Relaxation times do not capture logical qubit dynamics

Amit Kumar Pal; Philipp Schindler; Alexander Erhard; Ángel Rivas; Miguel-Angel Martin-Delgado; Rainer Blatt; Thomas Monz; Markus Müller

doi:10.22331/q-2022-01-24-632

Relaxation times do not capture logical qubit dynamics

Amit Kumar Pal^1,2,3, Philipp Schindler⁴, Alexander Erhard⁴, Ángel Rivas^5,6, Miguel-Angel Martin-Delgado^5,6, Rainer Blatt^4,7, Thomas Monz^4,8, and Markus Müller^2,9,10

¹Department of Physics, Indian Institute of Technology Palakkad, Palakkad 678557, India
²Department of Physics, College of Science, Swansea University, Singleton Park, Swansea - SA2 8PP, United Kingdom
³Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warszawa, Poland
⁴Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria
⁵Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain
⁶CCS -Center for Computational Simulation, Campus de Montegancedo UPM, 28660 Boadilla del Monte, Madrid, Spain.
⁷Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria
⁸Alpine Quantum Technologies GmbH, 6020 Innsbruck, Austria
⁹Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
¹⁰Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany

Published:	2022-01-24, volume 6, page 632
Eprint:	arXiv:2012.07911v2
Doi:	https://doi.org/10.22331/q-2022-01-24-632
Citation:	Quantum 6, 632 (2022).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits. We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits. We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits. Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.

Popular summary

Relaxation describes processes that bring a system into equilibrium. A perfect qubit is a closed system that is not able to relax. In practice we do not have a perfect system and the qubit can couple to its environment which can be described by relaxation due to coupling of a perfect qubit to a classical bath. There are various experimental noise processes that couple the qubit to the environment, where in equilibrium all quantum correlations are lost in the bath and the state can be described by classical physics. Under the action of a noise process, the value of an observable that measures quantum correlations will therefore decay with time. Typically, the relaxation of physical qubits is expressed as an exponential decay, only characterized by the relaxation time. This simple picture does not necessarily hold for multiple qubits in experiments where noise processes are spatially correlated. We demonstrate that for small quantum error correction codes, spatially correlated noise can yield complex relaxation behavior that cannot be described by a single relaxation time.

► BibTeX data

@article{Pal2022relaxationtimesdo,
  doi = {10.22331/q-2022-01-24-632},
  url = {https://doi.org/10.22331/q-2022-01-24-632},
  title = {Relaxation times do not capture logical qubit dynamics},
  author = {Pal, Amit Kumar and Schindler, Philipp and Erhard, Alexander and Rivas, {\'{A}}ngel and Martin-Delgado, Miguel-Angel and Blatt, Rainer and Monz, Thomas and M{\"{u}}ller, Markus},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {632},
  month = jan,
  year = {2022}
}

► References

[1] A. Abragam. The principles of nuclear magnetism. Oxford, Clarendon Press, 1961.

[2] Dorit Aharonov, Alexei Kitaev, and John Preskill. Fault-tolerant quantum computation with long-range correlated noise. Phys. Rev. Lett., 96: 050504, Feb 2006. 10.1103/PhysRevLett.96.050504. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.050504.
https://doi.org/10.1103/PhysRevLett.96.050504

[3] 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. URL https://link.aps.org/doi/10.1103/PhysRevX.7.041061.
https://doi.org/10.1103/PhysRevX.7.041061

[4] H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002.

[5] J Chiaverini, D Leibfried, T Schaetz, MD Barrett, RB Blakestad, J Britton, WM Itano, JD Jost, E Knill, C Langer, R Ozeri, and DJ Wineland. Realization of quantum error correction. Nature, 432 (7017): 602—605, December 2004. ISSN 0028-0836. 10.1038/nature03074.
https://doi.org/10.1038/nature03074

[6] James P. Clemens, Shabnam Siddiqui, and Julio Gea-Banacloche. Quantum error correction against correlated noise. Phys. Rev. A, 69: 062313, Jun 2004. 10.1103/PhysRevA.69.062313. URL https://link.aps.org/doi/10.1103/PhysRevA.69.062313.
https://doi.org/10.1103/PhysRevA.69.062313

