Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.
Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and $\ell_2$-norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
Process tomography demands a huge number of measurements. However, many relevant time evolutions exhibit additional structure (approximately low Kraus rank). A central result of our paper states that significantly fewer measurements suffice for a full characterization of such evolutions. This includes unitary time evolution -- as given by the Schrödinger equation -- where our method achieves a quadratic reduction of the measurement effort over conventional characterization methods. Our reconstruction technique is conceptually simple and does not require any prior assumptions. Moreover, it is supported by analytic convergence guarantees, as well as extensive numerical demonstrations.
An important application concerns the implementation of quantum gates which form the basic building blocks of quantum circuits. Realistic implementations inevitably suffer from errors. The resulting (noisy) time evolution can be measured. In this way, one can learn what errors were made in order to correct them. Our proposed method is geared towards achieving this goal with substantially fewer measurements than traditional estimation techniques. In this way, it complements single-parameter performance certificates such as randomized benchmarking.
 I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, J. Mod. Opt. 44, 2455 (1997), quant-ph/9610001.
 M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Quantum-process tomography: Resource analysis of different strategies, Phys. Rev. A 77, 032322 (2008), quant-ph/0702131.
 M. Kliesch, R. Kueng, J. Eisert, and D. Gross, Improving compressed sensing with the diamond norm, IEEE Trans. Inf. Th. 62, 7445 (2016), arXiv:1511.01513.
 F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016), arXiv:1208.0692.
 F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Efficient quantum pseudorandomness, Phys. Rev. Lett. 116, 170502 (2016a), arXiv:1605.00713.
 Y. Nakata, C. Hirche, M. Koashi, and A. Winter, Efficient quantum pseudorandomness with nearly time-independent hamiltonian dynamics, Phys. Rev. X 7, 021006 (2017), arXiv:1609.07021.
 E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A. H. Werner, and J. Eisert, Mixing properties of stochastic quantum Hamiltonians, Comm. Math. Phys. 355, 905 (2017), arXiv:1606.01914.
 J. Helsen, J. J. Wallman, and S. Wehner, Representations of the multi-qubit clifford group, J. Math. Phys. 59 (2018), 10.1063/1.4997688, arXiv:1609.08188.
 D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Experimental quantum error correction, Phys. Rev. Lett. 81, 2152 (1998), quant-ph/9802018.
 B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O'Brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher-dimensional Hilbert spaces, Nature Phys. 5, 134 (2009).
 T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Realization of the quantum Toffoli gate with trapped ions, Phys. Rev. Lett. 102, 040501 (2009), arXiv:0804.0082.
 A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, Implementation of a Toffoli gate with superconducting circuits, Nature 481, 170 (2012), arXiv:1108.3966.
 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. 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), arXiv:0910.5498.
 C. H. Baldwin, A. Kalev, and I. H. Deutsch, Quantum process tomography of unitary and near-unitary maps, Phys. Rev. A 90, 012110 (2014), arXiv:1404.2877.
 S. Kimmel and Y. K. Liu, Phase retrieval using unitary 2-designs, in International Conference on Sampling Theory and Applications (SampTA) (2017) pp. 345–349, arXiv:1510.08887.
 R. Kueng, H. Rauhut, and U. Terstiege, Low rank matrix recovery from rank one measurements, Appl. Comp. Harm. Anal. 42, 88 (2017), arXiv:1410.6913.
 A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, Compressed sensing quantum process tomography for superconducting quantum gates, Phys. Rev. B 90, 144504 (2014), arXiv:1407.0761.
 M. Kabanava, R. Kueng, H. Rauhut, and U. Terstiege, Stable low-rank matrix recovery via null space properties, Information and Inference: A Journal of the IMA 5, 405 (2016), arXiv:1507.07184.
 R. Kueng and P. Jung, Robust nonnegative sparse recovery and the nullspace property of 0/1 measurements, IEEE Trans. Inf. Theory 64, 689 (2018), arXiv:1603.07997.
 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), arXiv:0909.3304.
 Y.-K. Liu, in Adv. Neural Inf. Process. Syst. 24, edited by J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. Weinberger (Curran Associates, Inc., 2011) pp. 1638–1646, arXiv:1103.2816.
 J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomography, Phys. Rev. Lett. 90, 193601 (2003), quant-ph/0303038.
 S. Kimmel, M. P. da Silva, C. A. Ryan, B. R. Johnson, and T. Ohki, Robust extraction of tomographic information via randomized benchmarking, Phys. Rev. X 4, 011050 (2014), arXiv:1306.2348.
 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.
 E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theor. 52, 489 (2006).
