 2020. 10.1146/​annurev-conmatphys-031119-050605.

[17] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and John M. Martinis. State preservation by repetitive error detection in a superconducting quantum circuit. Nature, 519: 66, Mar 2015. 10.1038/​nature14270.

[18] Maika Takita, A. D. Córcoles, Easwar Magesan, Baleegh Abdo, Markus Brink, Andrew Cross, Jerry M. Chow, and Jay M. Gambetta. Demonstration of weight-four parity measurements in the surface code architecture. Phys. Rev. Lett., 117: 210505, Nov 2016. 10.1103/​PhysRevLett.117.210505.

[19] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, Steven Girvin, Liang Jiang, Mazyar Mirrahimi, M. Devoret, and R. Schoelkopf. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature, 536: 441, 07 2016a. 10.1038/​nature18949.

[20] Christian Kraglund Andersen, Ants Remm, Stefania Lazar, Sebastian Krinner, Nathan Lacroix, Graham J. Norris, Mihai Gabureac, Christopher Eichler, and Andreas Wallraff. Repeated quantum error detection in a surface code. Nature Physics, 16: 875, 2020. 10.1038/​s41567-020-0920-y.

[21] Jingfu Zhang, Raymond Laflamme, and Dieter Suter. Experimental implementation of encoded logical qubit operations in a perfect quantum error correcting code. Phys. Rev. Lett., 109: 100503, 2012. 10.1103/​PhysRevLett.109.100503.

[22] E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne. Benchmarking quantum computers: The five-qubit error correction code. Phys. Rev. Lett., 86 (18): 5811, 2001. 10.1103/​PhysRevLett.86.5811.

[23] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbrüggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du, P. Neumann, and et al. Quantum error correction in a solid-state hybrid spin register. Nature, 506 (7487): 204, Feb 2014. 10.1038/​nature12919.

[24] Thomas Unden, Priya Balasubramanian, Daniel Louzon, Yuval Vinkler, Martin B. Plenio, Matthew Markham, Daniel Twitchen, Alastair Stacey, Igor Lovchinsky, Alexander O. Sushkov, Mikhail D. Lukin, Alex Retzker, Boris Naydenov, Liam P. McGuinness, and Fedor Jelezko. Quantum Metrology Enhanced by Repetitive Quantum Error Correction. Physical Review Letters, 116: 230502, 2016. 10.1103/​PhysRevLett.116.230502.

[25] B M Terhal, J Conrad, and C Vuillot. Towards scalable bosonic quantum error correction. Quantum Science and Technology, 5 (4): 043001, Jul 2020. 10.1088/​2058-9565/​ab98a5.

[26] Kyungjoo Noh and Christopher Chamberland. Fault-tolerant bosonic quantum error correction with the surface–gottesman-kitaev-preskill code. Physical Review A, 101 (1), Jan 2020. 10.1103/​PhysRevA.101.012316.

[27] C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home. Encoding a qubit in a trapped-ion mechanical oscillator. Nature, 566 (7745): 513, Feb 2019. 10.1038/​s41586-019-0960-6.

[28] Brennan de Neeve, Thanh Long Nguyen, Tanja Behrle, and Jonathan Home. Error correction of a logical grid state qubit by dissipative pumping, 2020. URL https:/​/​​abs/​2010.09681.

[29] C. Flühmann, V. Negnevitsky, M. Marinelli, and J. P. Home. Sequential Modular Position and Momentum Measurements of a Trapped Ion Mechanical Oscillator. Physical Review X, 8: 021001, 2018. 10.1103/​PhysRevX.8.021001.

[30] P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584: 368, 2020. 10.1038/​s41586-020-2603-3.

[31] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, S. M. Girvin, L. Jiang, Mazyar Mirrahimi, M. H. Devoret, and R. J. Schoelkopf. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature, 536: 441, 2016b. 10.1038/​nature18949.

[32] Alexander Erhard, Joel J. Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban A. Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, and Rainer Blatt. Characterizing large-scale quantum computers via cycle benchmarking. Nature Communications, 10 (1), Nov 2019. 10.1038/​s41467-019-13068-7.

[33] Christopher E Langer. High Fidelity Quantum Information Processing with Trapped Ions. PhD thesis, University of Colorado at Boulder, 2006.

[34] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Physical Review Letters, 117 (6): 060504, Aug 2016. 10.1103/​PhysRevLett.117.060504.