[7] Joshua Combes, Christopher Granade, Christopher Ferrie, and Steven T Flammia. Logical randomized benchmarking. arXiv:1702.03688, 2017. URL https://arxiv.org/abs/1702.03688.
arXiv:1702.03688

[8] Dripto M. Debroy, Muyuan Li, Shilin Huang, and Kenneth R. Brown. Logical performance of 9 qubit compass codes in ion traps with crosstalk errors. arXiv:1910.08495, 2019. URL https://arxiv.org/abs/1910.08495.
arXiv:1910.08495

[9] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. J. Math. Phys., 43 (9): 4452–4505, 2002. 10.1063/1.1499754. URL http://dx.doi.org/10.1063/1.1499754.
https://doi.org/10.1063/1.1499754

[10] C. K. Andersen et al. Repeated quantum error detection in a surface code. 2019. URL arXiv:1912.09410. 10.1038/s41567-020-0920-y.
https://doi.org/10.1038/s41567-020-0920-y
arXiv:1912.09410

[11] Steven T. Flammia and Yi-Kai Liu. Direct fidelity estimation from few pauli measurements. Phys. Rev. Lett., 106: 230501, Jun 2011. 10.1103/PhysRevLett.106.230501. URL https://link.aps.org/doi/10.1103/PhysRevLett.106.230501.
https://doi.org/10.1103/PhysRevLett.106.230501

[12] Jay M. Gambetta, Jerry M. Chow, and Matthias Steffen. Building logical qubits in a superconducting quantum computing system. Nature Phys. J. Quant. Inf., 3 (1): 2, 2017. ISSN 2056-6387. 10.1038/s41534-016-0004-0. URL https://doi.org/10.1038/s41534-016-0004-0.
https://doi.org/10.1038/s41534-016-0004-0

[13] M. Grassl, Th. Beth, and T. Pellizzari. Codes for the quantum erasure channel. Phys. Rev. A, 56: 33–38, 1997. 10.1103/PhysRevA.56.33.
https://doi.org/10.1103/PhysRevA.56.33

[14] H. Häffner, F. Schmidt-Kaler, W. Hänsel, C. F. Roos, T. Körber, M. Chwalla, M. Riebe, J. Benhelm, U. D. Rapol, C. Becher, and R. Blatt. Robust entanglement. Applied Physics B, 81 (2): 151–153, 2005. 10.1007/s00340-005-1917-z. URL https://doi.org/10.1007/s00340-005-1917-z.
https://doi.org/10.1007/s00340-005-1917-z

[15] 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, 2015. URL https://doi.org/10.1038/nature14270.
https://doi.org/10.1038/nature14270

[16] D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland. A decoherence-free quantum memory using trapped ions. Science, 291 (5506): 1013–1015, 2001. 10.1126/science.1057357. URL https://doi.org/10.1126/science.1057357.
https://doi.org/10.1126/science.1057357

[17] Rochus Klesse and Sandra Frank. Quantum error correction in spatially correlated quantum noise. Phys. Rev. Lett., 95: 230503, Nov 2005. 10.1103/PhysRevLett.95.230503. URL https://link.aps.org/doi/10.1103/PhysRevLett.95.230503.
https://doi.org/10.1103/PhysRevLett.95.230503

[18] Thaddeus D Ladd, Fedor Jelezko, Raymond Laflamme, Yasunobu Nakamura, Christopher Monroe, and Jeremy L O'Brien. Quantum Computing. Nature, 464 (7285): 45–53, sep 2010. ISSN 0028-0836. URL http://dx.doi.org/10.1038/nature08812.
https://doi.org/10.1038/nature08812

[19] D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett., 81: 2594–2597, Sep 1998. 10.1103/PhysRevLett.81.2594. URL https://link.aps.org/doi/10.1103/PhysRevLett.81.2594.
https://doi.org/10.1103/PhysRevLett.81.2594

[20] Daniel A. Lidar. Quantum Error Correction. Cambridge University Press, 2013.

