Guaranteed recovery of quantum processes from few measurements

Martin Kliesch1,2, Richard Kueng3,4,5, Jens Eisert4,6,7, and David Gross3,8

1Institute for Theoretical Physics, Heinrich Heine University Düsseldorf, Germany
2Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland
3Institute for Theoretical Physics, University of Cologne, Germany
4Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
5Institute for Quantum Information and Matter, California Institute of Technology, USA
6Department of Mathematics and Computer Science, Freie Universität Berlin, Germany
7Helmholtz-Zentrum Berlin für Materialien und Energie, Germany
8Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Australia

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Abstract

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.

Characterizing the time evolution of a quantum system is a fundamental task both in theory and practice. Prospective quantum technologies, in particular, require precise control of both the systems and their evolutions (e.g., ``quantum gates’’). The task of inferring the state of a system from measurements is well-understood (``state tomography’’). In contrast, much less is known about characterizing evolutions from observations alone (``process tomography’’). In this work, we provide a practical procedure for this task.

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.

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► References

[1] 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.
https:/​/​doi.org/​10.1080/​09500349708231894
arXiv:quant-ph/9610001

[2] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.77.032322
arXiv:quant-ph/0702131

[3] 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.
https:/​/​doi.org/​10.1109/​TIT.2016.2606500
arXiv:1511.01513

[4] 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.
https:/​/​doi.org/​10.1007/​s00220-016-2706-8
arXiv:1208.0692

[5] F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Efficient quantum pseudorandomness, Phys. Rev. Lett. 116, 170502 (2016a), arXiv:1605.00713.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.170502
arXiv:1605.00713

[6] 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.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021006
arXiv:1609.07021

[7] A. Harrow and S. Mehraban, Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates, arXiv:1809.06957.
arXiv:1809.06957

[8] 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.
https:/​/​doi.org/​10.1007/​s00220-017-2950-6
arXiv:1606.01914

[9] H. Zhu, R. Kueng, M. Grassl, and D. Gross, The Clifford group fails gracefully to be a unitary 4-design, arXiv:1609.08172.
arXiv:1609.08172

[10] 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.
https:/​/​doi.org/​10.1063/​1.4997688
arXiv:1609.08188

[11] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.81.2152
arXiv:quant-ph/9802018

[12] 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).
https:/​/​doi.org/​10.1038/​nphys1150

[13] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.102.040501
arXiv:0804.0082

[14] 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.
https:/​/​doi.org/​10.1038/​nature10713
arXiv:1108.3966

[15] 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.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​9/​095022
arXiv:1205.2300

[16] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.100401
arXiv:0910.5498

[17] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.90.012110
arXiv:arXiv:1404.2877

[18] 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.
https:/​/​doi.org/​10.1109/​SAMPTA.2017.8024414
arXiv:1510.08887

[19] 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.
https:/​/​doi.org/​10.1016/​j.acha.2015.07.007
arXiv:1410.6913

[20] 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.
https:/​/​doi.org/​10.1103/​PhysRevB.90.144504
arXiv:1407.0761

[21] 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.
https:/​/​doi.org/​10.1093/​imaiai/​iaw014
arXiv:1507.07184

[22] 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.
https:/​/​doi.org/​10.1109/​TIT.2017.2746620
arXiv:1603.07997

[23] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.150401
arXiv:0909.3304

[24] D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inf. Th. 57, 1548 (2011), arXiv:0910.1879.
https:/​/​doi.org/​10.1109/​TIT.2011.2104999
arXiv:0910.1879

[25] 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.
arXiv:1103.2816
http:/​/​papers.nips.cc/​paper/​4222-universal-low-rank-matrix-recovery-from-pauli-measurements.pdf

[26] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.90.193601
arXiv:quant-ph/0303038

[27] 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.
https:/​/​doi.org/​10.1103/​PhysRevX.4.011050
arXiv:1306.2348

[28] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.121.170502
arXiv:1803.00572

[29] 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).
https:/​/​doi.org/​10.1109/​TIT.2005.862083

[30] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Th. 52, 1289 (2006).
https:/​/​doi.org/​10.1109/​TIT.2006.871582

[31] 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.
https:/​/​doi.org/​10.1103/​PhysRevB.82.184515
arXiv:1006.5084

[32] 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.
https:/​/​doi.org/​10.1038/​nature13407
arXiv:arXiv:1401.4416

[33] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.114.240401
arXiv:arXiv:1410.3891

[34] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.81.032126
arXiv:0912.2101

[35] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012).
https:/​/​doi.org/​10.1038/​nphys2252

[36] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.77.052334
arXiv:quant-ph/0612106