[35] T. P. Harty, M. A. Sepiol, D. T. C. Allcock, C. J. Ballance, J. E. Tarlton, and D. M. Lucas. High-fidelity trapped-ion quantum logic using near-field microwaves. Phys. Rev. Lett., 117: 140501, Sep 2016. 10.1103/​PhysRevLett.117.140501.

[36] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland. High-fidelity universal gate set for ${^{9}\mathrm{Be}}^{+}$ ion qubits. Phys. Rev. Lett., 117: 060505, Aug 2016. 10.1103/​PhysRevLett.117.060505.

[37] Ye Wang, Mark Um, Junhua Zhang, Shuoming An, Ming Lyu, Jing-Ning Zhang, L.-M. Duan, Dahyun Yum, and Kihwan Kim. Single-qubit quantum memory exceeding ten-minute coherence time. Nature Photonics, 11 (10): 646, Sep 2017. 10.1038/​s41566-017-0007-1.

[38] V. M. Schäfer, C. J. Ballance, K. Thirumalai, L. J. Stephenson, T. G. Ballance, A. M. Steane, and D. M. Lucas. Fast quantum logic gates with trapped-ion qubits. Nature, 555: 75, 2018. 10.1038/​nature25737.

[39] C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim. Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Phys. Rev. A, 89: 022317, Feb 2014. 10.1103/​PhysRevA.89.022317.

[40] Naomi H. Nickerson, Joseph F. Fitzsimons, and Simon C. Benjamin. Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links. Physical Review X, 4: 041041, 2014. 10.1103/​PhysRevX.4.041041.

[41] D. Kielpinski, C. Monroe, and D. J. Wineland. Architecture for a large-scale ion-trap quantum computer. Nature, 417 (6890): 709, Jun 2002. 10.1038/​nature00784.

[42] Kaushal Vidyut, Bjoern Lekitsch, A. Stahl, J. Hilder, Daniel Pijn, C. Schmiegelow, Alejandro Bermudez, M. Müller, Ferdinand Schmidt-Kaler, and U. Poschinger. Shuttling-based trapped-ion quantum information processing. AVS Quantum Science, 2: 014101, 02 2020. 10.1116/​1.5126186.

[43] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, and et al. Demonstration of the trapped-ion quantum ccd computer architecture. Nature, 592 (7853): 209–213, Apr 2021. 10.1038/​s41586-021-03318-4.

[44] H. Kaufmann, T. Ruster, C. T. Schmiegelow, M. A. Luda, V. Kaushal, J. Schulz, D. Von Lindenfels, F. Schmidt-Kaler, and U. G. Poschinger. Scalable Creation of Long-Lived Multipartite Entanglement. Physical Review Letters, 119: 150503, 2017. 10.1103/​PhysRevLett.119.150503.

[45] Yong Wan, Daniel Kienzler, Stephen D. Erickson, Karl H. Mayer, Ting Rei Tan, Jenny J. Wu, Hilma M. Vasconcelos, Scott Glancy, Emanuel Knill, David J. Wineland, Andrew C. Wilson, and Dietrich Leibfried. Quantum gate teleportation between separated qubits in a trapped-ion processor. Science, 364 (6443): 875, 2019. 10.1126/​science.aaw9415.

[46] K. A. Landsman, Y. Wu, P. H. Leung, D. Zhu, N. M. Linke, K. R. Brown, L. Duan, and C. Monroe. Two-qubit entangling gates within arbitrarily long chains of trapped ions. Physical Review A, 100 (2): 022332, Aug 2019. 10.1103/​PhysRevA.100.022332.

[47] Sangtaek Kim. Acousto-Optic Devices for Optical Signal Processing and Quantum Computing. PhD thesis, USA, 2008.

[48] Philipp Schindler. Private communication, 2020.

[49] Norbert M. Linke, Dmitri Maslov, Martin Roetteler, Shantanu Debnath, Caroline Figgatt, Kevin A. Landsman, Kenneth Wright, and Christopher Monroe. Experimental comparison of two quantum computing architectures. Proceedings of the National Academy of Sciences, 114 (13): 3305, Mar 2017. 10.1073/​pnas.1618020114.

[50] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2, Jan 2003. 10.1016/​s0003-4916(02)00018-0.

[51] A. Yu. Kitaev. Quantum Error Correction with Imperfect Gates, page 181. Springer US, Boston, MA, 1997. 10.1007/​978-1-4615-5923-8_19.

