Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Technical Review
  • Published:

Quantum certification and benchmarking

Nature Reviews Physics volume 2, pages 382–390 (2020)Cite this article

Subjects

Abstract

With the rapid development of quantum technologies, a pressing need has emerged for a wide array of tools for the certification and characterization of quantum devices. Such tools are critical because the powerful applications of quantum information science will only be realized if stringent levels of precision of components can be reached and their functioning guaranteed. This Technical Review provides a brief overview of the known characterization methods for certification, benchmarking and tomographic reconstruction of quantum states and processes, and outlines their applications in quantum computing, simulation and communication.

Key points

  • To ensure the correct functioning of a quantum device, its components must be certified and benchmarked.

  • Certification, benchmarking and characterization tasks are particularly demanding in quantum simulation and computing applications.

  • The most common tools for certification and benchmarking are surveyed and assessed according to the information that may be extracted from the protocol, the assumptions underlying the protocol, and its complexity in terms of samples, measurements and post-processing.

  • We highlight particularly important concepts, protocols and applications and list key figures of merit — information gain, complexity and underlying assumptions — for several protocols.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic of a classification scheme for some of the certification protocols discussed in this Technical Review.

Similar content being viewed by others

References

  1. Acin, A. et al. The European quantum technologies roadmap. New J. Phys. 20, 080201 (2018).

    Google Scholar 

  2. Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: a vision for the road ahead. Science 362, aam9288 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  3. Kimble, H. J. The quantum internet. Nature 453, 1023–1030 (2008).

    ADS  Google Scholar 

  4. Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys. 8, 264–266 (2012).

    Google Scholar 

  5. Preskill, J. Quantum computing and the entanglement frontier. Preprint at https://arxiv.org/abs/1203.5813 (2012).

  6. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    ADS  Google Scholar 

  7. Campbell, E. T., Terhal, B. M. & Vuillot, C. Roads towards fault-tolerant universal quantum computation. Nature 549, 172–179 (2017).

    ADS  Google Scholar 

  8. Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).

    Google Scholar 

  9. Gheorghiu, A., Kapourniotis, T. & Kashefi, E. Verification of quantum computation: an overview of existing approaches. Th. Comp. Sys. 63, 715–808 (2019).

    MathSciNet  MATH  Google Scholar 

  10. Fitzsimons, J. F. Private quantum computation: an introduction to blind quantum computing and related protocols. NPJ Quant. Inf. 3, 23 (2017).

    ADS  Google Scholar 

  11. Hradil, Z. Quantum-state estimation. Phys. Rev. A 55, 1561–1564 (1997).

    ADS  MathSciNet  Google Scholar 

  12. James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001).

    ADS  Google Scholar 

  13. Hradil, Z., Rehacek, J., Fiurasek, J. & Jezek, M. in Quantum State Estimation, 59–112 (Springer, 2004).

  14. Blume-Kohout, R. Optimal, reliable estimation of quantum states. New J. Phys. 12, 043034 (2010).

    ADS  MATH  Google Scholar 

  15. Ferrie, C. High posterior density ellipsoids of quantum states. New J. Phys. 16, 023006 (2014).

    ADS  MathSciNet  Google Scholar 

  16. Blume-Kohout, R. Robust error bars for quantum tomography. Preprint at https://arxiv.org/abs/1202.5270 (2012).

  17. Christandl, M. & Renner, R. Reliable quantum state tomography. Phys. Rev. Lett. 109, 120403 (2012).

    ADS  Google Scholar 

  18. Wang, J., Scholz, V. B. & Renner, R. Confidence polytopes in quantum state tomography. Phys. Rev. Lett. 122, 190401 (2019).

    ADS  Google Scholar 

  19. Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J. Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010).

    ADS  Google Scholar 

  20. Kalev, A., Kosut, R. L. & Deutsch, I. H. Quantum tomography protocols with positivity are compressed sensing protocols. NPJ Quant. Inf. 1, 15018 (2015).

    ADS  Google Scholar 

  21. Guta, M., Kahn, J., Kueng, R. & Tropp, J. A. Fast state tomography with optimal error bounds. Preprint at https://arxiv.org/abs/1809.11162 (2018).

  22. Flammia, S. T., Gross, D., Liu, Y.-K. & Eisert, J. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys. 14, 095022 (2012).