[37] 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.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​1/​015024
arXiv:1204.5735

[38] I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
https:/​/​doi.org/​10.1038/​nphys2259

[39] 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.
https:/​/​doi.org/​10.1103/​RevModPhys.79.135
arXiv:arXiv:quant-ph/0512071

[40] 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.
https:/​/​doi.org/​10.1126/​science.aab3642
arXiv:arXiv:1505.01182

[41] 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.
https:/​/​doi.org/​10.1088/​1367-2630/​aa60ed
arXiv:1506.06220

[42] 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.
https:/​/​doi.org/​10.1007/​s00220-016-2706-8
arXiv:1208.0692

[43] 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

[44] 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

[45] A. Gilchrist, N. K. Langford, and M. A. Nielsen, Distance measures to compare real and ideal quantum processes, Phys. Rev. A 71, 062310 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.062310

[46] J. Watrous, Semidefinite programs for completely bounded norms, Theory of Computing 5, 217 (2009), arXiv:0901.4709.
https:/​/​doi.org/​10.4086/​toc.2009.v005a011
arXiv:0901.4709

[47] A. Ben-Aroya and A. Ta-Shma, On the complexity of approximating the diamond norm, arXiv:0902.3397.
arXiv:0902.3397

[48] J. Watrous, Simpler semidefinite programs for completely bounded norms, arXiv:1207.5726.
arXiv:1207.5726

[49] P. Delsarte, J. Goethals, and J. Seidel, Spherical codes and designs, Geom. Dedicata 6, 363 (1977).
https:/​/​doi.org/​10.1007/​BF03187604

[50] 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.
https:/​/​doi.org/​10.1063/​1.1737053
arXiv:quant-ph/0310075

[51] 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.
https:/​/​doi.org/​10.1109/​CCC.2007.26
arXiv:quant-ph/0701126

[52] 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.
https:/​/​doi.org/​10.1063/​1.2716992
arXiv:quant-ph/0611002

[53] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.80.012304
arXiv:quant-ph/0606161

[54] E. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math. 9, 717 (2009), arXiv:0805.4471.
https:/​/​doi.org/​10.1007/​s10208-009-9045-5
arXiv:0805.4471

[55] 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.
https:/​/​doi.org/​10.1137/​070697835
arXiv:0706.4138

[56] 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.
https:/​/​doi.org/​10.1109/​TIT.2011.2111771
arXiv:1001.0339

[57] A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and quantum computation, Vol. 47 (American Mathematical Society, 2002).

[58] J. Watrous, CS 766 Theory of quantum information, Available online at https:/​/​cs.uwaterloo.ca/​ watrous/​LectureNotes.html (2011).
https:/​/​cs.uwaterloo.ca/​~watrous/​LectureNotes.html

[59] 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.
https:/​/​doi.org/​10.1109/​SAMPTA.2015.7148921

[60] M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, http:/​/​cvxr.com/​cvx (2014).
http:/​/​cvxr.com/​cvx

[61] 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,.
https:/​/​doi.org/​10.1007/​978-1-84800-155-8

[62] 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.
https:/​/​doi.org/​10.1007/​978-3-319-19749-4_2
arXiv:1405.1102

[63] 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.
https:/​/​doi.org/​10.1109/​TIT.2018.2861887
arXiv:1612.07931

[64] 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.
https:/​/​doi.org/​10.1088/​2058-9565/​aa6ae2
arXiv:1611.01189

[65] 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.
https:/​/​doi.org/​10.1109/​TIT.2017.2719044
arXiv:1508.01797

[66] Y. Shi, Both Toffoli and controlled-NOT need little help to do universal quantum computation, quant-ph/​0205115.
arXiv:quant-ph/0205115

[67] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.52.3457
arXiv:quant-ph/9503016

[68] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).

[69] A. Y. Kitaev, Quantum computations: algorithms and error correction, Russian Math. Surv. 52, 1191 (1997).
https:/​/​doi.org/​10.1070/​RM1997v052n06ABEH002155

[70] 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.
https:/​/​doi.org/​10.1098/​rspa.1998.0166
arXiv:quant-ph/9702058

[71] 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.
https:/​/​doi.org/​10.1145/​258533.258579

[72] S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few Pauli measurements, Phys. Rev. Lett. 106, 230501 (2011), arXiv:1104.4695.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.230501
arXiv:1104.4695

[73] 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

[74] J. J. Wallman and S. T. Flammia, Randomized benchmarking with confidence, New J. Phys. 16, 103032 (2014), arXiv:1404.6025.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​10/​103032
arXiv:1404.6025

[75] 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.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.170502
arXiv:1510.05653

[76] S. Mendelson, Learning without concentration, J. ACM 62, 21:1 (2015), arXiv:1401.0304.
https:/​/​doi.org/​10.1145/​2699439
arXiv:1401.0304

[77] 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.
https:/​/​doi.org/​10.1093/​imrn/​rnv096
arXiv:1312.3580

[78] R. Goodman and N. R. Wallach, Representations and invariants of the classical groups, Vol. 68 (Cambridge University Press, 2000).