[52] H Bombin and M A Martin-Delgado. Topological quantum distillation. Phys. Rev. Lett., 97 (18): 180501, 2006. 10.1103/​PhysRevLett.97.180501.

[53] Daniel A. Lidar and Todd A. Brun. Quantum error correction. Cambridge University Press, 2013. 10.1017/​CBO9781139034807.

[54] Barbara M. Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics, 87 (2): 307, Apr 2015. 10.1103/​revmodphys.87.307.

[55] A M Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77 (5): 793, 1996. 10.1103/​PhysRevLett.77.793.

[56] David P. DiVincenzo and Peter W. Shor. Fault-tolerant error correction with efficient quantum codes. Phys. Rev. Lett., 77: 3260, 1996. 10.1103/​PhysRevLett.77.3260.

[57] David P. DiVincenzo and Panos Aliferis. Effective Fault-Tolerant Quantum Computation with Slow Measurements. Phys. Rev. Lett., 98: 020501, 2007. 10.1103/​PhysRevLett.98.020501.

[58] Muyuan Li, Mauricio Gutiérrez, Stanley E. David, Alonzo Hernandez, and Kenneth R. Brown. Fault tolerance with bare ancillary qubits for a [[7,1,3]] code. Phys. Rev. A, 96: 032341, Sep 2017. 10.1103/​PhysRevA.96.032341.

[59] Christopher Chamberland and Michael E. Beverland. Flag fault-tolerant error correction with arbitrary distance codes. Quantum, 2: 53, Feb 2018. 10.22331/​q-2018-02-08-53.

[60] Christopher Chamberland and Kyungjoo Noh. Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits. npj Quantum Information, 6 (1), Oct 2020. 10.1038/​s41534-020-00319-5.

[61] Christopher Chamberland, Aleksander Kubica, Theodore J Yoder, and Guanyu Zhu. Triangular color codes on trivalent graphs with flag qubits. New Journal of Physics, 22 (2): 023019, Feb 2020. 10.1088/​1367-2630/​ab68fd.

[62] Rui Chao and Ben W. Reichardt. Quantum Error Correction with only Two Extra Qubits. Physical Review Letters, 121: 050502, 2018. 10.1103/​PhysRevLett.121.050502.

[63] Rui Chao and Ben Reichardt. Fault-tolerant quantum computation with few qubits. npj Quantum Information, 4, 05 2017. 10.1038/​s41534-018-0085-z.

[64] Ben W Reichardt. Fault-tolerant quantum error correction for steane’s seven-qubit color code with few or no extra qubits. Quantum Science and Technology, 6 (1): 015007, Dec 2020. 10.1088/​2058-9565/​abc6f4.

[65] Colin J Trout, Muyuan Li, Mauricio Gutiérrez, Yukai Wu, Sheng-Tao Wang, Luming Duan, and Kenneth R Brown. Simulating the performance of a distance-3 surface code in a linear ion trap. New Journal of Physics, 20 (4): 043038, apr 2018. 10.1088/​1367-2630/​aab341.

[66] M. Gutiérrez, M. Müller, and A. Bermúdez. Transversality and lattice surgery: Exploring realistic routes toward coupled logical qubits with trapped-ion quantum processors. Phys. Rev. A, 99: 022330, Feb 2019. 10.1103/​PhysRevA.99.022330.

[67] A. Bermudez, X. Xu, M. Gutiérrez, S. C. Benjamin, and M. Müller. Fault-tolerant protection of near-term trapped-ion topological qubits under realistic noise sources. Physical Review A, 100: 062307, 2019. 10.1103/​PhysRevA.100.062307.

[68] Natalie C. Brown and Kenneth R. Brown. Leakage mitigation for quantum error correction using a mixed qubit scheme. Phys. Rev. A, 100: 032325, Sep 2019. 10.1103/​PhysRevA.100.032325.

[69] Swarnadeep Majumder, Leonardo Andreta de Castro, and Kenneth R. Brown. Real-time calibration with spectator qubits. npj Quantum Information, 6 (1): 19, Feb 2020. 10.1038/​s41534-020-0251-y.

[70] Dripto M. Debroy, Muyuan Li, Shilin Huang, and Kenneth R. Brown. Logical performance of 9 qubit compass codes in ion traps with crosstalk errors. Quantum Science and Technology, 5 (3), 2020. 10.1088/​2058-9565/​ab7e80.

[71] Mohan Sarovar, Timothy Proctor, Kenneth Rudinger, Kevin Young, Erik Nielsen, and Robin Blume-Kohout. Detecting crosstalk errors in quantum information processors. Quantum, 4: 321, 2020. 10.22331/​q-2020-09-11-321.