    ADS  Google Scholar 

  23. Kliesch, M., Kueng, R., Eisert, J. & Gross, D. Guaranteed recovery of quantum processes from few measurements. Quantum 3, 171 (2019).

    Google Scholar 

  24. Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).

    ADS  Google Scholar 

  25. Hübener, R., Mari, A. & Eisert, J. Wick’s theorem for matrix product states. Phys. Rev. Lett. 110, 040401 (2013).

    ADS  Google Scholar 

  26. Baumgratz, T., Gross, D., Cramer, M. & Plenio, M. B. Scalable reconstruction of density matrices. Phys. Rev. Lett. 111, 020401 (2013).

    ADS  Google Scholar 

  27. Ohliger, M., Nesme, V. & Eisert, J. Efficient and feasible state tomography of quantum many-body systems. New J. Phys. 15, 015024 (2013).

    ADS  Google Scholar 

  28. Torlai, G. et al. Many-body quantum state tomography with neural networks. Nat. Phys. 14, 447–450 (2018).

    Google Scholar 

  29. Carrasquilla, J., Torlai, G., Melko, R. G. & Aolita, L. Reconstructing quantum states with generative models. Nat. Mach. Intell. 1, 155–161 (2019).

    Google Scholar 

  30. Aaronson, S. The learnability of quantum states. Proc. Roy. Soc. A 463, 3089–3114 (2007).

    ADS  MathSciNet  MATH  Google Scholar 

  31. Rocchetto, A. Stabiliser states are efficiently PAC-learnable. Quant. Inf. Comp. 541–552 (2018).

  32. Rocchetto, A. et al. Experimental learning of quantum states. Sci. Adv. 5, eaau1946 (2019).

    ADS  Google Scholar 

  33. Holzäpfel, M., Baumgratz, T., Cramer, M. & Plenio, M. B. Scalable reconstruction of unitary processes and Hamiltonians. Phys. Rev. A 91, 042129 (2015).

    ADS  Google Scholar 

  34. Granade, C. E., Ferrie, C., Wiebe, N. & Cory, D. G. Robust online Hamiltonian learning. New J. Phys. 14, 103013 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  35. Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).

    ADS  Google Scholar 

  36. Reich, D. M., Gualdi, G. & Koch, C. P. Optimal strategies for estimating the average fidelity of quantum gates. Phys. Rev. Lett. 111, 200401 (2013-12).

  37. Pallister, S., Linden, N. & Montanaro, A. Optimal verification of entangled states with local measurements. Phys. Rev. Lett. 120, 170502 (2018).

    ADS  MathSciNet  Google Scholar 

  38. Aolita, L., Gogolin, C., Kliesch, M. & Eisert, J. Reliable quantum certification for photonic quantum technologies. Nat. Commun. 6, 8498 (2015).

    ADS  Google Scholar 

  39. Gluza, M., Kliesch, M., Eisert, J. & Aolita, L. Fidelity witnesses for fermionic quantum simulations. Phys. Rev. Lett. 120, 190501 (2018).

    ADS  Google Scholar 

  40. Hangleiter, D., Kliesch, M., Schwarz, M. & Eisert, J. Direct certification of a class of quantum simulations. Quantum Sci. Technol. 2, 015004 (2017).

    ADS  Google Scholar 

  41. Jurcevic, P. et al. Observation of entanglement propagation in a quantum many-body system. Nature 511, 202–205 (2014).

    ADS  Google Scholar 

  42. Eisert, J., Brandao, F. G. S. L. & Audenaert, K. M. R. Quantitative entanglement witnesses. New J. Phys. 9, 46 (2007).

    ADS  MathSciNet  Google Scholar 

  43. Audenaert, K. M. R. & Plenio, M. B. When are correlations quantum? New J. Phys. 8, 266 (2006).

    ADS  Google Scholar 

  44. Guehne, O., Reimpell, M. & Werner, R. F. Estimating entanglement measures in experiments. Phys. Rev. Lett. 98, 110502 (2007).

    ADS  Google Scholar 

  45. Brydges, T. et al. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).

    ADS  Google Scholar 

  46. Trotzky, S. et al. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nat. Phys. 8, 325–330 (2012).

    Google Scholar 

  47. Trotzky, S. et al. Suppression of the critical temperature for superfluidity near the Mott transition: validating a quantum simulator. Nat. Phys. 6, 998–1004 (2010).