[21] Daniel A. Lidar, Dave Bacon, Julia Kempe, and K. B. Whaley. Decoherence-free subspaces for multiple-qubit errors. i. characterization. Phys. Rev. A, 63: 022306, Jan 2001a. 10.1103/PhysRevA.63.022306. URL https://link.aps.org/doi/10.1103/PhysRevA.63.022306.
https://doi.org/10.1103/PhysRevA.63.022306

[22] Daniel A. Lidar, Dave Bacon, Julia Kempe, and K. B. Whaley. Decoherence-free subspaces for multiple-qubit errors. ii. universal, fault-tolerant quantum computation. Phys. Rev. A, 63: 022307, Jan 2001b. 10.1103/PhysRevA.63.022307. URL https://link.aps.org/doi/10.1103/PhysRevA.63.022307.
https://doi.org/10.1103/PhysRevA.63.022307

[23] Norbert M. Linke, Mauricio Gutierrez, Kevin A. Landsman, Caroline Figgatt, Shantanu Debnath, Kenneth R. Brown, and Christopher Monroe. Fault-tolerant quantum error detection. Sci. Adv., 3 (10): e1701074, 2017. 10.1126/sciadv.1701074.
https://doi.org/10.1126/sciadv.1701074

[24] Salvatore Lorenzo, Francesco Plastina, and Mauro Paternostro. Geometrical characterization of non-markovianity. Phys. Rev. A, 88: 020102(R), Aug 2013. 10.1103/PhysRevA.88.020102. URL https://journals.aps.org/pra/abstract/10.1103/PhysRevA.88.020102.
https://doi.org/10.1103/PhysRevA.88.020102

[25] Esteban A Martinez, Thomas Monz, Daniel Nigg, Philipp Schindler, and Rainer Blatt. Compiling quantum algorithms for architectures with multi-qubit gates. New Journal of Physics, 18 (6): 063029, jun 2016. 10.1088/1367-2630/18/6/063029. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/18/6/063029

[26] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010.

[27] 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 (6194): 302–305, 2014. ISSN 0036-8075. 10.1126/science.1253742. URL http://science.sciencemag.org/content/345/6194/302.
https://doi.org/10.1126/science.1253742
http://science.sciencemag.org/content/345/6194/302

[28] E. Novais and Harold U. Baranger. Decoherence by correlated noise and quantum error correction. Phys. Rev. Lett., 97: 040501, Jul 2006. 10.1103/PhysRevLett.97.040501. URL https://link.aps.org/doi/10.1103/PhysRevLett.97.040501.
https://doi.org/10.1103/PhysRevLett.97.040501

[29] E. Novais and Eduardo R. Mucciolo. Surface code threshold in the presence of correlated errors. Phys. Rev. Lett., 110: 010502, Jan 2013. 10.1103/PhysRevLett.110.010502. URL https://link.aps.org/doi/10.1103/PhysRevLett.110.010502.
https://doi.org/10.1103/PhysRevLett.110.010502

[30] Amit Kumar Pal, Philipp Schindler, Alexander Erhard, Ángel Rivas, Miguel-Angel Martin-Delgado, Rainer Blatt, Thomas Monz, and Markus Müller. Supplementary Information: Relaxation times do not capture logical qubit dynamics, December 2020. URL https://doi.org/10.5281/zenodo.4321279.
https://doi.org/10.5281/zenodo.4321279

[31] Matteo Paris and Jaroslav Rehacek. Quantum state estimation, volume 649. Springer Science & Business Media, 2004. URL https://dx.doi.org/10.1007/b98673.
https://doi.org/10.1007/b98673

[32] Lukas Postler, Ángel Rivas, Philipp Schindler, Alexander Erhard, Roman Stricker, Daniel Nigg, Thomas Monz, Rainer Blatt, and Markus Müller. Experimental quantification of spatial correlations in quantum dynamics. Quantum, 2: 90, September 2018. ISSN 2521-327X. 10.22331/q-2018-09-03-90. URL https://doi.org/10.22331/q-2018-09-03-90.
https://doi.org/10.22331/q-2018-09-03-90

[33] J. Preskill. Fault-Tolerant Quantum Computation, pages 213–269. World Scientific, 2011. 10.1142/9789812385253_0008.
https://doi.org/10.1142/9789812385253_0008

[34] John Preskill. Sufficient condition on noise correlations for scalable quantum computing. Quant. Inf. Comput., 13: 181–194, 2013.