[72] Daniel A. Lidar. Review of Decoherence-Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling, page 295. John Wiley & Sons, Inc., Feb 2014. 10.1002/​9781118742631.

[73] Debbie W. Leung, Isaac L. Chuang, Fumiko Yamaguchi, and Yoshihisa Yamamoto. Efficient implementation of coupled logic gates for quantum computation. Physical Review A, 61 (4), Mar 2000. 10.1103/​PhysRevA.61.042310.

[74] David Hayes, Steven T Flammia, and Michael J Biercuk. Programmable quantum simulation by dynamic hamiltonian engineering. New Journal of Physics, 16 (8): 083027, Aug 2014. 10.1088/​1367-2630/​16/​8/​083027.

[75] A.F. Varón C. Piltz, T. Sriarunothai and C. Wunderlich. A trapped-ion-based quantum byte with 10-5 next-neighbour cross-talk. Nature Communications, 5 (1): 4679, 2014. 10.1038/​ncomms5679.

[76] Prakash Murali, David C. Mckay, Margaret Martonosi, and Ali Javadi-Abhari. Software mitigation of crosstalk on noisy intermediate-scale quantum computers. Proceedings of the Twenty-Fifth International Conference on Architectural Support for Programming Languages and Operating Systems, page 1001, Mar 2020. 10.1145/​3373376.3378477.

[77] L. M. K. Vandersypen and I. L. Chuang. Nmr techniques for quantum control and computation. Reviews of Modern Physics, 76 (4): 1037, Jan 2005. 10.1103/​revmodphys.76.1037.

[78] Matthew Reagor, Christopher B. Osborn, Nikolas Tezak, Alexa Staley, Guenevere Prawiroatmodjo, Michael Scheer, Nasser Alidoust, Eyob A. Sete, Nicolas Didier, Marcus P. da Silva, and et al. Demonstration of universal parametric entangling gates on a multi-qubit lattice. Science Advances, 4 (2): eaao3603, Feb 2018. 10.1126/​sciadv.aao3603.

[79] Robin Harper, Steven T. Flammia, and Joel J. Wallman. Efficient learning of quantum noise. Nature Physics, Aug 2020. 10.1038/​s41567-020-0992-8.

[80] D. P. L. Aude Craik, N. M. Linke, M. A. Sepiol, T. P. Harty, J. F. Goodwin, C. J. Ballance, D. N. Stacey, A. M. Steane, D. M. Lucas, and D. T. C. Allcock. High-fidelity spatial and polarization addressing of ca+43 qubits using near-field microwave control. Physical Review A, 95 (2), Feb 2017. 10.1103/​PhysRevA.95.022337.

[81] Lorenza Viola, Emanuel Knill, and Seth Lloyd. Dynamical decoupling of open quantum systems. Physical Review Letters, 82 (12): 2417, Mar 1999. 10.1103/​physrevlett.82.2417.

[82] Michael H. Goerz, K. Birgitta Whaley, and Christiane P. Koch. Hybrid optimization schemes for quantum control. EPJ Quantum Technology, 2: 21, 2015. 10.1140/​epjqt/​s40507-015-0034-0.

[83] Hideo Mabuchi and Navin Khaneja. Principles and applications of control in quantum systems. International Journal of Robust and Nonlinear Control, 15: 647, 2005. 10.1002/​rnc.1016.

[84] Raj Chakrabarti and Herschel Rabitz. Quantum control landscapes. International Reviews in Physical Chemistry, 26 (4): 671, 2007. 10.1080/​01442350701633300.

[85] Howard M. Wiseman and Gerard J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2009. 10.1017/​CBO9780511813948.

[86] Philipp Schindler, Daniel Nigg, Thomas Monz, Julio T. Barreiro, Esteban Martinez, Shannon X. Wang, Stephan Quint, Matthias F. Brandl, Volckmar Nebendahl, Christian F. Roos, Michael Chwalla, Markus Hennrich, and Rainer Blatt. A quantum information processor with trapped ions. New Journal of Physics, 15, 2013. 10.1088/​1367-2630/​15/​12/​123012.

[87] H. C. Nägerl, D. Leibfried, H. Rohde, G. Thalhammer, J. Eschner, F. Schmidt-Kaler, and R. Blatt. Laser addressing of individual ions in a linear ion trap. Phys. Rev. A, 60: 145, Jul 1999. 10.1103/​PhysRevA.60.145.

