Fast quantum measurement tomography with optimal error bounds

Leonardo Zambrano; Sergi Ramos-Calderer; Richard Kueng

doi:10.22331/q-2026-07-15-2162

Fast quantum measurement tomography with optimal error bounds

Leonardo Zambrano¹, Sergi Ramos-Calderer^2,3, and Richard Kueng⁴

¹ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
²Centre for Quantum Technologies, National University of Singapore, Singapore.
³Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE
⁴Department of Quantum Information and Computation at Kepler (QUICK), Johannes Kepler University Linz, 4040 Linz, Austria

Published:	2026-07-15, volume 10, page 2162
Editor:	Jiangwei Shang
Eprint:	arXiv:2507.04500v3
Doi:	https://doi.org/10.22331/q-2026-07-15-2162
Citation:	Quantum 10, 2162 (2026).

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Abstract

We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost and still achieves optimal sample complexity. Given measurement data from a known probe state ensemble, we first apply least-squares estimation to produce an unconstrained approximation of the POVM, and then project this estimate onto the set of valid quantum measurements. For a POVM with $L$ outcomes acting on a $d$-dimensional system, we show that the protocol requires $\mathcal{O}\left((d^3+d^2L)/\epsilon^2\right)$ samples to achieve error $\epsilon$ in worst-case distance, and $\mathcal{O}(d^2 L/\epsilon^2)$ samples in average-case distance. We further establish two matching sample complexity lower bounds of $\Omega((d^3 + d^2 L) /\epsilon^2)$ and $\Omega(d^2 L/\epsilon^2)$ for any non-adaptive, single-copy POVM tomography protocol. Hence, our projected least squares POVM tomography is sample-optimal in both the dimension and the number of outcomes for both distances. Our method admits an analytic form when using global or local 2-designs as probe ensembles and enables rigorous non-asymptotic error guarantees. Finally, we also complement our findings with empirical performance studies carried out on a noisy superconducting quantum computer with flux-tunable transmon qubits.

► BibTeX data

@article{Zambrano2026fastquantum,
  doi = {10.22331/q-2026-07-15-2162},
  url = {https://doi.org/10.22331/q-2026-07-15-2162},
  title = {Fast quantum measurement tomography with optimal error bounds},
  author = {Zambrano, Leonardo and Ramos-Calderer, Sergi and Kueng, Richard},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2162},
  month = jul,
  year = {2026}
}

► References

[1] K. Temme, S. Bravyi and J. M. Gambetta, Error Mitigation for Short-Depth Quantum Circuits, Phys. Rev. Lett. 119, 180509 (2017).
https://doi.org/10.1103/PhysRevLett.119.180509

[2] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[3] S. Endo, Z. Cai, S. C. Benjamin and X. Yuan, Hybrid quantum-classical algorithms and quantum error mitigation, J. Phys. Soc. Jpn. 90, 032001 (2021).
https://doi.org/10.7566/JPSJ.90.032001

[4] Y. Kim et al., Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
https://doi.org/10.1038/s41586-023-06096-3

[5] Y. Li and S. C. Benjamin, Efficient Variational Quantum Simulator Incorporating Active Error Minimization, Phys. Rev. X 7, 021050 (2017).
https://doi.org/10.1103/PhysRevX.7.021050

[6] Y. Chen, M. Farahzad, S. Yoo and T.-C. Wei, Detector tomography on IBM quantum computers and mitigation of an imperfect measurement, Phys. Rev. A 100, 052315 (2019).
https://doi.org/10.1103/PhysRevA.100.052315

[7] F. B. Maciejewski, Z. Zimborás and M. Oszmaniec, Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography, Quantum 4, 257 (2020).
https://doi.org/10.22331/q-2020-04-24-257

[8] 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).
https://doi.org/10.1103/PhysRevA.103.042605

[9] E. van den Berg, Z. K. Minev and K. Temme, Model-free readout-error mitigation for quantum expectation values, Phys. Rev. A 105, 032620 (2022).
https://doi.org/10.1103/PhysRevA.105.032620

[10] L. Funcke et al., Measurement error mitigation in quantum computers through classical bit-flip correction, Phys. Rev. A 105, 062404 (2022).
https://doi.org/10.1103/PhysRevA.105.062404

