Extracting tomographic information about quantum states is a crucial task in the quest towards devising high-precision quantum devices. Current schemes typically require measurement devices for tomography that are a priori calibrated to high precision. Ironically, the accuracy of the measurement calibration is fundamentally limited by the accuracy of state preparation, establishing a vicious cycle. Here, we prove that this cycle can be broken and the dependence on the measurement device's calibration significantly relaxed. We show that exploiting the natural low-rank structure of quantum states of interest suffices to arrive at a highly scalable `blind' tomography scheme with a classically efficient post-processing algorithm. We further improve the efficiency of our scheme by making use of the sparse structure of the calibrations. This is achieved by relaxing the blind quantum tomography problem to the de-mixing of a sparse sum of low-rank matrices. We prove that the proposed algorithm recovers a low-rank quantum state and the calibration provided that the measurement model exhibits a restricted isometry property. For generic measurements, we show that it requires a close-to-optimal number of measurement settings. Complementing these conceptual and mathematical insights, we numerically demonstrate that robust blind quantum tomography is possible in a practical setting inspired by an implementation of trapped ions.
In our work, we give a mathematical proof that this is possible if the quantum states are sufficiently pure. We assume that measured data depend linearly on the state and a set of calibration parameters that model small deviations from a calibration baseline. We develop an algorithm that can be efficiently run on a classical computer and prove that given the data as an input, the algorithm converges to the correct state and calibration parameters provided the measurement is suitably unstructured. The number of required measurement scales close to optimally in the degrees of freedom of the problem. Since a pure state is described by fewer parameters than an arbitrary quantum state, our structure assumption allows us to additionally estimate calibration parameters without increasing the number of measurements. Furthermore, we numerically demonstrate that our method works in a setting that is motivated by ion traps experiments. Altogether, we develop a method that we expect to be useful in practice for high-precision state tomography based on the theoretical and numerical evidence.
 A. Acin, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm. ``The European quantum technologies roadmap''. New J. Phys. 20, 080201 (2017). arXiv:1712.03773.
 J. Eisert, D. Hangleiter, N. Walk, I. Roth, R. Markham, D.and Parekh, U. Chabaud, and E. Kashefi. ``Quantum certification and benchmarking''. Nature Rev. Phys. 2, 382–390 (2020). arXiv:1910.06343.
 S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven. ``Characterizing quantum supremacy in near-term devices''. Nature Phys. 14, 595–600 (2018). arXiv:1608.00263.
 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.
 E. Magesan, J. M. Gambetta, and J. Emerson. ``Scalable and robust randomized benchmarking of quantum processes''. Phys. Rev. Lett. 106, 180504 (2011). arXiv:1009.3639.
 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.
 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'' (2013). arXiv:1310.4492.
 A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James. ``Self-calibrating quantum state tomography''. New J. Phys. 14, 085003 (2012).
 D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert. ``Quantum state tomography via compressed sensing''. Phys. Rev. Lett. 105, 150401 (2010).
 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). arXiv:1205.2300.
 A. Kalev, R. L. Kosut, and I. H. Deutsch. ``Quantum tomography protocols with positivity are compressed sensing protocols''. npj Quant. Inf. 1, 15018 (2015). arXiv:1502.00536.
 C. A. Riofrio, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert. ``Experimental quantum compressed sensing for a seven-qubit system''. Nature Comm. 8, 15305 (2017).
 A. Steffens, C. A. Riofrio, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert. ``Experimentally exploring compressed sensing quantum tomography''. Quant. Sc. Tech. 2, 025005 (2017).
 I. Roth, M. Kliesch, A. Flinth, G. Wunder, and J. Eisert. ``Reliable recovery of hierarchically sparse signals for Gaussian and Kronecker product measurements''. IEEE Trans. Sig. Proc. 68, 4002–4016 (2020). arXiv:1612.07806.
 H.-W. Li, Z.-Q. Yin, Y.-C. Wu, X.-B. Zou, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han. ``Semi-device-independent random-number expansion without entanglement''. Phys. Rev. A 84, 034301 (2011).
 H.-W. Li, M. Pawłowski, Z.-Q. Yin, G.-C. Guo, and Z.-F. Han. ``Semi-device-independent randomness certification using $n\rightarrow1$ quantum random access codes''. Phys. Rev. A 85, 052308 (2012).
 R. Gallego, N. Brunner, C. Hadley, and A. Acin. ``Device-independent tests of classical and quantum dimensions''. Phys. Rev. Lett. 105, 230501 (2010).
 D. Mogilevtsev, J. Řeháček, and Z. Hradil. ``Self-calibration for self-consistent tomography''. New J. Phys. 14, 095001 (2012).
 J. Řeháček, D. Mogilevtsev, and Z. Hradil. ``Operational tomography: Fitting of data patterns''. Phys. Rev. Lett. 105, 010402 (2010).
 L. Motka, B. Stoklasa, J. Rehacek, Z. Hradil, V. Karasek, D. Mogilevtsev, G. Harder, C. Silberhorn, and L. L. Sánchez-Soto. ``Efficient algorithm for optimizing data-pattern tomography''. Phys. Rev. A 89, 054102 (2014).
 C. Ferrie. ``Quantum model averaging''. New J. Phys. 16, 093035 (2014).
 R. Blume-Kohout, J. K. Gamble, E. Nielsen, K. Rudinger, J. Mizrahi, K. Fortier, and P. Maunz. ``Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography''. Nature Comm. 8, 14485 (2017). arXiv:1605.07674.
 P. Cerfontaine, R. Otten, and H. Bluhm. ``Self-consistent calibration of quantum-gate sets''. Phys. Rev. Appl. 13, 044071 (2020). arXiv:1906.00950.