[88] K. Mølmer and A. Sørensen. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett., 82 (9): 1835, 1999. 10.1103/​PhysRevLett.82.1835.

[89] Anders Sørensen and Klaus Mølmer. Quantum computation with ions in thermal motion. Physical Review Letters, 82: 1971, 1999. 10.1103/​PhysRevLett.82.1971.

[90] Christian F. Roos. Ion trap quantum gates with amplitude-modulated laser beams. New Journal of Physics, 10, 2008. 10.1088/​1367-2630/​10/​1/​013002.

[91] V. Nebendahl. Optimized quantum error-correction codes for experiments. Phys. Rev. A, 91: 022332, 2015. 10.1103/​PhysRevA.91.022332.

[92] Dmitri Maslov and Martin Roetteler. Shorter stabilizer circuits via bruhat decomposition and quantum circuit transformations. IEEE Transactions on Information Theory, 64 (7): 4729, Jul 2018. 10.1109/​TIT.2018.2825602.

[93] P. Oscar Boykin, Tal Mor, Matthew Pulver, Vwani Roychowdhury, and Farrokh Vatan. On universal and fault-tolerant quantum computing: A novel basis and a new constructive proof of universality for shor's basis. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, FOCS '99, page 486. IEEE Computer Society, 1999. 10.1109/​SFFCS.1999.814621.

[94] J. True Merrill, S. Charles Doret, Grahame Vittorini, J. P. Addison, and Kenneth R. Brown. Transformed composite sequences for improved qubit addressing. Phys. Rev. A, 90: 040301, Oct 2014. 10.1103/​PhysRevA.90.040301.

[95] Mauricio Gutierrez, Lukas Svec, Alexander Vargo, and Kenneth R. Brown. Approximation of real error channels by Clifford channels and Pauli measurements. Phys. Rev. A, 87: 030302, 2013. 10.1103/​PhysRevA.87.030302.

[96] Austin G. Fowler. Coping with qubit leakage in topological codes. Phys. Rev. A, 88: 042308, Oct 2013. 10.1103/​PhysRevA.88.042308.

[97] A. R. Calderbank and P. W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54 (2): 1098, 1996. 10.1103/​PhysRevA.54.1098.

[98] E. Dennis, A. Kitaev, A. Landahl, and Preskill J. Topological quantum memory. J. Math. Phys., 43 (9): 4452, 2002. 10.1063/​1.1499754.

[99] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71: 022316, 2005. 10.1103/​PhysRevA.71.022316.

[100] Andrew J. Landahl and Ciaran Ryan-Anderson. Quantum computing by color-code lattice surgery, 2014. URL https:/​/​​abs/​1407.5103.

[101] Sergey Bravyi and Andrew Cross. Doubled color codes, 2015. URL https:/​/​​abs/​1509.03239.

[102] Tomas Jochym-O'Connor and Stephen D. Bartlett. Stacked codes: Universal fault-tolerant quantum computation in a two-dimensional layout. Phys. Rev. A, 93: 022323, Feb 2016. 10.1103/​PhysRevA.93.022323.

[103] Cody Jones, Peter Brooks, and Jim Harrington. Gauge color codes in two dimensions. Phys. Rev. A, 93: 052332, May 2016. 10.1103/​PhysRevA.93.052332.

[104] A. M. Steane. Active Stabilization, Quantum Computation, and Quantum State Synthesis. Phys. Rev. Lett., 78: 2252, Mar 1997. 10.1103/​PhysRevLett.78.2252.

[105] E. Knill. Quantum computing with realistically noisy devices. Nature, 434: 39, Mar 2005. 10.1038/​nature03350.

[106] Ciaran Ryan-Anderson. Quantum algorithms, architecture, and error correction, 2018a. URL https:/​/​​abs/​1812.04735.

[107] Ciaran Ryan-Anderson. Pecos: Performance estimator of codes on surfaces. https:/​/​​PECOS-packages/​PECOS, 2018b.

[108] Damian S. Steiger, Thomas Häner, and Matthias Troyer. ProjectQ: an open source software framework for quantum computing. Quantum, 2: 49, January 2018. 10.22331/​q-2018-01-31-49.

[109] Thomas Häner, Damian S Steiger, Krysta Svore, and Matthias Troyer. A software methodology for compiling quantum programs. Quantum Science and Technology, 3 (2): 020501, feb 2018. 10.1088/​2058-9565/​aaa5cc.

[110] Anders Sørensen and Klaus Mølmer. Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A, 62: 022311, Jul 2000. 10.1103/​PhysRevA.62.022311.

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[1] Lukas Postler, Sascha Heuβen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D. Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus Müller, and Thomas Monz, "Demonstration of fault-tolerant universal quantum gate operations", Nature 605 7911, 675 (2022).

[2] Martin Ringbauer, Michael Meth, Lukas Postler, Roman Stricker, Rainer Blatt, Philipp Schindler, and Thomas Monz, "A universal qudit quantum processor with trapped ions", Nature Physics 18 9, 1053 (2022).

[3] David Schwerdt, Yotam Shapira, Tom Manovitz, and Roee Ozeri, "Comparing two-qubit and multiqubit gates within the toric code", Physical Review A 105 2, 022612 (2022).

[4] J. Hilder, D. Pijn, O. Onishchenko, A. Stahl, M. Orth, B. Lekitsch, A. Rodriguez-Blanco, M. Müller, F. Schmidt-Kaler, and U. G. Poschinger, "Fault-Tolerant Parity Readout on a Shuttling-Based Trapped-Ion Quantum Computer", Physical Review X 12 1, 011032 (2022).

[5] Pedro Figueroa-Romero, Kavan Modi, Robert J. Harris, Thomas M. Stace, and Min-Hsiu Hsieh, "Randomized Benchmarking for Non-Markovian Noise", PRX Quantum 2 4, 040351 (2021).

[6] J. A. Montañez-Barrera, R. T. Holladay, G. P. Beretta, and Michael R. von Spakovsky, "Method for generating randomly perturbed density operators subject to different sets of constraints", Quantum Information Processing 21 9, 314 (2022).

[7] Charles H. Baldwin, Karl Mayer, Natalie C. Brown, Ciarán Ryan-Anderson, and David Hayes, "Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations", Quantum 6, 707 (2022).

[8] Bogdan M. Mihalcea, "Quasienergy operators and generalized squeezed states for systems of trapped ions", Annals of Physics 442, 168926 (2022).

[9] Mummadi Swathi and Bhawana Rudra, "A Novel Approach for Asymmetric Quantum Error Correction With Syndrome Measurement", IEEE Access 10, 44669 (2022).

[10] Ye-Hong Chen, Roberto Stassi, Wei Qin, Adam Miranowicz, and Franco Nori, "Fault-Tolerant Multiqubit Geometric Entangling Gates Using Photonic Cat-State Qubits", Physical Review Applied 18 2, 024076 (2022).

[11] C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, "Realization of Real-Time Fault-Tolerant Quantum Error Correction", Physical Review X 11 4, 041058 (2021).

[12] I. A. Simakov, I. S. Besedin, and A. V. Ustinov, "Simulation of the five-qubit quantum error correction code on superconducting qubits", Physical Review A 105 3, 032409 (2022).

[13] Stephen D. Erickson, Jenny J. Wu, Pan-Yu Hou, Daniel C. Cole, Shawn Geller, Alex Kwiatkowski, Scott Glancy, Emanuel Knill, Daniel H. Slichter, Andrew C. Wilson, and Dietrich Leibfried, "High-Fidelity Indirect Readout of Trapped-Ion Hyperfine Qubits", Physical Review Letters 128 16, 160503 (2022).

[14] Roman Stricker, Davide Vodola, Alexander Erhard, Lukas Postler, Michael Meth, Martin Ringbauer, Philipp Schindler, Rainer Blatt, Markus Müller, and Thomas Monz, "Characterizing Quantum Instruments: From Nondemolition Measurements to Quantum Error Correction", PRX Quantum 3 3, 030318 (2022).

[15] Chung-You Shih, Sainath Motlakunta, Nikhil Kotibhaskar, Manas Sajjan, Roland Hablützel, and Rajibul Islam, "Reprogrammable and high-precision holographic optical addressing of trapped ions for scalable quantum control", npj Quantum Information 7, 57 (2021).

[16] Seungchan Seo, Jiheon Seong, and Joonwoo Bae, "Mitigation of Crosstalk Errors in a Quantum Measurement and Its Applications", arXiv:2112.10651.

The above citations are from Crossref's cited-by service (last updated successfully 2022-09-24 16:12:24) and SAO/NASA ADS (last updated successfully 2022-09-24 16:12:25). The list may be incomplete as not all publishers provide suitable and complete citation data.