[11] K. Korhonen et al., Practical techniques for high-precision measurements on near-term quantum hardware and applications in molecular energy estimation, npj Quantum Inf. 11, 110 (2025).
https://doi.org/10.1038/s41534-025-01066-1

[12] H.-Y. Huang, R. Kueng and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
https://doi.org/10.1038/s41567-020-0932-7

[13] S. Chen, W. Yu, P. Zeng and S. T. Flammia, Robust shadow estimation, PRX Quantum 2, 030348 (2021).
https://doi.org/10.1103/PRXQuantum.2.030348

[14] D. E. Koh and S. Grewal, Classical Shadows With Noise, Quantum 6, 776 (2022).
https://doi.org/10.22331/q-2022-08-16-776

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

[16] J. Fiurášek, Maximum-likelihood estimation of quantum measurement, Phys. Rev. A 64, 024102 (2001).
https://doi.org/10.1103/PhysRevA.64.024102

[17] G. M. D'Ariano, L. Maccone and P. L. Presti, Quantum Calibration of Measurement Instrumentation, Phys. Rev. Lett. 93, 250407 (2004).
https://doi.org/10.1103/PhysRevLett.93.250407

[18] J. S. Lundeen et al., Tomography of quantum detectors, Nat. Phys. 5, 27 (2009).
https://doi.org/10.1038/nphys1133

[19] A. Feito et al., Measuring measurement: theory and practice, New J. Phys. 11, 093038 (2009).
https://doi.org/10.1088/1367-2630/11/9/093038

[20] L. Zhang et al., Recursive quantum detector tomography, New J. Phys. 14, 115005 (2012).
https://doi.org/10.1088/1367-2630/14/11/115005

[21] M. Cattaneo et al., Self-consistent quantum measurement tomography based on semidefinite programming, Phys. Rev. Res. 5, 033154 (2023).
https://doi.org/10.1103/PhysRevResearch.5.033154

[22] S. Grandi, A. Zavatta, M. Bellini and M. G. Paris, Experimental quantum tomography of a homodyne detector, New J. Phys. 19, 053015 (2017).
https://doi.org/10.1088/1367-2630/aa6f2c

[23] Y. Wang et al., Two-stage estimation for quantum detector tomography: Error analysis, numerical and experimental results, IEEE Trans. Inf. Theory 67, 2293 (2021).
https://doi.org/10.1109/TIT.2021.3062596

[24] S. Xiao et al., On the regularization and optimization in quantum detector tomography, Automatica 155, 111124 (2023).
https://doi.org/10.1016/j.automatica.2023.111124

[25] E. Nielsen et al., Gate set tomography, Quantum 5, 557 (2021).
https://doi.org/10.22331/q-2021-10-05-557

[26] R. O'Donnell and J. Wright, Efficient quantum tomography, in Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16 (Association for Computing Machinery, New York, NY, USA, 2016) p. 899–912.
https://doi.org/10.1145/2897518.2897544

[27] J. Haah, A. W. Harrow, Z. Ji, X. Wu and N. Yu, Sample-Optimal Tomography of Quantum States, IEEE Trans. Inf. Theory 63, 5628 (2017).
https://doi.org/10.1109/TIT.2017.2719044

[28] R. Kueng, H. Rauhut and U. Terstiege, Low rank matrix recovery from rank one measurements, Appl. Comput. Harmon. Anal. 42, 88 (2017).
https://doi.org/10.1016/j.acha.2015.07.007

[29] M. Guţă, J. Kahn, R. Kueng and J. A. Tropp, Fast state tomography with optimal error bounds, J. Phys. A: Math. Theor 53, 204001 (2020).
https://doi.org/10.1088/1751-8121/ab8111

[30] T. Surawy-Stepney, J. Kahn, R. Kueng and M. Guta, Projected least-squares quantum process tomography, Quantum 6, 844 (2022).
https://doi.org/10.22331/q-2022-10-20-844

[31] S. Chen, J. Li, B. Huang and A. Liu, Tight bounds for quantum state certification with incoherent measurements, in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) ( organization IEEE, 2022) pp. 1205–1213.
https://doi.org/10.1109/FOCS54457.2022.00118

[32] A. Lowe and A. Nayak, Lower bounds for learning quantum states with single-copy measurements, arXiv:2207.14438 (2022).
https://doi.org/10.1145/3717450
arXiv:2207.14438

[33] A. Anshu and S. Arunachalam, A survey on the complexity of learning quantum states, Nat. Rev. Phys. 6, 59 (2024).
https://doi.org/10.1038/s42254-023-00662-4

[34] L. Zambrano, Quantum tomography for non-iid sources, arXiv:2602.22057 (2026).
https://doi.org/10.48550/arXiv.2602.22057
arXiv:2602.22057

[35] M. Navascués and S. Popescu, How Energy Conservation Limits Our Measurements, Phys. Rev. Lett. 112, 140502 (2014).
https://doi.org/10.1103/PhysRevLett.112.140502

[36] Z. Puchała, L. Pawela, A. Krawiec and R. Kukulski, Strategies for optimal single-shot discrimination of quantum measurements, Phys. Rev. A 98, 042103 (2018).
https://doi.org/10.1103/PhysRevA.98.042103

[37] F. B. Maciejewski, Z. Puchała and M. Oszmaniec, Exploring Quantum Average-Case Distances: Proofs, Properties, and Examples, IEEE Trans. Inf. Theory 69, 4600 (2023).
https://doi.org/10.1109/TIT.2023.3250100

[38] L. Zambrano, T. Parella-Dilmé, A. Acín and D. Farina, Certification of quantum properties with imperfect measurements, arXiv:2601.16570 (2026).
https://doi.org/10.48550/arXiv.2601.16570
arXiv:2601.16570

[39] J. Barberà-Rodríguez, L. Zambrano, A. Acín and D. Farina, Boosting projective methods for quantum process and detector tomography, Physical Review Research 7, 013208 (2025).
https://doi.org/10.1103/PhysRevResearch.7.013208

[40] 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).
https://doi.org/10.1103/PhysRevA.80.012304

[41] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171 (2004).
https://doi.org/10.1063/1.1737053

[42] A. Klappenecker and M. Rotteler, Mutually unbiased bases are complex projective 2-designs, in Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. (organization IEEE, 2005) pp. 1740–1744.
https://doi.org/10.1109/ISIT.2005.1523643

[43] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38, 49 (1996).
https://doi.org/10.1137/1038003

[44] R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science (Cambridge University Press, 2018).
https://doi.org/10.1017/9781108231596

[45] J. Scarlett and V. Cevher, An introductory guide to Fano's inequality with applications in statistical estimation, arXiv:1901.00555 (2019).
arXiv:1901.00555

[46] S. T. Flammia, D. Gross, Y.-K. Liu and J. Eisert, Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators, New J. Phys. 14, 095022 (2012).
https://doi.org/10.1088/1367-2630/14/9/095022

[47] I. Roth et al., Recovering quantum gates from few average gate fidelities, Phys. Rev. Lett. 121, 170502 (2018).
https://doi.org/10.1103/PhysRevLett.121.170502

[48] S. Efthymiou et al., Qibolab: an open-source hybrid quantum operating system, Quantum 8, 1247 (2024).
https://doi.org/10.22331/q-2024-02-12-1247

[49] A. Pasquale et al., Qibocal: an open-source framework for calibration of self-hosted quantum devices, arXiv:2410.00101 (2024).
https://doi.org/10.48550/arXiv.2410.00101
arXiv:2410.00101

[50] Z. Jiang, A. Kalev, W. Mruczkiewicz and H. Neven, Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning, Quantum 4, 276 (2020).
https://doi.org/10.22331/q-2020-06-04-276

[51] Z. You, Q. Liu and Y. Zhou, Circuit optimization of qubit IC-POVMs for shadow estimation, arXiv:2409.05676 (2024).
https://doi.org/10.48550/arXiv.2409.05676
arXiv:2409.05676

[52] I. Pinelis, Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab. , 1679 (1994).
https://doi.org/10.1214/aop/1176988477

[53] D. Martinez-Taboada and A. Ramdas, Empirical Bernstein in smooth Banach spaces, arXiv:2409.06060 (2024).
https://doi.org/10.48550/arXiv.2409.06060
arXiv:2409.06060

[54] E. Meckes and M. Meckes, Spectral measures of powers of random matrices, ECP 18, 1 (2013).
https://doi.org/10.1214/ECP.v18-2551

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[16] Dongdong Zhang, "Stable Qubit Readout and the Identifiability of Population Change", arXiv:2606.30462, (2026).

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