Classical certification of quantum gates under the dimension assumption

Jan Nöller; Nikolai Miklin; Martin Kliesch; Mariami Gachechiladze

doi:10.22331/q-2025-08-08-1825

Classical certification of quantum gates under the dimension assumption

Jan Nöller¹, Nikolai Miklin², Martin Kliesch², and Mariami Gachechiladze¹

¹Department of Computer Science, Technical University of Darmstadt, Darmstadt, 64289 Germany
²Institute for Quantum Inspired and Quantum Optimization, Hamburg University of Technology, Germany

Published:	2025-08-08, volume 9, page 1825
Editor:	Remigiusz Augusiak
Eprint:	arXiv:2401.17006v4
Doi:	https://doi.org/10.22331/q-2025-08-08-1825
Citation:	Quantum 9, 1825 (2025).

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Abstract

The rapid advancement of quantum hardware necessitates the development of reliable methods to certify its correct functioning. However, existing certification tests fall short, as they either suffer from systematic errors or do not guarantee that only a correctly functioning quantum device can pass the test. We introduce a certification method for quantum gates tailored for a practical server-user scenario, where a classical user tests the results of exact quantum computations performed by a quantum server. This method is free from the systematic state preparation and measurement (SPAM) errors. For single-qubit gates, including those that form a universal set for single-qubit quantum computation, we demonstrate that our approach offers soundness guarantees based solely on the dimension assumption. Additionally, for a highly-relevant phase gate – which corresponds experimentally to a $\pi/2$-pulse – we prove that the method's sample complexity scales as $\mathrm{O}(\varepsilon^{-1})$ relative to the average gate infidelity $\varepsilon$. By combining the SPAM-error-free and sound notion of certification with practical applicability, our approach paves the way for promising research into efficient and reliable certification methods for full-scale quantum computation.

► BibTeX data

@article{Noller2025classical,
  doi = {10.22331/q-2025-08-08-1825},
  url = {https://doi.org/10.22331/q-2025-08-08-1825},
  title = {Classical certification of quantum gates under the dimension assumption},
  author = {N{\"{o}}ller, Jan and Miklin, Nikolai and Kliesch, Martin and Gachechiladze, Mariami},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1825},
  month = aug,
  year = {2025}
}

► References

[1] J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020), arXiv:1910.06343 [quant-ph].
https://doi.org/10.1038/s42254-020-0186-4
arXiv:1910.06343

[2] M. Kliesch and I. Roth, Theory of quantum system certification, PRX Quantum 2, 010201 (2021), tutorial, arXiv:2010.05925 [quant-ph].
https://doi.org/10.1103/PRXQuantum.2.010201
arXiv:2010.05925

[3] I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44, 2455 (1997), arXiv:quant-ph/9610001.
https://doi.org/10.1080/09500349708231894
arXiv:quant-ph/9610001

[4] M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Quantum-process tomography: Resource analysis of different strategies, 77, 032322 (2008), arXiv:quant-ph/0702131.
https://doi.org/10.1103/PhysRevA.77.032322
arXiv:quant-ph/0702131

[5] Y.-C. Liu, J. Shang, X.-D. Yu, and X. Zhang, Efficient verification of quantum processes, 101, 042315 (2020), arXiv:1910.13730 [quant-ph].
https://doi.org/10.1103/PhysRevA.101.042315
arXiv:1910.13730

[6] H. Zhu and H. Zhang, Efficient verification of quantum gates with local operations, 101, 042316 (2020), arXiv:1910.14032 [quant-ph].
https://doi.org/10.1103/PhysRevA.101.042316
arXiv:1910.14032

[7] P. Zeng, Y. Zhou, and Z. Liu, Quantum gate verification and its application in property testing, Physical Review Research 2, 023306 (2020), arXiv:1911.06855 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.2.023306
arXiv:1911.06855

[8] 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), arXiv:1211.0322 [quant-ph].
https://doi.org/10.1103/PhysRevA.87.062119
arXiv:1211.0322

[9] R. Blume-Kohout, J. King 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 [quant-ph].
https://doi.org/10.48550/arXiv.1310.4492
arXiv:1310.4492

[10] R. Brieger, I. Roth, and M. Kliesch, Compressive gate set tomography, PRX Quantum 4, 010325 (2023), arXiv:2112.05176 [quant-ph].
https://doi.org/10.1103/PRXQuantum.4.010325
arXiv:2112.05176