    Google Scholar 

  48. Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasi-random optical lattice. Science 349, 842–845 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  49. Braun, S. et al. Emergence of coherence and the dynamics of quantum phase transitions. Proc. Natl. Acad. Sci. USA 112, 3641–3646 (2015).

    ADS  Google Scholar 

  50. Kokail, C. et al. Self-verifying variational quantum simulation of the lattice Schwinger model. Nature 569, 355–360 (2019).

    ADS  Google Scholar 

  51. Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124, 010504 (2020).

    ADS  Google Scholar 

  52. Takeuchi, Y. & Morimae, T. Verification of many-qubit states. Phys. Rev. X 8, 021060 (2018).

    Google Scholar 

  53. Markham, D. & Krause, A. A simple protocol for certifying graph states and applications in quantum networks. Preprint at https://arxiv.org/abs/1801.05057 (2018).

  54. Takeuchi, Y., Mantri, A., Morimae, T., Mizutani, A. & Fitzsimons, J. F. Resource-efficient verification of quantum computing using Serfling’s bound. NPJ Quantum Inf. 5, 1–8 (2019).

    Google Scholar 

  55. Chabaud, U., Douce, T., Grosshans, F., Kashefi, E. & Markham, D. Building trust for continuous variable quantum states. Preprint at https://arxiv.org/abs/1905.12700 (2019).

  56. Merkel, S. T. et al. Self-consistent quantum process tomography. Phys. Rev. A 87, 062119 (2013).

    ADS  Google Scholar 

  57. Blume-Kohout, R. et al. Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit. Preprint at https://arxiv.org/abs/1310.4492 (2013).

  58. Blume-Kohout, R. &, et al. Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography. Nat. Commun. 8, 14485 (2017).

    Google Scholar 

  59. Branczyk, A. M. et al. Self-calibrating quantum state tomography. New J. Phys. 14, 085003 (2012).

    ADS  Google Scholar 

  60. Mogilevtsev, D., Rehacek, J. & Hradil, Z. Self-calibration for self-consistent tomography. New J. Phys. 14, 095001 (2012).

    ADS  Google Scholar 

  61. Sim, J. Y., Shang, J., Ng, H. K. & Englert, B.-G. Proper error bars for self-calibrating quantum tomography. Preprint at https://arxiv.org/abs/1904.11202 (2019).

  62. Motka, L. et al. Efficient algorithm for optimizing data-pattern tomography. Phys. Rev. A 89, 054102 (2014).

    ADS  Google Scholar 

  63. Rehacek, J., Mogilevtsev, D. & Hradil, Z. Operational tomography: fitting of data patterns. Phys. Rev. Lett. 105, 010402 (2010).

    ADS  Google Scholar 

  64. Ferrie, C. et al. Quantum model averaging. New J. Phys. 16, 093035 (2014).

    ADS  MathSciNet  Google Scholar 

  65. Emerson, J., Alicki, R. & Zyczkowski, K. Scalable noise estimation with random unitary operators. J. Opt. B 7, S347–S352 (2005).

    ADS  MathSciNet  Google Scholar 

  66. Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).

    ADS  Google Scholar 

  67. Lévi, B., López, C. C., Emerson, J. & Cory, D. G. Efficient error characterization in quantum information processing. Phys. Rev. A 75, 022314 (2007).

    ADS  Google Scholar 

  68. Magesan, E., Gambetta, J. M. & Emerson, J. Robust randomized benchmarking of quantum processes. Phys. Rev. Lett. 106, 042311 (2011).

    Google Scholar 

  69. Scarani, V. et al. The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009).

    ADS  Google Scholar 

  70. Debnath, S. et al. Demonstration of a small programmable quantum computer with atomic qubits. Nature 536, 63–66 (2016).

    ADS  Google Scholar 

  71. Linke, N. M. et al. Experimental comparison of two quantum computing architectures. Proc. Natl. Acad. Sci. USA 114, 3305–3310 (2017).

    Google Scholar 

  72. Mayers, D. & Yao, A. Self testing quantum apparatus. Quantum Inf. Comput. 4, 273 (2004).

    MathSciNet  MATH  Google Scholar 

  73. McKague, M. in Theory of Quantum Computation, Communication, and Cryptography, 104–120 (Springer, 2011).

  74. Sekatski, P., Bancal, J.-D., Wagner, S. & Sangouard, N. Certifying the building blocks of quantum computers from Bell’s theorem. Phys. Rev. Lett. 121, 180505 (2018).