[35] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/q-2018-08-06-79. URL https://doi.org/10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[36] Matthew D Reed, Leonardo DiCarlo, Simon E Nigg, Luyan Sun, Luigi Frunzio, Steven M Girvin, and Robert J Schoelkopf. Realization of three-qubit quantum error correction with superconducting circuits. Nature, 482 (7385): 382–385, 2012. URL https://doi.org/10.1038/nature10786.
https://doi.org/10.1038/nature10786

[37] Á. Rivas and M. Müller. Quantifying spatial correlations of general quantum dynamics. New Journal of Physics, 17 (6): 062001, 2015. 10.1088/1367-2630/17/7/079602.
https://doi.org/10.1088/1367-2630/17/7/079602

[38] Ángel Rivas, Susana F. Huelga, and Martin B. Plenio. Quantum non-markovianity: characterization, quantification and detection. Rep. Mod. Phys., 77: 094001, Aug 2014. 10.1088/0034-4885/77/9/094001. URL https://iopscience.iop.org/article/10.1088/0034-4885/77/9/094001/meta.
https://doi.org/10.1088/0034-4885/77/9/094001

[39] 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): 1059–1061, 2011. 10.1126/science.1203329.
https://doi.org/10.1126/science.1203329

[40] 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 (12): 123012, dec 2013. 10.1088/1367-2630/15/12/123012. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/15/12/123012

[41] A. Shabani. Correlated errors can lead to better performance of quantum codes. Phys. Rev. A, 77: 022323, Feb 2008. 10.1103/PhysRevA.77.022323. URL https://link.aps.org/doi/10.1103/PhysRevA.77.022323.
https://doi.org/10.1103/PhysRevA.77.022323

[42] P. W. Shor. Fault-tolerant quantum computation, pages 56–65. IEEE, Oct 1996. 10.1109/SFCS.1996.548464.
https://doi.org/10.1109/SFCS.1996.548464

[43] Barbara M. Terhal. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307–346, Apr 2015. 10.1103/RevModPhys.87.307. URL https://link.aps.org/doi/10.1103/RevModPhys.87.307.
https://doi.org/10.1103/RevModPhys.87.307

[44] Gerald Waldherr, Y Wang, S Zaiser, M Jamali, T Schulte-Herbrüggen, H Abe, T Ohshima, J Isoya, JF Du, P Neumann, et al. Quantum error correction in a solid-state hybrid spin register. Nature, 506 (7487): 204–207, 2014. URL https://doi.org/10.1038/nature12919.
https://doi.org/10.1038/nature12919

[45] P. Zanardi and M. Rasetti. Noiseless quantum codes. Phys. Rev. Lett., 79: 3306–3309, Oct 1997. 10.1103/PhysRevLett.79.3306. URL https://link.aps.org/doi/10.1103/PhysRevLett.79.3306.
https://doi.org/10.1103/PhysRevLett.79.3306

Cited by

[1] Garima Rajpoot, Komal Kumari, and Sudhir Ranjan Jain, "Quantum error correction beyond the toric code: dynamical systems meet encoding", The European Physical Journal Special Topics (2023).

[2] Sascha Heußen, Lukas Postler, Manuel Rispler, Ivan Pogorelov, Christian D. Marciniak, Thomas Monz, Philipp Schindler, and Markus Müller, "Strategies for a practical advantage of fault-tolerant circuit design in noisy trapped-ion quantum computers", Physical Review A 107 4, 042422 (2023).

[3] Robin Blume-Kohout, Marcus P. da Silva, Erik Nielsen, Timothy Proctor, Kenneth Rudinger, Mohan Sarovar, and Kevin Young, "A Taxonomy of Small Markovian Errors", PRX Quantum 3 2, 020335 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-14 17:54:01) and SAO/NASA ADS (last updated successfully 2024-05-14 17:54:01). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.