[11] J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, J. Opt. B 7, S347 (2005), arXiv:quant-ph/0503243.
https://doi.org/10.1088/1464-4266/7/10/021
arXiv:arXiv:quant-ph/0503243

[12] B. Lévi, C. C. López, J. Emerson, and D. G. Cory, Efficient error characterization in quantum information processing, 75, 022314 (2007), arXiv:quant-ph/0608246.
https://doi.org/10.1103/PhysRevA.75.022314
arXiv:quant-ph/0608246

[13] C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80, 012304 (2009), arXiv:quant-ph/0606161.
https://doi.org/10.1103/PhysRevA.80.012304
arXiv:quant-ph/0606161

[14] 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), arXiv:0707.0685 [quant-ph].
https://doi.org/10.1126/science.1145699
arXiv:0707.0685

[15] 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), arXiv:0707.0963 [quant-ph].
https://doi.org/10.1103/PhysRevA.77.012307
arXiv:0707.0963

[16] E. Magesan, J. M. Gambetta, and J. Emerson, Characterizing quantum gates via randomized benchmarking, Phys. Rev. A 85, 042311 (2012), arXiv:1109.6887 [quant-ph].
https://doi.org/10.1103/PhysRevA.85.042311
arXiv:1109.6887

[17] J. Helsen, I. Roth, E. Onorati, A. H. Werner, and J. Eisert, A general framework for randomized benchmarking, PRX Quantum 3, 020357 (2022), arXiv:2010.07974 [quant-ph].
https://doi.org/10.1103/PRXQuantum.3.020357
arXiv:2010.07974

[18] M. Heinrich, M. Kliesch, and I. Roth, Randomized benchmarking with random quantum circuits, arXiv:2212.06181 [quant-ph] (2022).
https://doi.org/10.48550/arXiv.2212.06181
arXiv:2212.06181

[19] D. Mayers and A. Yao, Self testing quantum apparatus, Quantum Info. Comput. 4, 273–286 (2004), arXiv:quant-ph/0307205.
https://doi.org/10.26421/QIC4.4-3
arXiv:quant-ph/0307205

[20] I. Šupić and J. Bowles, Self-testing of quantum systems: a review, Quantum 4, 337 (2020), arXiv:1904.10042 [quant-ph].
https://doi.org/10.22331/q-2020-09-30-337
arXiv:1904.10042

[21] P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, Certifying the building blocks of quantum computers from Bell's theorem, Phys. Rev. Lett. 121, 180505 (2018), arXiv:1802.02170 [quant-ph].
https://doi.org/10.1103/PhysRevLett.121.180505
arXiv:1802.02170

[22] S. Wagner, J.-D. Bancal, N. Sangouard, and P. Sekatski, Device-independent characterization of quantum instruments, Quantum 4, 243 (2020), arXiv:1812.02628 [quant-ph].
https://doi.org/10.22331/q-2020-03-19-243
arXiv:1812.02628

[23] S. Sarkar, Model-independent inference of quantum interaction from statistics, Phys. Rev. A 110, L020402 (2024), arXiv:2402.08003 [quant-ph].
https://doi.org/10.1103/PhysRevA.110.L020402
arXiv:2402.08003

[24] F. Magniez, D. Mayers, M. Mosca, and H. Ollivier, Self-testing of quantum circuits, in Automata, Languages and Programming, edited by M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp. 72–83, arXiv:quant-ph/0512111.
https://doi.org/10.1007/11786986_8
arXiv:quant-ph/0512111

[25] B. Reichardt, F. Unger, and U. Vazirani, Classical command of quantum systems, Nature 496, 456 (2013), arXiv:1209.0449 [quant-ph].
https://doi.org/10.1038/nature12035
arXiv:1209.0449

[26] T. Metger and T. Vidick, Self-testing of a single quantum device under computational assumptions, Quantum 5, 544 (2021), arXiv:2001.09161 [quant-ph].
https://doi.org/10.22331/q-2021-09-16-544
arXiv:2001.09161

[27] U. Mahadev, Classical verification of quantum computations, in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (2018) pp. 259–267, arXiv:1804.01082 [quant-ph].
https://doi.org/10.1109/FOCS.2018.00033
arXiv:1804.01082