 D. Gross. ``Recovering low-rank matrices from few coefficients in any basis''. IEEE Trans. Inf. Th. 57, 1548–1566 (2011). arXiv:0910.1879.
 Y.-K. Liu. ``Universal low-rank matrix recovery from Pauli measurements''. Adv. Neural Inf. Process. Syst. 24, 1638–1646 (2011). arXiv:1103.2816.
 R. Kueng. ``Low rank matrix recovery from few orthonormal basis measurements''. In Sampling Theory and Applications (SampTA), 2015 International Conference on. Pages 402–406. (2015).
 A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White. ``Efficient measurement of quantum dynamics via compressive sensing''. Phys. Rev. Lett. 106, 100401 (2011).
 S. Kimmel and Y. K. Liu. ``Phase retrieval using unitary 2-designs''. In 2017 International Conference on Sampling Theory and Applications (SampTA). Pages 345–349. (2017). arXiv:1510.08887.
 I. Roth, R. Kueng, S. Kimmel, Y.-K. Liu, D. Gross, J. Eisert, and M. Kliesch. ``Recovering quantum gates from few average gate fidelities''. Phys. Rev. Lett. 121, 170502 (2018). arXiv:1803.00572.
 I. Roth, M. Kliesch, G. Wunder, and J. Eisert. ``Reliable recovery of hierarchically sparse signals''. In Proceedings of the third ``international traveling workshop on interactions between sparse models and technology'' (iTWIST'16). (2016). arXiv:1609.04167.
 S. Oymak, A. Jalali, M. Fazel, Y. C. Eldar, and B. Hassibi. ``Simultaneously structured models with application to sparse and low-rank matrices''. IEEE Trans. Inf. Th. 61, 2886–2908 (2015).
 Q. Berthet and P. Rigollet. ``Complexity theoretic lower bounds for sparse principal component detection''. In Conference on Learning Theory. Pages 1046–1066. (2013). url: http://proceedings.mlr.press/v30/Berthet13.html.
 M. Brennan and G. Bresler. ``Optimal average-case reductions to sparse PCA: From weak assumptions to strong hardness''. In 32nd Annual Conference on Learning Theory. Volume 99 of Proceedings of Machine Learning Research. (2019). arXiv:1902.07380.
 G. Wunder, I. Roth, R. Fritschek, B. Groß, and J. Eisert. ``Secure massive IoT using hierarchical fast blind deconvolution''. In 2018 IEEE Wireless Communications and Networking Conference Workshops, WCNC 2018 Workshops, Barcelona, Spain, April 15-18, 2018. Pages 119–124. (2018). arXiv:1801.09628.
 P. Sprechmann, I. Ramirez, G. Sapiro, and Y. Eldar. ``Collaborative hierarchical sparse modeling''. In 2010 44th Annual Conference on Information Sciences and Systems (CISS). Pages 1–6. (2010).
 P. Sprechmann, I. Ramirez, G. Sapiro, and Y. C. Eldar. ``C-HiLasso: A collaborative hierarchical sparse modeling framework''. IEEE Trans. Sig. Proc. 59, 4183–4198 (2011).
 I. Roth, A. Flinth, R. Kueng, J. Eisert, and G. Wunder. ``Hierarchical restricted isometry property for Kronecker product measurements''. In 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton). Pages 632–638. (2018).
 A. Flinth, B. Groß, I. Roth, J. Eisert, and G. Wunder. ``Hierarchical isometry properties of hierarchical measurements''. Appl. Comp. Harm. An. 58, 27–49 (2022). arXiv:2005.10379.
 G. Wunder, I. Roth, R. Fritschek, and J. Eisert. ``HiHTP: A custom-tailored hierarchical sparse detector for massive MTC''. In 2017 51st Asilomar Conference on Signals, Systems, and Computers. Pages 1929–1934. (2017).
 G. Wunder, I. Roth, M. Barzegar, A. Flinth, S. Haghighatshoar, G. Caire, and G. Kutyniok. ``Hierarchical sparse channel estimation for massive mimo''. In WSA 2018; 22nd International ITG Workshop on Smart Antennas. Pages 1–8. VDE (2018).
 G. Wunder, S. Stefanatos, A. Flinth, I. Roth, and G. Caire. ``Low-overhead hierarchically-sparse channel estimation for multiuser wideband massive MIMO''. IEEE Trans. Wire. Comm. 18, 2186–2199 (2019).
 N. Halko, P.-G. Martinsson, and J. A. Tropp. ``Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions''. SIAM Rev. 53, 217–288 (2011).
 K. Wei, J.-F. Cai, T. F. Chan, and S. Leung. ``Guarantees of Riemannian optimization for low rank matrix recovery''. SIAM J. Mat. An. App. 37, 1198–1222 (2016).
 T. Blumensath and M. E. Davies. ``Sampling theorems for signals from the union of finite-dimensional linear subspaces''. IEEE Trans. Inf. Theory 55, 1872–1882 (2009).
 A. S. Bandeira, E. Dobriban, D. G. Mixon, and W. F. Sawin. ``Certifying the restricted isometry property is hard''. IEEE Trans. Inf. Th. 59, 3448–3450 (2013).
 J. Wilkens, D. Hangleiter, and I. Roth (2020). Gitlab repository at https://gitlab.com/wilkensJ/blind-quantum-tomography.
 E. J. Candes and Y. Plan. ``Tight oracle inequalities for low-rank matrix recovery from a minimal optnumber of noisy random measurements''. IEEE Trans. Inf. Th. 57, 2342–2359 (2011).
 Raphael Brieger, Ingo Roth, and Martin Kliesch, "Compressive Gate Set Tomography", PRX Quantum 4 1, 010325 (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-09-22 21:57:37). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref's cited-by service no data on citing works was found (last attempt 2023-09-22 21:57:36).
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.