    ADS  Google Scholar 

  75. Reichardt, B. W., Unger, F. & Vazirani, U. Classical command of quantum systems. Nature 496, 456–460 (2013).

    ADS  Google Scholar 

  76. Natarajan, A. & Vidick, T. Low-degree testing for quantum states. Preprint at https://arxiv.org/abs/1801.03821 (2018).

  77. Supic, I. & Bowles, J. Self-testing of quantum systems: a review. Preprint at https://arxiv.org/abs/1904.10042 (2019).

  78. Fitzsimons, J. F. & Kashefi, E. Unconditionally verifiable blind computation. Phys. Rev. A 96, 012303 (2017).

    ADS  Google Scholar 

  79. Coladangelo, A., Grilo, A., Jeffery, S. & Vidick, T. Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources. Preprint at https://arxiv.org/abs/1708.07359 (2017).

  80. Mahadev, U. Classical verification of quantum computations. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), 259–267 (IEEE, 2018).

  81. Gheorghiu, A. & Vidick, T. Computationally-secure and composable remote state preparation. Preprint at https://arxiv.org/abs/1904.06320 (2019).

  82. Alagic, G., Childs, A. M. & Hung, S.-H. Two-message verification of quantum computation. Preprint at https://arxiv.org/abs/1911.08101 (2019).

  83. Regev, O. et al. The Learning with Errors problem (invited survey). Proc. 25th Annual IEEE Conf. on Computational Complexity (CCC ‘10), https://doi.org/10.1109/CCC.2010.26 (2010).

  84. Aaronson, S. & Arkhipov, A. BosonSampling is far from uniform. Preprint at https://arxiv.org/abs/1309.7460 (2013).

  85. Carolan, J. et al. On the experimental verification of quantum complexity in linear optics. Nat. Photon. 8, 621–626 (2014).

    ADS  Google Scholar 

  86. Spagnolo, N. et al. Efficient experimental validation of photonic boson sampling against the uniform distribution. Nat. Photon. 8, 615–620 (2014).

    ADS  Google Scholar 

  87. Aaronson, S. & Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. Preprint at https://arxiv.org/abs/1612.05903 (2016).

  88. Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595–600 (2018).

    Google Scholar 

  89. Bouland, A., Fefferman, B., Nirkhe, C. & Vazirani, U. On the complexity and verification of quantum random circuit sampling. Nat. Phys. 15, 159–163 (2019).

    Google Scholar 

  90. Valiant, G. & Valiant, P. An automatic inequality prover and instance optimal identity testing. SIAM J. Comput. 46, 429–455 (2017).

    MathSciNet  MATH  Google Scholar 

  91. Hangleiter, D., Kliesch, M., Eisert, J. & Gogolin, C. Sample complexity of device-independently certified ‘quantum supremacy’. Phys. Rev. Lett. 122, 210502 (2019).

    ADS  Google Scholar 

  92. Bermejo-Vega, J., Hangleiter, D., Schwarz, M., Raussendorf, R. & Eisert, J. Architectures for quantum simulation showing a quantum speedup. Phys. Rev. X 8, 021010 (2018).

    Google Scholar 

  93. Harper, R., Hincks, I., Ferrie, C., Flammia, S. T. & Wallman, J. J. Statistical analysis of randomized benchmarking. Phys. Rev. A 99, 052350 (2019).

    ADS  Google Scholar 

  94. Baumgratz, T., Nüßeler, A., Cramer, M. & Plenio, M. B. A scalable maximum likelihood method for quantum state tomography. New J. Phys. 15, 125004 (2013).

    ADS  Google Scholar 

  95. Portmann, C. & Renner, R. Cryptographic security of quantum key distribution. Preprint at https://arxiv.org/abs/1409.3525 (2014).

  96. Watrous, J. Simpler semidefinite programs for completely bounded norms. Preprint at https://arxiv.org/abs/1207.5726 (2012).

  97. Schumacher, B. Sending entanglement through noisy quantum channels. Phys. Rev. A 54, 2614–2628 (1996).

    ADS  Google Scholar 

  98. Kueng, R., Long, D. M., Doherty, A. C. & Flammia, S. T. Comparing experiments to the fault-tolerance threshold. Phys. Rev. Lett. 117, 170502 (2016).

    ADS  Google Scholar 

  99. Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Preprint at https://arxiv.org/abs/1811.12926 (2018).