[28] R. Stricker, J. Carrasco, M. Ringbauer, L. Postler, M. Meth, C. Edmunds, P. Schindler, R. Blatt, P. Zoller, B. Kraus, and T. Monz, Towards experimental classical verification of quantum computation, Quantum Science and Technology 9, 02LT01 (2024), arXiv:2203.07395 [quant-ph].
https://doi.org/10.1088/2058-9565/ad2986
arXiv:2203.07395

[29] H.-Y. R. Huang, S. T. Flammia, and J. Preskill, Foundations for learning from noisy quantum experiments (2022), presented at QIP 2022, Padedena, California, arXiv:2204.13691 [quant-ph].
https://doi.org/10.48550/arXiv.2204.13691
arXiv:2204.13691

[30] K. Mohan, A. Tavakoli, and N. Brunner, Sequential random access codes and self-testing of quantum measurement instruments, New Journal of Physics 21, 083034 (2019), arXiv:1905.06726 [quant-ph].
https://doi.org/10.1088/1367-2630/ab3773
arXiv:1905.06726

[31] N. Miklin, J. J. Borkała, and M. Pawłowski, Semi-device-independent self-testing of unsharp measurements, Phys. Rev. Res. 2, 033014 (2020), arXiv:1903.12533 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.2.033014
arXiv:1903.12533

[32] A. Tavakoli, M. Smania, T. Vértesi, N. Brunner, and M. Bourennane, Self-testing nonprojective quantum measurements in prepare-and-measure experiments, Science Advances 6, eaaw6664 (2020), arXiv:1811.12712 [quant-ph].
https://doi.org/10.1126/sciadv.aaw6664
arXiv:1811.12712

[33] N. Miklin and M. Oszmaniec, A universal scheme for robust self-testing in the prepare-and-measure scenario, Quantum 5, 424 (2021), arXiv:2003.01032 [quant-ph].
https://doi.org/10.22331/q-2021-04-06-424
arXiv:2003.01032

[34] H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Experimental characterization of unsharp qubit observables and sequential measurement incompatibility via quantum random access codes, Phys. Rev. Lett. 125, 080403 (2020), arXiv:2001.04768 [quant-ph].
https://doi.org/10.1103/PhysRevLett.125.080403
arXiv:2001.04768

[35] M. Navascués, K. F. Pál, T. Vértesi, and M. Araújo, Self-testing in prepare-and-measure scenarios and a robust version of Wigner's theorem, Phys. Rev. Lett. 131, 250802 (2023), arXiv:2306.00730 [quant-ph].
https://doi.org/10.1103/PhysRevLett.131.250802
arXiv:2306.00730

[36] E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg+Teubner Verlag, 1931).
https://doi.org/10.1007/978-3-663-02555-9

[37] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
https://doi.org/10.1103/RevModPhys.75.281

[38] M.-D. Choi, Completely positive linear maps on complex matrices, Lin. Alg. App. 10, 285 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0

[39] A. Jamiolkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972).
https://doi.org/10.1016/0034-4877(72)90011-0

[40] A. Gočanin, I. Šupić, and B. Dakić, Sample-efficient device-independent quantum state verification and certification, PRX Quantum 3, 010317 (2022), arXiv:2105.05832 [quant-ph].
https://doi.org/10.1103/PRXQuantum.3.010317
arXiv:2105.05832

[41] W. van Dam, F. Magniez, M. Mosca, and M. Santha, Self-testing of universal and fault-tolerant sets of quantum gates, in Proceedings of the thirty-second annual ACM symposium on Theory of computing, STOC00 (ACM, 2000) arXiv:quant-ph/9904108.
https://doi.org/10.1145/335305.335402
arXiv:quant-ph/9904108

[42] R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1985).
https://doi.org/10.1017/cbo9780511810817

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[2] Noah Kaufmann, Maria Quadeer, and David Elkouss, "Estimating Bell diagonal states with separable measurements", Physical Review A 112 4, 042434 (2025).

[3] Mirjam Weilenmann, Costantino Budroni, and Miguel Navascues, "Memory attacks in network nonlocality and self-testing", Quantum 9, 1735 (2025).

[4] Jan Nöller, Nikolai Miklin, Martin Kliesch, and Mariami Gachechiladze, "Sound certification of memory-bounded quantum computers", arXiv:2411.04215, (2024).

[5] Anna Schroeder, Lucas B. Vieira, Jan Nöller, Nikolai Miklin, and Mariami Gachechiladze, "Certifying Quantum Gates via Automata Advantage", arXiv:2510.09575, (2025).

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