 T. Yamamoto, M. Neeley, E. Lucero, R. C. Bialczak, J. Kelly, M. Lenander, M. Mariantoni, A. D. O'Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, A. N. Cleland, and J. M. Martinis, Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates using superconducting phase qubits, Phys. Rev. B 82, 184515 (2010), arXiv:1006.5084.
 D. Kim, Z. Shi, C. B. Simmons, D. R. Ward, J. R. Prance, T. S. Koh, J. K. Gamble, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, and M. A. Eriksson, Quantum control and process tomography of a semiconductor quantum dot hybrid qubit, Nature 511, 70 (2014), arXiv:1401.4416.
 B. E. Anderson, H. Sosa-Martinez, C. A. Riofrio, I. H. Deutsch, and P. S. Jessen, Accurate and robust unitary transformation of a high-dimensional quantum system, Phys. Rev. Lett. 114, 240401 (2015), arXiv:1410.3891.
 S. T. Merkel, C. A. Riofrío, S. T. Flammia, and I. H. Deutsch, Random unitary maps for quantum state reconstruction, Phys. Rev. A 81, 032126 (2010), arXiv:0912.2101.
 N. Timoney, V. Elman, S. Glaser, C. Weiss, M. Johanning, W. Neuhauser, and C. Wunderlich, Error-resistant single-qubit gates with trapped ions, Phys. Rev. A 77, 052334 (2008), quant-ph/0612106.
 M. Ohliger, V. Nesme, and J. Eisert, Efficient and feasible state tomography of quantum many-body systems, New J. Phys. 15, 015024 (2013), arXiv:1204.5735.
 P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007), arXiv:quant-ph/0512071.
 J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Universal linear optics, Science 349, 711 (2015), arXiv:1505.01182.
 N. J. Russell, L. Chakhmakhchyan, J. L. O'Brien, and A. Laing, Direct dialling of Haar random unitary matrices, New J. Phys. 19, 033007 (2017), arXiv:1506.06220.
 F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Comm. Math. Phys. 346, 397 (2016b), arXiv:1208.0692.
 J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171 (2004), quant-ph/0310075.
 A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world, in Computational Complexity, 2007. CCC '07. Twenty-Second Annual IEEE Conference on (2007) pp. 129–140, quant-ph/0701126.
 D. Gross, K. M. R. Audenaert, and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys. 48, 052104 (2007), quant-ph/0611002.
 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), quant-ph/0606161.
 B. Recht, M. Fazel, and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev. 52, 471 (2010), arXiv:0706.4138.
 E. J. Candès and Y. Plan, Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements, IEEE Trans. Inform. Theory 57, 2342 (2011), arXiv:1001.0339.
 A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and quantum computation, Vol. 47 (American Mathematical Society, 2002).
 J. Watrous, CS 766 Theory of quantum information, Available online at https://cs.uwaterloo.ca/ watrous/LectureNotes.html (2011).
 R. Kueng, Low rank matrix recovery from few orthonormal basis measurements, in Sampling Theory and Applications (SampTA), 2015 International Conference on (2015) pp. 402–406.
 M. Grant and S. Boyd, in Recent advances in learning and control, Lecture Notes in Control and Information Sciences, edited by V. Blondel, S. Boyd, and H. Kimura (Springer-Verlag Limited, 2008) pp. 95–110,.
 J. A. Tropp, Convex recovery of a structured signal from independent random linear measurements, in Sampling Theory, a Renaissance, edited by E. G. Pfander (Springer, 2015) pp. 67–101, arXiv:1405.1102.
 U. Michel, M. Kliesch, R. Kueng, and D. Gross, Note on the saturation of the norm inequalities between diamond and nuclear norm, IEEE Trans. Inf. Theory 64, 7443 (2018), arXiv:1612.07931.
 A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, Experimentally exploring compressed sensing quantum tomography, Quantum Sci. Technol. 2, 025005 (2017), arXiv:1611.01189.
 J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, IEEE Trans Inf. Th. 63, 5628 (2017), arXiv:1508.01797.
 A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995), quant-ph/9503016.
 M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).
 A. Y. Kitaev, Quantum computations: algorithms and error correction, Russian Math. Surv. 52, 1191 (1997).
 E. Knill, R. Laflamme, and W. H. Zurek, Resilient quantum computation: error models and thresholds, Proc. R. Soc. A 454, 365 (1998), arXiv:quant-ph/9702058.
 D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error rate, in 29th ACM Symp. on Theory of Computing (STOC) (New York, 1997) pp. 176–188.
 S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few Pauli measurements, Phys. Rev. Lett. 106, 230501 (2011), arXiv:1104.4695.
 J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, J. Opt. B 7, S347 (2005), arXiv:quant-ph/0503243.
 J. J. Wallman and S. T. Flammia, Randomized benchmarking with confidence, New J. Phys. 16, 103032 (2014), arXiv:1404.6025.
 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), arXiv:1510.05653.
 V. Koltchinskii and S. Mendelson, Bounding the smallest singular value of a random matrix without concentration, International Mathematics Research Notices , rnv096 (2015), arXiv:1312.3580.
 R. Goodman and N. R. Wallach, Representations and invariants of the classical groups, Vol. 68 (Cambridge University Press, 2000).
 J. A. Tropp, User-friendly tools for random matrices: An introduction, Tech. Rep. (DTIC Document, 2012).
 U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011), January 2011 Special Issue, arXiv:1008.3477.
 V. Chandrasekaran, B. Recht, P. Parrilo, and A. Willsky, The convex geometry of linear inverse problems, Found. Comput. Math. 12, 805 (2012), arXiv:1012.0621.
 S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. 3, 1 (2011).
 A. Kyrillidis, A. Kalev, D. Park, S. Bhojanapalli, C. Caramanis, and S. Sanghavi, Provable compressed sensing quantum state tomography via non-convex methods, npj Quantum Information 4, 36 (2018), arXiv:1711.02524.
 A. Acharya, T. Kypraios, and M. Guta, Statistically efficient tomography of low rank states with incomplete measurements, New J. Phys. 18, 043018 (2016), arXiv:1510.03229.
 A. Acharya and M. Guta, Statistical analysis of compressive low rank tomography with random measurements, J. Phys. A 50, 195301 (2017), arXiv:1609.03758.
 E. J. Candès, X. Li, and M. Soltanolkotabi, Phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal. 39, 277 (2015), arXiv:1310.3240.
 D. Gross, F. Krahmer, and R. Kueng, Improved recovery guarantees for phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal. 41, 37 (2016), arXiv:1402.6286.
 M. Ziman, Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments, Phys. Rev. A 77, 062112 (2008), arXiv:0802.3862.
 R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing: Theory and Applications (Cambridge University Press, 2012) pp. 210–268, arXiv:1011.3027.
 S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing (Springer, 2013).
 Symmetric group:s4, http://groupprops.subwiki.org/wiki/Symmetric_group:S4 (2016a), accessed: 2016-08-17.
 Linear representation theory of symmetric group:S4, http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_group:S4 (2016b), accessed: 2016-08-17.
 B. Simon, Representations of finite and compact groups, 10 (Am. Math. Soc., 1996).
 Ariadna E. Venegas-Li, Alexandra M. Jurgens, and James P. Crutchfield, "Measurement-induced randomness and structure in controlled qubit processes", Physical Review E 102 4, 040102 (2020).
 Jens Eisert, Dominik Hangleiter, Nathan Walk, Ingo Roth, Damian Markham, Rhea Parekh, Ulysse Chabaud, and Elham Kashefi, "Quantum certification and benchmarking", Nature Reviews Physics 2 7, 382 (2020).
 Ye-Chao Liu, Jiangwei Shang, Xiao-Dong Yu, and Xiangdong Zhang, "Efficient verification of quantum processes", Physical Review A 101 4, 042315 (2020).
 I. Roth, R. Kueng, S. Kimmel, Y. -K. Liu, D. Gross, J. Eisert, and M. Kliesch, "Recovering Quantum Gates from Few Average Gate Fidelities", Physical Review Letters 121 17, 170502 (2018).
 Joel J. Wallman, "Randomized benchmarking with gate-dependent noise", arXiv:1703.09835.
 Jian-Feng Cai and Ke Wei, "Exploiting the structure effectively and efficiently in low rank matrix recovery", arXiv:1809.03652.
 Fernando G. S. L. Brandão, Wissam Chemissany, Nicholas Hunter-Jones, Richard Kueng, and John Preskill, "Models of quantum complexity growth", arXiv:1912.04297.
 N. Milazzo, D. Braun, and O. Giraud, "Optimal measurement strategies for fast entanglement detection", Physical Review A 100 1, 012328 (2019).
 Thomas Strohmer and Ke Wei, "Painless Breakups -- Efficient Demixing of Low Rank Matrices", arXiv:1703.09848.
The above citations are from Crossref's cited-by service (last updated successfully 2020-10-20 18:50:25) and SAO/NASA ADS (last updated successfully 2020-10-20 18:50:26). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.