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

Noah Kaufmann1,2,*, Ivan Rojkov1, and Florentin Reiter1,3,†

  • *Contact author: noah.kaufmann@nbi.ku.dk, kaufmann.noah@hotmail.com
  • †Contact author: freiter@phys.ethz.ch
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Phys. Rev. Applied 23, 034014 – Published 7 March, 2025

DOI: https://doi.org/10.1103/PhysRevApplied.23.034014

Abstract

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

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References (83)

  1. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  2. C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott, Correlated charge noise and relaxation errors in superconducting qubits, Nature 594, 369 (2021).
  3. S. Krinner, S. Lazar, A. Remm, C. K. Andersen, N. Lacroix, G. J. Norris, C. Hellings, M. Gabureac, C. Eichler, and A. Wallraff, Benchmarking coherent errors in controlled-phase gates due to spectator qubits, Phys. Rev. Appl. 14, 024042 (2020).
  4. S. Resch and U. R. Karpuzcu, Benchmarking quantum computers and the impact of quantum noise, ACM Comput Surv (CSUR) 54, 1 (2021).
  5. E. Nielsen, K. Rudinger, T. Proctor, A. Russo, K. Young, and R. Blume-Kohout, Probing quantum processor performance with pyGSTi, Quantum Sci. Technol. 5, 044002 (2020).
  6. W. Jiang, J. Xiong, and Y. Shi, A co-design framework of neural networks and quantum circuits towards quantum advantage, Nat. Commun. 12, 1 (2021).
  7. R. Harper, S. T. Flammia, and J. J. Wallman, Efficient learning of quantum noise, Nat. Phys. 16, 1184 (2020).
  8. K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
  9. J. Wallman, C. Granade, R. Harper, and S. T. Flammia, Estimating the coherence of noise, New J. Phys. 17, 113020 (2015).
  10. J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, R. Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak, and A. Morello, Storing quantum information for 30 seconds in a nanoelectronic device, Nat. Nanotechnol. 9, 986 (2014).
  11. M. A. Fogarty, M. Veldhorst, R. Harper, C. H. Yang, S. D. Bartlett, S. T. Flammia, and A. S. Dzurak, Nonexponential fidelity decay in randomized benchmarking with low-frequency noise, Phys. Rev. A 92, 022326 (2015).
  12. D. Suter and G. A. Alvarez, Colloquium: Protecting quantum information against environmental noise, Rev. Mod. Phys. 88, 041001 (2016).
  13. H. Ball, W. D. Oliver, and M. J. Biercuk, The role of master clock stability in quantum information processing, npj Quantum Inf. 2, 1 (2016).
  14. J. Yoneda, K. Takeda, T. Otsuka, T. Nakajima, M. R. Delbecq, G. Allison, T. Honda, T. Kodera, S. Oda, and Y. Hoshi et al., A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%, Nat. Nanotechnol. 13, 102 (2018).
  15. I. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, J. Mod. Opt. 44, 2455 (1997).
  16. J. F. Poyatos, J. I. Cirac, and P. Zoller, Complete characterization of a quantum process: The two-bit quantum gate, Phys. Rev. Lett. 78, 390 (1997).
  17. R. Blume-Kohout, J. K. Gamble, E. Nielsen, J. Mizrahi, J. D. Sterk, and P. Maunz, Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit, arXiv:1310.4492.
  18. S. T. Merkel, J. M. Gambetta, J. A. Smolin, S. Poletto, A. D. Córcoles, B. R. Johnson, C. A. Ryan, and M. Steffen, Self-consistent quantum process tomography, Phys. Rev. A 87, 062119 (2013).
  19. H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Scalable multiparticle entanglement of trapped ions, Nature 438, 643 (2005).
  20. J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclassical Opt. 7, S347 (2005).
  21. E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
  22. E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011).
  23. M. Sarovar, T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, Detecting crosstalk errors in quantum information processors, Quantum 4, 321 (2020).
  24. E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nat. Phys. 19, 1116 (2023).
  25. R. Harper, W. Yu, and S. T. Flammia, Fast estimation of sparse quantum noise, PRX Quantum 2, 010322 (2021).
  26. M. L. Dahlhauser and T. S. Humble, Benchmarking characterization methods for noisy quantum circuits, Phys. Rev. A 109, 042620 (2024).
  27. O. Kern, G. Alber, and D. L. Shepelyansky, Quantum error correction of coherent errors by randomization, EPJ D 32, 153 (2005).
  28. J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A 94, 052325 (2016).