[79] J. A. Tropp, User-friendly tools for random matrices: An introduction, Tech. Rep. (DTIC Document, 2012).
http:/​/​www.dtic.mil/​cgi-bin/​GetTRDoc?AD=ADA576100

[80] 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.
https:/​/​doi.org/​10.1016/​j.aop.2010.09.012
arXiv:1008.3477

[81] S. Gharibian, Z. Landau, S. W. Shin, and G. Wang, Tensor network non-zero testing, Quant. Inf. Comp. 15, 885 (2015), arXiv:1406.5279.
arXiv:1406.5279

[82] 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.
https:/​/​doi.org/​10.1007/​s10208-012-9135-7
arXiv:1012.0621

[83] R. A. Low, Pseudo-randomness and learning in quantum computation, Ph.D. thesis, University of Bristol (2010), arXiv:1006.5227.
arXiv:1006.5227

[84] E. Knill, Fermionic linear optics and matchgates, quant-ph/​0108033.
arXiv:quant-ph/0108033

[85] J.-F. Cai, E. J. Candes, and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Opt. 20, 1956 (2010).
https:/​/​doi.org/​10.1137/​080738970

[86] S. Chatterjee, Matrix estimation by universal singular value thresholding, Ann. Statist. 43, 177 (2015), arXiv:1212.1247.
https:/​/​doi.org/​10.1214/​14-AOS1272
arXiv:1212.1247

[87] 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).
https:/​/​doi.org/​10.1561/​2200000016

[88] 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.
https:/​/​doi.org/​10.1038/​s41534-018-0080-4
arXiv:1711.02524

[89] V. Voroninski, Quantum tomography from few full-rank observables, arXiv:1309.7669.
arXiv:1309.7669

[90] 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.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​4/​043018
arXiv:1510.03229

[91] A. Acharya and M. Guta, Statistical analysis of compressive low rank tomography with random measurements, J. Phys. A 50, 195301 (2017), arXiv:1609.03758.
https:/​/​doi.org/​10.1088/​1751-8121/​aa682e
arXiv:1609.03758

[92] E. J. Candès, X. Li, and M. Soltanolkotabi, Phase retrieval from coded diffraction patterns, Appl. Comput. Harmon. Anal. 39, 277 (2015), arXiv:1310.3240.
https:/​/​doi.org/​10.1016/​j.acha.2014.09.004
arXiv:1310.3240

[93] 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.
https:/​/​doi.org/​10.1016/​j.acha.2015.05.004
arXiv:1402.6286

[94] 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.
https:/​/​doi.org/​10.1103/​PhysRevA.77.062112
arXiv:0802.3862

[95] 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.
arXiv:1011.3027

[96] S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing (Springer, 2013).

[97] Symmetric group:s4, http:/​/​groupprops.subwiki.org/​wiki/​Symmetric_group:S4 (2016a), accessed: 2016-08-17.
http:/​/​groupprops.subwiki.org/​wiki/​Symmetric_group:S4

[98] Linear representation theory of symmetric group:S4, http:/​/​groupprops.subwiki.org/​wiki/​Linear_representation_theory_of_symmetric_group:S4 (2016b), accessed: 2016-08-17.
http:/​/​groupprops.subwiki.org/​wiki/​Linear_representation_theory_of_symmetric_group:S4

[99] B. Simon, Representations of finite and compact groups, 10 (Am. Math. Soc., 1996).

Cited by

[1] 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).

[2] Jian-Feng Cai and Ke Wei, "Exploiting the structure effectively and efficiently in low rank matrix recovery", arXiv:1809.03652.

[3] Joel J. Wallman, "Randomized benchmarking with gate-dependent noise", arXiv:1703.09835.

[4] N. Milazzo, D. Braun, and O. Giraud, "Optimal measurement strategies for fast entanglement detection", Physical Review A 100 1, 012328 (2019).

[5] Thomas Strohmer and Ke Wei, "Painless Breakups -- Efficient Demixing of Low Rank Matrices", arXiv:1703.09848.

The above citations are from SAO/NASA ADS (last updated 2019-12-06 20:25:52). 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 2019-12-06 20:25:50).