  100. Streltsov, A., Adesso, G. & Plenio, M. B. Colloquium: Quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017).

    ADS  MathSciNet  Google Scholar 

  101. Guehne, O. & Toth, G. Entanglement detection. Phys. Rep. 474, 1 (2009).

    ADS  MathSciNet  Google Scholar 

  102. Mari, A., Kieling, K., Nielsen, B. M., Polzik, E. & Eisert, J. Directly estimating non-classicality. Phys. Rev. Lett. 106, 010403 (2011).

    ADS  Google Scholar 

  103. Guta, M., Kypraios, T. & Dryden, I. Rank-based model selection for multiple ions quantum tomography. New J. Phys. 14, 105002 (2012).

    ADS  Google Scholar 

  104. Phillips, D. S. et al. Benchmarking of Gaussian boson sampling using two-point correlators. Phys. Rev. A 99, 023836 (2019).

    ADS  Google Scholar 

  105. Ferracin, S., Kapourniotis, T. & Datta, A. Verifying quantum computations on noisy intermediate-scale quantum devices. Preprint at https://arxiv.org/abs/1811.09709 (2018).

  106. Knill, E. et al. Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008).

    ADS  Google Scholar 

  107. Barends, R. et al. Rolling quantum dice with a superconducting qubit. Phys. Rev. A 90, 030303 (2014).

    ADS  Google Scholar 

  108. Carignan-Dugas, A., Wallman, J. J. & Emerson, J. Characterizing universal gate sets via dihedral benchmarking. Phys. Rev. A 92, 060302 (2015).

    ADS  Google Scholar 

  109. Cross, A. W., Magesan, E., Bishop, L. S., Smolin, J. A. & Gambetta, J. M. Scalable randomized benchmarking of non-Clifford gates. NPJ Quant. Inf. 2, 16012 (2016).

    ADS  Google Scholar 

  110. Onorati, E., Werner, A. H. & Eisert, J. Randomized benchmarking for individual quantum gates. Phys. Rev. Lett. 123, 060501 (2019).

    ADS  Google Scholar 

  111. Helsen, J., Xue, X., Vandersypen, L. M. K. & Wehner, S. A new class of efficient randomized benchmarking protocols. NPJ Quant. Inf. 5, 1–9 (2019).

    Google Scholar 

  112. Erhard, A. et al. Characterizing large-scale quantum computers via cycle benchmarking. Nat. Commun. 10, 5347 (2019).

    ADS  Google Scholar 

  113. Wallman, J. J. Randomized benchmarking with gate-dependent noise. Quantum 2, 47 (2018).

    Google Scholar 

  114. Wallman, J. J. & Flammia, S. T. Randomized benchmarking with confidence. New. J. Phys. 16, 103032 (2014).

    ADS  Google Scholar 

  115. Helsen, J., Wallman, J. J., Flammia, S. T. & Wehner, S. Multiqubit randomized benchmarking using few samples. Phys. Rev. A 100, 032304 (2019).

    ADS  Google Scholar 

  116. Magesan, E. et al. Efficient measurement of quantum gate error by interleaved randomized benchmarking. Phys. Rev. Lett. 109, 080505 (2012).

    ADS  MATH  Google Scholar 

  117. Wallman, J., Granade, C., Harper, R. & Flammia, S. T. Estimating the coherence of noise. New J. Phys. 17, 113020 (2015).

    ADS  Google Scholar 

  118. Gambetta, J. M. et al. Characterization of addressability by simultaneous randomized benchmarking. Phys. Rev. Lett. 109, 240504 (2012).

    ADS  Google Scholar 

  119. Wallman, J. J., Barnhill, M. & Emerson, J. Robust characterization of loss rates. Phys. Rev. Lett. 115, 060501 (2015).

    ADS  Google Scholar 

  120. Wallman, J. J., Barnhill, M. & Emerson, J. Robust characterization of leakage errors. New J. Phys. 18, 043021 (2016).

    ADS  Google Scholar 

  121. Kimmel, S., da Silva, M. P., Ryan, C. A., Johnson, B. R. & Ohki, T. Robust extraction of tomographic information via randomized benchmarking. Phys. Rev. X 4, 011050 (2014).

    Google Scholar 

  122. Roth, I. et al. Recovering quantum gates from few average gate fidelities. Phys. Rev. Lett. 121, 170502 (2018).

    ADS  Google Scholar 

  123. Flammia, S. T. & Wallman, J. J. Efficient estimation of Pauli channels. Preprint at https://arxiv.org/abs/1907.12976 (2019).

