Projected Least-Squares Quantum Process Tomography

Abstract

We propose and investigate a new method of quantum process tomography (QPT) which we call projected least squares (PLS). In short, PLS consists of first computing the least-squares estimator of the Choi matrix of an unknown channel, and subsequently projecting it onto the convex set of Choi matrices. We consider four experimental setups including direct QPT with Pauli eigenvectors as input and Pauli measurements, and ancilla-assisted QPT with mutually unbiased bases (MUB) measurements. In each case, we provide a closed form solution for the least-squares estimator of the Choi matrix. We propose a novel, two-step method for projecting these estimators onto the set of matrices representing physical quantum channels, and a fast numerical implementation in the form of the hyperplane intersection projection algorithm. We provide rigorous, non-asymptotic concentration bounds, sampling complexities and confidence regions for the Frobenius and trace-norm error of the estimators. For the Frobenius error, the bounds are linear in the rank of the Choi matrix, and for low ranks, they improve the error rates of the least squares estimator by a factor $d^2$, where $d$ is the system dimension. We illustrate the method with numerical experiments involving channels on systems with up to 7 qubits, and find that PLS has highly competitive accuracy and computational tractability.

Featured image: Illustration of process tomography on a noisy 7-qubit quantum Fourier transform using Pauli input states and Pauli measurements

► BibTeX data

@article{SurawyStepney2022projectedleast, doi = {10.22331/q-2022-10-20-844}, url = {https://doi.org/10.22331/q-2022-10-20-844}, title = {Projected {L}east-{S}quares {Q}uantum {P}rocess {T}omography}, author = {Surawy-Stepney, Trystan and Kahn, Jonas and Kueng, Richard and Guta, Madalin}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {6}, pages = {844}, month = oct, year = {2022} }

► References

[1] M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, and R. Blatt. “Process tomography of ion trap quantum gates”. Phys. Rev. Lett. 97, 220407 (2006).
https://doi.org/10.1103/PhysRevLett.97.220407

[2] Y. S. Weinstein, T. F. Havel, J. Emerson, N. Boulant, M. Saraceno, S. Lloyd, and D. G. Cory. “Quantum process tomography of the quantum fourier transform”. The Journal of Chemical Physics 121, 6117–6133 (2004).
https://doi.org/10.1063/1.1785151

[3] J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White. “Quantum process tomography of a controlled-not gate”. Phys. Rev. Lett. 93, 080502 (2004).
https://doi.org/10.1103/PhysRevLett.93.080502

[4] L. A. Pachón, A. H. Marcus, and A. Aspuru-Guzik. “Quantum process tomography by 2d fluorescence spectroscopy”. The Journal of Chemical Physics 142, 212442 (2015).
https://doi.org/10.1063/1.4919954

[5] R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, and et al. “Quantum process tomography of a universal entangling gate implemented with josephson phase qubits”. Nature Physics 6, 409–413 (2010).
https://doi.org/10.1038/nphys1639

[6] M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, and J. Wrachtrup. “Quantum process tomography and linblad estimation of a solid-state qubit”. New Journal of Physics 8, 33–33 (2006).
https://doi.org/10.1088/1367-2630/8/3/033

[7] I. L. Chuang and M. A. Nielsen. “Prescription for experimental determination of the dynamics of a quantum black box”. Journal of Modern Optics 44, 2455–2467 (1997).
https://doi.org/10.1080/09500349708231894

[8] J. F. Poyatos, J. I. Cirac, and P. Zoller. “Complete characterization of a quantum process: The two-bit quantum gate”. Phys. Rev. Lett. 78, 390–393 (1997).
https://doi.org/10.1103/PhysRevLett.78.390

[9] M. D. Choi. “Completely positive linear maps on complex matrices”. Linear Algebra and its Applications 10, 285 – 290 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0

[10] A. Jamiołkowski. “Linear transformations which preserve trace and positive semidefiniteness of operators”. Reports on Mathematical Physics 3, 275 – 278 (1972).
https://doi.org/10.1016/0034-4877(72)90011-0

[11] D. W. Leung. “Towards robust quantum computation”. Thesis (PhD). STANFORD UNIVERSITY, Source DAI-B 61/11, p. 5911, 225 pages (2000). arXiv:cs/0012017.
https://doi.org/10.48550/arXiv.cs/0012017
arXiv:cs/0012017

[12] G. M. D’Ariano and P. Lo Presti. “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation”. Phys. Rev. Lett. 86, 4195–4198 (2001).
https://doi.org/10.1103/PhysRevLett.86.4195

[13] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White. “Measurement of qubits”. Phys. Rev. A 64, 052312 (2001).
https://doi.org/10.1103/PhysRevA.64.052312

[14] A. I. Lvovsky. “Iterative maximum-likelihood reconstruction in quantum homodyne tomography”. Journal of Optics B: Quantum and Semiclassical Optics 6, S556–S559 (2004).
https://doi.org/10.1088/1464-4266/6/6/014

[15] R. Blume-Kohout. “Hedged maximum likelihood quantum state estimation”. Phys. Rev. Lett. 105, 200504 (2010).
https://doi.org/10.1103/PhysRevLett.105.200504

[16] J. A. Smolin, J. M. Gambetta, and G. Smith. “Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise”. Phys. Rev. Lett. 108, 070502 (2012).
https://doi.org/10.1103/PhysRevLett.108.070502

[17] L. Granade, J. Combes, and D. G. Cory. “Practical bayesian tomography”. New Journal of Physics 18, 033024 (2016).
https://doi.org/10.1088/1367-2630/18/3/033024

[18] M. Christandl, R. König, and R. Renner. “Postselection technique for quantum channels with applications to quantum cryptography”. Phys. Rev. Lett. 102, 020504 (2009).
https://doi.org/10.1103/PhysRevLett.102.020504

[19] R. Blume-Kohout. “Optimal, reliable estimation of quantum states”. New Journal of Physics 12, 043034 (2010).
https://doi.org/10.1088/1367-2630/12/4/043034

[20] C. Granade, C. Ferrie, and S. T. Flammia. “Practical adaptive quantum tomography”. New Journal of Physics 19, 113017 (2017).
https://doi.org/10.1088/1367-2630/aa8fe6

[21] H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt. “Scalable multiparticle entanglement of trapped ions”. Nature 438, 643–646 (2005).
https://doi.org/10.1038/nature04279

[22] M. Christandl and R. Renner. “Reliable quantum state tomography”. Phys. Rev. Lett. 109, 120403 (2012).
https://doi.org/10.1103/PhysRevLett.109.120403

[23] P. Faist and R. Renner. “Practical and reliable error bars in quantum tomography”. Phys. Rev. Lett. 117, 010404 (2016).
https://doi.org/10.1103/PhysRevLett.117.010404

[24] L. P. Thinh, P. Faist, J. Helsen, D. Elkouss, and S. Wehner. “Practical and reliable error bars for quantum process tomography”. Phys. Rev. A 99, 052311 (2019).
https://doi.org/10.1103/PhysRevA.99.052311

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

[26] 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).
https://doi.org/10.1103/PhysRevLett.121.170502

[27] M. Kliesch, R. Kueng, J. Eisert, and D. Gross. “Guaranteed recovery of quantum processes from few measurements”. Quantum 3, 171 (2019).
https://doi.org/10.22331/q-2019-08-12-171

[28] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu. “Efficient quantum state tomography”. Nature communications 1, 149 (2009).
https://doi.org/10.1038/ncomms1147

[29] T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio. “Scalable reconstruction of density matrices”. Phys. Rev. Lett. 111, 020401 (2013).
https://doi.org/10.1103/PhysRevLett.111.020401

[30] B. P. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempe, P. Jurcevic, I. Dhand, A. S. Buyskikh, A. J. Daley, M. Cramer, M. B. Plenio, R. Blatt, and C. F. Roos. “Efficient tomography of a quantum many-body system”. Nature Physics 13, 1158–1162 (2017).
https://doi.org/10.1038/nphys4244

[31] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, , and G. Carleo. “Neural network quantum state tomography”. Nature Physics 14, 447–450 (2018).
https://doi.org/10.1038/s41567-018-0048-5

[32] G. Torlai, C. J. Wood, A. Acharya, G. Carleo, J. Carrasquilla, and L. Aolita. “Quantum process tomography with unsupervised learning and tensor networks” (2020). arXiv:2006.02424.
arXiv:2006.02424

[33] J. M. Chow, J. M. Gambetta, L. Tornberg, Jens Koch, Lev S. Bishop, A. A. Houck, B. R. Johnson, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. “Randomized benchmarking and process tomography for gate errors in a solid-state qubit”. Phys. Rev. Lett. 102, 090502 (2009).
https://doi.org/10.1103/PhysRevLett.102.090502

[34] 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”. Physical Review A 77 (2008).
https://doi.org/10.1103/physreva.77.012307

[35] L. Steffen, M. P. da Silva, A. Fedorov, M. Baur, and A. Wallraff. “Experimental monte carlo quantum process certification”. Phys. Rev. Lett. 108, 260506 (2012).
https://doi.org/10.1103/PhysRevLett.108.260506

[36] M. P. da Silva, O. Landon-Cardinal, and D. Poulin. “Practical characterization of quantum devices without tomography”. Phys. Rev. Lett. 107, 210404 (2011).
https://doi.org/10.1103/PhysRevLett.107.210404

[37] M. Guţă, J. Kahn, R. Kueng, and J. A. Tropp. “Fast state tomography with optimal error bounds”. Journal of Physics A: Mathematical and Theoretical 53, 204001 (2020).
https://doi.org/10.1088/1751-8121/ab8111

[38] G. C. Knee, E. Bolduc, J. Leach, and E. M. Gauger. “Quantum process tomography via completely positive and trace-preserving projection”. Phys. Rev. A 98, 062336 (2018).
https://doi.org/10.1103/PhysRevA.98.062336

[39] R. L. Dykstra. “An algorithm for restricted least squares regression”. Journal of the American Statistical Association 78, 837–842 (1983). url: http://www.jstor.org/stable/2288193.
http://www.jstor.org/stable/2288193

[40] M. Paris and J. Rehacek, editors. “Quantum state estimation”. Volume 649 of Lecture Notes in Physics. Springer. (2004).
https://doi.org/10.1007/b98673

[41] T. Sugiyama, P. S. Turner, and M. Murao. “Precision-guaranteed quantum tomography”. Phys. Rev. Lett. 111, 160406 (2013).
https://doi.org/10.1103/PhysRevLett.111.160406

[42] M. Kliesch and I. Roth. “Theory of quantum system certification”. PRX Quantum 2, 010201 (2021).
https://doi.org/10.1103/PRXQuantum.2.010201

[43] J. Kahn and M. Guţă. “Local asymptotic normality for finite dimensional quantum systems”. Communications in Mathematical Physics 289, 597–652 (2009).
https://doi.org/10.1007/s00220-009-0787-3

[44] R. O’Donnell and J. Wright. “Efficient quantum tomography”. In Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing. Pages 899–912. STOC ’16New York, NY, USA (2016). ACM.
https://doi.org/10.1145/2897518.2897544

[45] J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu. “Sample-optimal tomography of quantum states”. IEEE T. Inform. Theory 63, 5628–5641 (2017).
https://doi.org/10.1109/TIT.2017.2719044

[46] V. Giovannetti, S. Lloyd, and L. Maccone. “Advances in quantum metrology”. Nature Photonics 5, 222–229 (2011).
https://doi.org/10.1038/nphoton.2011.35

[47] S. Zhou and L. Jiang. “Asymptotic theory of quantum channel estimation”. PRX Quantum 2, 010343 (2021).
https://doi.org/10.1103/PRXQuantum.2.010343

[48] J. Schwinger. “Unitary operator bases”. Proc. Nat. Acad. Sci. U.S.A. 46, 570–579 (1960).
https://doi.org/10.1073/pnas.46.4.570

[49] W. K. Wootters and B. D. Fields. “Optimal state-determination by mutually unbiased measurements”. Ann. Physics 191, 363–381 (1989).
https://doi.org/10.1016/0003-4916(89)90322-9

[50] D. Gross, F. Krahmer, and R. Kueng. “A partial derandomization of phaselift using spherical designs”. J. Fourier Anal. Appl. 21, 229–266 (2015).
https://doi.org/10.1007/s00041-014-9361-2

[51] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves. “Symmetric informationally complete quantum measurements”. Journal of Mathematical Physics 45, 2171–2180 (2004).
https://doi.org/10.1063/1.1737053

[52] 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

[53] F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki. “Local random quantum circuits are approximate polynomial-designs”. Comm. Math. Phys. 346, 397–434 (2016).
https://doi.org/10.1007/s00220-016-2706-8

[54] N. Hunter-Jones. “Unitary designs from statistical mechanics in random quantum circuits” (2019). arXiv:1905.12053.
arXiv:1905.12053

[55] Jonas Haferkamp and Nicholas Hunter-Jones. “Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions”. Phys. Rev. A 104, 022417 (2021).
https://doi.org/10.1103/PhysRevA.104.022417

[56] H. H. Bauschke, P. L. Combettes, et al. “Convex analysis and monotone operator theory in hilbert spaces”. Volume 408. Springer. (2011).
https://doi.org/10.1007/978-1-4419-9467-7

[57] A. Gilchrist, N. K. Langford, and M. A. Nielsen. “Distance measures to compare real and ideal quantum processes”. Physical Review A 71 (2005).
https://doi.org/10.1103/physreva.71.062310

[58] J. Watrous. “Unital channels and majorization”. Page 201–249. Cambridge University Press. (2018).
https://doi.org/10.1017/9781316848142.005

[59] J. J. Wallman. “Bounding experimental quantum error rates relative to fault-tolerant thresholds” (2015).

[60] R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia. “Comparing experiments to the fault-tolerance threshold”. Phys. Rev. Lett. 117, 170502 (2016).
https://doi.org/10.1103/PhysRevLett.117.170502

[61] R. Kueng, H. Rauhut, and U. Terstiege. “Low rank matrix recovery from rank one measurements”. Appl. Comput. Harmon. Anal. 42, 88–116 (2017).
https://doi.org/10.1016/j.acha.2015.07.007

[62] H. Y. Huang, R. Kueng, and J. Preskill. “Predicting many properties of a quantum system from very few measurements”. Nat. Phys. 16, 1050––1057 (2020). url: doi.org/10.1038/s41567-020-0932-7.
https://doi.org/10.1038/s41567-020-0932-7

[63] D. S. França, F. G. S. L. Brandão, and R. Kueng. “Fast and Robust Quantum State Tomography from Few Basis Measurements”. In Min-Hsiu Hsieh, editor, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Volume 197 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1–7:13. Dagstuhl, Germany (2021). Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.TQC.2021.7

[64] J. Kahn. “Hyperplane intersection projection”. https://github.com/Hannoskaj/Hyperplane_Intersection_Projection (2021).
https://github.com/Hannoskaj/Hyperplane_Intersection_Projection

[65] J. Kahn, M Guţă, R. Kueng, and T. Surawy-Stepney. “Hyperplane intersection projection” (To be written).

[66] M. Slater. “Lagrange multipliers revisited”. Cowles Commission Discussion Paper No. 403 (1950). url: cowles.yale.edu/sites/default/files/files/pub/d00/d0080.pdf.
https://cowles.yale.edu/sites/default/files/files/pub/d00/d0080.pdf

[67] S. Boyd and L. Vandenberghe. “Convex optimization”. Cambridge University Press. (2009).
https://doi.org/10.1017/CBO9780511804441

[68] Charles George Broyden. “The convergence of a class of double-rank minimization algorithms 1. general considerations”. IMA Journal of Applied Mathematics 6, 76–90 (1970).
https://doi.org/10.1093/imamat/6.1.76

[69] David F Shanno. “Conditioning of quasi-newton methods for function minimization”. Mathematics of computation 24, 647–656 (1970).
https://doi.org/10.1090/S0025-5718-1970-0274029-X

[70] Roger Fletcher. “A new approach to variable metric algorithms”. The computer journal 13, 317–322 (1970).
https://doi.org/10.1093/comjnl/13.3.317

[71] Donald Goldfarb. “A family of variable-metric methods derived by variational means”. Mathematics of computation 24, 23–26 (1970).
https://doi.org/10.2307/2004873

[72] Yurii Nesterov. “A method for unconstrained convex minimization problem with the rate of convergence o ($1/k^{2}$)”. In Doklady an ussr. Volume 269, pages 543–547. (1983). url: cir.nii.ac.jp/crid/1570572699326076416.
https://cir.nii.ac.jp/crid/1570572699326076416

[73] Alp Yurtsever, Joel A. Tropp, Olivier Fercoq, Madeleine Udell, and Volkan Cevher. “Scalable semidefinite programming”. SIAM Journal on Mathematics of Data Science 3, 171–200 (2021).
https://doi.org/10.1137/19M1305045

[74] P. Tol. “Colour schemes” (2018).

[75] M. A. Nielsen and I. L. Chuang. “Quantum computation and quantum information: 10th anniversary edition”. Cambridge University Press. (2010).
https://doi.org/10.1017/CBO9780511976667

[76] J. A. Tropp. “User-friendly tail bounds for sums of random matrices”. Foundations of Computational Mathematics 12, 389–434 (2012).
https://doi.org/10.1007/s10208-011-9099-z

[77] J. L. W. V. Jensen. “Sur les fonctions convexes et les inégalités entre les valeurs moyennes”. Acta Math. 30, 175–193 (1906).
https://doi.org/10.1007/BF02418571

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[1] Hsin-Yuan Huang, Steven T. Flammia, and John Preskill, “Foundations for learning from noisy quantum experiments”, arXiv:2204.13691.

[2] Ryan Levy, Di Luo, and Bryan K. Clark, “Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers”, arXiv:2110.02965.

[3] Jonathan Kunjummen, Minh C. Tran, Daniel Carney, and Jacob M. Taylor, “Shadow process tomography of quantum channels”, arXiv:2110.03629.

[4] Christopher W. Warren, Jorge Fernández-Pendás, Shahnawaz Ahmed, Tahereh Abad, Andreas Bengtsson, Janka Biznárová, Kamanasish Debnath, Xiu Gu, Christian Križan, Amr Osman, Anita Fadavi Roudsari, Per Delsing, Göran Johansson, Anton Frisk Kockum, Giovanna Tancredi, and Jonas Bylander, “Extensive characterization of a family of efficient three-qubit gates at the coherence limit”, arXiv:2207.02938.

[5] Zuzana Gavorová, Matan Seidel, and Yonathan Touati, “Topological obstructions to implementing quantum if-clause”, arXiv:2011.10031.

[6] Shahnawaz Ahmed, Fernando Quijandría, and Anton Frisk Kockum, “Gradient-descent quantum process tomography by learning Kraus operators”, arXiv:2208.00812.

[7] Akshay Gaikwad, Arvind, and Kavita Dorai, “Efficient experimental characterization of quantum processes via compressed sensing on an NMR quantum processor”, arXiv:2109.13189.

[8] Guo-Dong Lu, Zhou Zhang, Yue Dai, Yu-Li Dong, and Cheng-Jie Zhang, “Not All Entangled States are Useful for Ancilla‑Assisted Quantum Process Tomography”, Annalen der Physik 534 5, 2100550 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2022-11-13 07:31:10). 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 2022-11-13 07:31:08).

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Generative Data Intelligence

Projected Least-Squares Quantum Process Tomography

Abstract

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