  29. J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest, J. Baugh, D. G. Cory, and R. Laflamme, Symmetrized characterization of noisy quantum processes, Science 317, 1893 (2007).
  30. O. Moussa, M. P. da Silva, C. A. Ryan, and R. Laflamme, Practical experimental certification of computational quantum gates using a twirling procedure, Phys. Rev. Lett. 109, 070504 (2012).
  31. M. R. Geller and Z. Zhou, Efficient error models for fault-tolerant architectures and the Pauli twirling approximation, Phys. Rev. A 88, 012314 (2013).
  32. IBM Quantum. https://quantum-computing.ibm.com/ (2021).
  33. Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017).
  34. E. Magesan, D. Puzzuoli, C. E. Granade, and D. G. Cory, Modeling quantum noise for efficient testing of fault-tolerant circuits, Phys. Rev. A 87, 012324 (2013).
  35. Y. R. Sanders, J. J. Wallman, and B. C. Sanders, Bounding quantum gate error rate based on reported average fidelity, New J. Phys. 18, 012002 (2015).
  36. S. J. Beale, J. J. Wallman, M. Gutiérrez, K. R. Brown, and R. Laflamme, Quantum error correction decoheres noise, Phys. Rev. Lett. 121, 190501 (2018).
  37. J. K. Iverson and J. Preskill, Coherence in logical quantum channels, New J. Phys. 22, 073066 (2020).
  38. E. Huang, A. C. Doherty, and S. Flammia, Performance of quantum error correction with coherent errors, Phys. Rev. A 99, 022313 (2019).
  39. D. Gottesman, The Heisenberg representation of quantum computers (Los Alamos National Lab. (LANL), Los Alamos, NM (United States), 1998).
  40. G. Cenedese, G. Benenti, and M. Bondani, Correcting coherent errors by random operation on actual quantum hardware, Entropy 25, 324 (2023).
  41. M. Ware, G. Ribeill, D. Ristè, C. A. Ryan, B. Johnson, and M. P. da Silva, Experimental Pauli-frame randomization on a superconducting qubit, Phys. Rev. A 103, 042604 (2021).
  42. S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, Mitigating measurement errors in multiqubit experiments, Phys. Rev. A 103, 042605 (2021).
  43. M. Gutiérrez, L. Svec, A. Vargo, and K. R. Brown, Approximation of realistic errors by Clifford channels and Pauli measurements, Phys. Rev. A 87, 030302 (2013).
  44. A. Carignan-Dugas, M. Alexander, and J. Emerson, A polar decomposition for quantum channels (with applications to bounding error propagation in quantum circuits), Quantum 3, 173 (2019).
  45. S. Kimmel, M. P. da Silva, C. A. Ryan, B. R. Johnson, and T. Ohki, Robust extraction of tomographic information via randomized benchmarking, Phys. Rev. X 4, 011050 (2014).
  46. S. J. Devitt, W. J. Munro, and K. Nemoto, Quantum error correction for beginners, Rep. Prog. Phys. 76, 076001 (2013).
  47. R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia, Comparing experiments to the fault-tolerance threshold, Phys. Rev. Lett. 117, 170502 (2016).
  48. Y. Suzuki, K. Fujii, and M. Koashi, Efficient simulation of quantum error correction under coherent error based on the nonunitary free-fermionic formalism, Phys. Rev. Lett. 119, 190503 (2017).
  49. D. Greenbaum, Introduction to quantum gate set tomography, arXiv:1509.02921.
  50. C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, Experimental quantum compressed sensing for a seven-qubit system, Nat. Commun. 8, 15305 (2017).
  51. H.-Y. Huang, R. Kueng, and J. Preskill, Efficient estimation of Pauli observables by derandomization, Phys. Rev. Lett. 127, 030503 (2021).
  52. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  53. C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, Experimental comparison of efficient tomography schemes for a six-qubit state, Phys. Rev. Lett. 113, 040503 (2014).
  54. G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Permutationally invariant quantum tomography, Phys. Rev. Lett. 105, 250403 (2010).
  55. F. Bloch, Nuclear induction, Phys. Rev. 70, 460 (1946).
  56. D. Bultrini, M. H. Gordon, E. López, and G. Sierra, Simple mitigation strategy for a systematic gate error in ibmq, J. Appl. Math. Phys. 9, 1215 (2021).
  57. N. Sundaresan, I. Lauer, E. Pritchett, E. Magesan, P. Jurcevic, and J. M. Gambetta, Reducing unitary and spectator errors in cross resonance with optimized rotary echoes, PRX Quantum 1, 020318 (2020).
  58. S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Phys. Rev. X 8, 031027 (2018).
  59. M. S. Blok, V. V. Ramasesh, T. Schuster, K. O’Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi, Quantum information scrambling on a superconducting qutrit processor, Phys. Rev. X 11, 021010 (2021).
  60. A. Morvan, V. V. Ramasesh, M. S. Blok, J. M. Kreikebaum, K. O’Brien, L. Chen, B. K. Mitchell, R. K. Naik, D. I. Santiago, and I. Siddiqi, Qutrit randomized benchmarking, Phys. Rev. Lett. 126, 210504 (2021).
  61. N. Goss, A. Morvan, B. Marinelli, B. K. Mitchell, L. B. Nguyen, R. K. Naik, L. Chen, C. Jünger, J. M. Kreikebaum, D. I. Santiago, J. J. Wallman, and I. Siddiqi, High-fidelity qutrit entangling gates for superconducting circuits, Nat. Commun. 13, 7481 (2022).
  62. S. Cao, D. Lall, M. Bakr, G. Campanaro, S. D. Fasciati, J. Wills, V. Chidambaram, B. Shteynas, I. Rungger, and P. J. Leek, Efficient characterization of qudit logical gates with gate set tomography using an error-free virtual z gate model, Phys. Rev. Lett. 133, 120802 (2024).
  63. N. Goss, S. Ferracin, A. Hashim, A. Carignan-Dugas, J. M. Kreikebaum, R. K. Naik, D. I. Santiago, and I. Siddiqi, Extending the computational reach of a superconducting qutrit processor, npj Quantum Inf. 10, 101 (2024).
  64. C. J. Wood and J. M. Gambetta, Quantification and characterization of leakage errors, Phys. Rev. A 97, 032306 (2018).
  65. J. J. Wallman, M. Barnhill, and J. Emerson, Robust characterization of leakage errors, New J. Phys. 18, 043021 (2016).
  66. M. Gutiérrez, C. Smith, L. Lulushi, S. Janardan, and K. R. Brown, Errors and pseudothresholds for incoherent and coherent noise, Phys. Rev. A 94, 042338 (2016).
  67. D. Greenbaum and Z. Dutton, Modeling coherent errors in quantum error correction, Quantum Sci. Technol. 3, 015007 (2017).
  68. D. M. Debroy, M. Li, M. Newman, and K. R. Brown, Stabilizer slicing: Coherent error cancellations in low-density parity-check stabilizer codes, Phys. Rev. Lett. 121, 250502 (2018).
  69. S. Majumder, C. G. Yale, T. D. Morris, D. S. Lobser, A. D. Burch, M. N. H. Chow, M. C. Revelle, S. M. Clark, and R. C. Pooser, Characterizing and mitigating coherent errors in a trapped ion quantum processor using hidden inverses, Quantum 7, 1006 (2023).
  70. B. Zhang, S. Majumder, P. H. Leung, S. Crain, Y. Wang, C. Fang, D. M. Debroy, J. Kim, and K. R. Brown, Hidden inverses: Coherent error cancellation at the circuit level, Phys. Rev. Appl. 17, 034074 (2022).
  71. J. Berberich, D. Fink, and C. Holm, Robustness of quantum algorithms against coherent control errors, Phys. Rev. A 109, 012417 (2024).
  72. D. C. Murphy and K. R. Brown, Controlling error orientation to improve quantum algorithm success rates, Phys. Rev. A 99, 032318 (2019).
  73. Z. Cai, X. Xu, and S. C. Benjamin, Mitigating coherent noise using Pauli conjugation, npj Quantum Inf. 6, 1 (2020).
  74. J. P. Barnes, C. J. Trout, D. Lucarelli, and B. D. Clader, Quantum error-correction failure distributions: Comparison of coherent and stochastic error models, Phys. Rev. A 95, 062338 (2017).
  75. P. Iyer and D. Poulin, A small quantum computer is needed to optimize fault-tolerant protocols, Quantum Sci. Technol. 3, 030504 (2018).
  76. J. J. Wallman and S. T. Flammia, Randomized benchmarking with confidence, New J. Phys. 16, 103032 (2014).
  77. S. Bravyi, M. Englbrecht, R. König, and N. Peard, Correcting coherent errors with surface codes, npj Quantum Inf. 4, 1 (2018).
  78. P. Iyer and D. Poulin, Hardness of decoding quantum stabilizer codes, IEEE Trans. Inf. Theory 61, 5209 (2015).
  79. X. Xue, B. D’Anjou, T. F. Watson, D. R. Ward, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, W. A. Coish, and L. M. K. Vandersypen, Repetitive quantum nondemolition measurement and soft decoding of a silicon spin qubit, Phys. Rev. X 10, 021006 (2020).
  80. C. A. Pattison, M. E. Beverland, M. P. Da Silva, and N. Delfosse, Improved quantum error correction using soft information, arXiv:2107.13589.
  81. N. Raveendran, N. Rengaswamy, A. K. Pradhan, and B. Vasić, in 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Broomfield, CO, USA, 2022), pp. 275–281.
  82. D. M. Debroy, L. Egan, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Cetina, C. Monroe, and K. R. Brown, Optimizing stabilizer parities for improved logical qubit memories, Phys. Rev. Lett. 127, 240501 (2021).
  83. Y. Ouyang, Avoiding coherent errors with rotated concatenated stabilizer codes, npj Quantum Inf. 7, 1 (2021).

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Phys. Rev. Applied 23, 034014– Published 7 March, 2025
  • Received 17 November 2024
  • Accepted 4 February 2025
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