  124. Franca, D. S. & Hashagen, A.-L. Approximate randomized benchmarking for finite groups. J. Phys. A 51, 395302 (2018).

  125. Proctor, T. J. et al. Direct randomized benchmarking for multiqubit devices. Phys. Rev. Lett. 123, 030503 (2019).

    ADS  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge discussions with D. Gross and J. Helsen, in addition to many other members of the scientific community. J.E. acknowledges funding from the DFG (CRC 183 project B01, EI 519/9-1, EI 519/14-1, EI 519/15-1, MATH+ project EF1-7, Deadalus, CRC 1114 project B06, FOR 2724), the BMWF (Q.Link.X), the BMWi (PlanQK), FQXi (2019-207756) and the Templeton Foundation. This work also received funding from the European Union (EU) Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS). N.W. acknowledges funding support from the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 750905. E.K. and D.M. acknowledge funding from the ANR project ANR-13-BS04-0014 COMB, E.K. by the EPSRC (EP/N003829/1).

Author information

Authors and Affiliations

  1. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany

    Jens Eisert, Dominik Hangleiter, Nathan Walk & Ingo Roth

  2. Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Jens Eisert

  3. Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, Germany

    Jens Eisert

  4. Paris Center for Quantum Computing, CNRS, Paris, France

    Damian Markham, Rhea Parekh, Ulysse Chabaud & Elham Kashefi

  5. Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6, Paris, France

    Damian Markham, Rhea Parekh, Ulysse Chabaud & Elham Kashefi

  6. School of Informatics, University of Edinburgh, Edinburgh, UK

    Elham Kashefi

Authors
  1. Jens Eisert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Hangleiter
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nathan Walk
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ingo Roth
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Damian Markham
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Rhea Parekh
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ulysse Chabaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Elham Kashefi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to all aspects of the article.

Corresponding authors

Correspondence to Jens Eisert or Dominik Hangleiter.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information

Nature Reviews Physics thanks Mehul Malik and the other, anonymous, reviewers for their contribution to the peer review of this work.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Noisy intermediate-scale quantum (NISQ) devices

A first generation of quantum computational devices, partially already realized, that consist of up to a few hundred qubits (intermediate scale), which are only partially controlled and not protected by quantum error correction (and hence are noisy).

Tomographic reconstruction

The process of reconstructing an unknown quantum state or process from the observed statistics of a well-chosen collection of measurements carried out on an ensemble of identically prepared systems.

Resampling techniques

Refers to a body of methods in statistics aimed at estimating the precision of sample statistics, of validating models by using random subsets or of estimating confidence regions by statistical means.

Bayesian prior

In Bayesian statistical inference, an initial estimate of the probability of a hypothesis (the Bayesian prior) is updated according to Bayes’ theorem as more information becomes available.

Compressed sensing

A field of applied mathematics that studies rigorous guarantees for algorithmic solutions to linear inverse problems under structure assumptions, such as sparsity and low-rankness.

PAC

‘Probably approximately correct’ (PAC) learning is an algorithm that, given sample inputs and outputs of an unknown function f, returns a candidate function (hypothesis) that with high probability closely approximates f when applied to unseen data.

Unitary 2-designs

A finite set of unitary gates with the property that averages over any quadratic polynomial with respect to this set are equal to the Haar measure average over the full unitary group, useful in particular when estimating average fidelities.

Sub-universal models of quantum computing

Computational tasks that are insufficient for an arbitrary or universal quantum computation but remain provably hard for classical computers. They are potentially feasible using NISQ devices.

Clifford gates

A group of quantum gates that map Pauli operators onto Pauli operators and play a key role specifically in fault-tolerant quantum computing.

Quantum twirling lemma

A lemma that reduces arbitrary quantum maps to convex combinations of local Pauli maps by using averages over the Pauli or the Clifford group.

Post-quantum-secure collision-resistant hash functions

Functions scrambling a larger space into a smaller space (hash) which rarely map distinct inputs to the same output (collision-resistant) and cannot be efficiently inverted by quantum computers (post-quantum-secure).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eisert, J., Hangleiter, D., Walk, N. et al. Quantum certification and benchmarking. Nat Rev Phys 2, 382–390 (2020). https://doi.org/10.1038/s42254-020-0186-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-020-0186-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing