A polar decomposition for quantum channels (with applications to bounding error propagation in quantum circuits)

Arnaud Carignan-Dugas1, Matthew Alexander1, and Joseph Emerson1,2

1Institute for Quantum Computing and the Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Inevitably, assessing the overall performance of a quantum computer must rely on characterizing some of its elementary constituents and, from this information, formulate a broader statement concerning more complex constructions thereof. However, given the vastitude of possible quantum errors as well as their coherent nature, accurately inferring the quality of composite operations is generally difficult. To navigate through this jumble, we introduce a non-physical simplification of quantum maps that we refer to as the leading Kraus (LK) approximation. The uncluttered parameterization of LK approximated maps naturally suggests the introduction of a unitary-decoherent polar factorization for quantum channels in any dimension. We then leverage this structural dichotomy to bound the evolution -- as circuits grow in depth -- of two of the most experimentally relevant figures of merit, namely the average process fidelity and the unitarity. We demonstrate that the leeway in the behavior of the process fidelity is essentially taken into account by physical unitary operations.

► BibTeX data

► References

[1] R. Barends, J. Kelly, A. Veitia, A. Megrant, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, E. Jeffrey, C. Neill, P. J. J. O'Malley, J. Mutus, C. Quintana, P. Roushan, D. Sank, J. Wenner, T. C. White, A. N. Korotkov, A. N. Cleland, and John M. Martinis. Rolling quantum dice with a superconducting qubit. Phys. Rev. A, 90: 030303, Sep 2014. 10.1103/​PhysRevA.90.030303. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.030303.

[2] Ingemar Bengtsson and Karol Zyczkowski. Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, 2006. ISBN 0521814510.

[3] F. Bloch. Nuclear induction. Phys. Rev., 70: 460–474, Oct 1946. 10.1103/​PhysRev.70.460. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRev.70.460.

[4] Robin Blume-Kohout, Hui Khoon Ng, David Poulin, and Lorenza Viola. Information-preserving structures: A general framework for quantum zero-error information. Physical Review A, 82 (6): 062306, Dec 2010. 10.1103/​PhysRevA.82.062306.

[5] P. S. Bourdon and H. T. Williams. Unital quantum operations on the bloch ball and bloch region. Phys. Rev. A, 69: 022314, Feb 2004. 10.1103/​PhysRevA.69.022314. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.69.022314.

[6] Winton G. Brown and Bryan Eastin. Randomized benchmarking with restricted gate sets. Physical Review A, 97: 062323, June 2018. 10.1103/​PhysRevA.97.062323.

[7] Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson. Characterizing universal gate sets via dihedral benchmarking. Physical Review A, 92: 060302, December 2015. 10.1103/​PhysRevA.92.060302.

[8] Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson. Bounding the average gate fidelity of composite channels using the unitarity. arXiv e-prints, art. arXiv:1610.05296, October 2016.

[9] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10 (3): 285 – 290, 1975. ISSN 0024-3795. https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​0024379575900750.

[10] Joshua Combes, Christopher Granade, Christopher Ferrie, and Steven T. Flammia. Logical Randomized Benchmarking. arXiv e-prints, art. arXiv:1702.03688, February 2017.

[11] Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin, and Jay M. Gambetta. Scalable randomised benchmarking of non-Clifford gates. npj Quantum Information, 2: 16012, April 2016. 10.1038/​npjqi.2016.12.

[12] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Physical Review A, 80 (1): 012304, July 2009. ISSN 1050-2947. 10.1103/​PhysRevA.80.012304. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.80.012304.

[13] Joseph Emerson, Robert Alicki, and Karol Życzkowski. Scalable noise estimation with random unitary operators. Journal of Optics B: Quantum and Semiclassical Optics, 7 (10): S347–S352, October 2005. ISSN 1464-4266. 10.1088/​1464-4266/​7/​10/​021. URL http:/​/​arxiv.org/​abs/​quant-ph/​0503243.

[14] R.R. Ernst, G. Bodenhausen, and A. Wokaun. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. International series of monographs on chemistry. Clarendon Press, 1987. ISBN 9780198556299. URL https:/​/​books.google.ca/​books?id=XndTnwEACAAJ.

[15] G. Feng, J. J. Wallman, B. Buonacorsi, F. H. Cho, D. K. Park, T. Xin, D. Lu, J. Baugh, and R. Laflamme. Estimating the Coherence of Noise in Quantum Control of a Solid-State Qubit. Physical Review Letters, 117 (26): 260501, December 2016. 10.1103/​PhysRevLett.117.260501.

[16] D. S. França and A. K. Hashagen. Approximate randomized benchmarking for finite groups. Journal of Physics A Mathematical General, 51 (39): 395302, Sep 2018. 10.1088/​1751-8121/​aad6fa.

[17] Akio Fujiwara and Paul Algoet. One-to-one parametrization of quantum channels. Phys. Rev. A, 59: 3290–3294, May 1999. 10.1103/​PhysRevA.59.3290. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.59.3290.

[18] J. P. Gaebler, A. M. Meier, T. R. Tan, R. Bowler, Y. Lin, D. Hanneke, J. D. Jost, J. P. Home, E. Knill, D. Leibfried, and D. J. Wineland. Randomized benchmarking of multiqubit gates. Phys. Rev. Lett., 108: 260503, Jun 2012. 10.1103/​PhysRevLett.108.260503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.108.260503.

[19] Jay M. Gambetta, A. D. Córcoles, S. T. Merkel, B. R. Johnson, John A. Smolin, Jerry M. Chow, Colm A. Ryan, Chad Rigetti, S. Poletto, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. Characterization of Addressability by Simultaneous Randomized Benchmarking. Phys. Rev. Lett., 109: 240504, Dec 2012. 10.1103/​PhysRevLett.109.240504.

[20] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of n‐level systems. Journal of Mathematical Physics, 17 (5): 821–825, 1976. 10.1063/​1.522979. URL https:/​/​aip.scitation.org/​doi/​abs/​10.1063/​1.522979.

[21] Christopher Granade, Christopher Ferrie, and D G Cory. Accelerated Randomized Benchmarking. New Journal of Physics, 17 (1): 1–6, January 2014. ISSN 13672630. 10.1088/​1367-2630/​17/​1/​013042. URL http:/​/​arxiv.org/​abs/​1404.5275.

[22] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman. Real Randomized Benchmarking. Quantum, 2: 85, August 2018. ISSN 2521-327X. 10.22331/​q-2018-08-22-85. URL https:/​/​doi.org/​10.22331/​q-2018-08-22-85.

[23] Timothy F. Havel. Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups. Journal of Mathematical Physics, 44: 534–557, February 2003. 10.1063/​1.1518555.

[24] Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, and Stephanie Wehner. A new class of efficient randomized benchmarking protocols. arXiv e-prints, art. arXiv:1806.02048, June 2018.

[25] 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 (1): 012307, January 2008. ISSN 1050-2947. 10.1103/​PhysRevA.77.012307. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.77.012307.

[26] K. Kraus, A. Böhm, J.D. Dollard, and W.H. Wootters. States, effects, and operations: fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin. Lecture notes in physics. Springer-Verlag, 1983. ISBN 9780387127323. URL https:/​/​books.google.ca/​books?id=fRBBAQAAIAAJ.

[27] G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48: 119–130, June 1976. 10.1007/​BF01608499.

[28] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson. Scalable and Robust Randomized Benchmarking of Quantum Processes. Physical Review Letters, 106 (18): 180504, May 2011. ISSN 0031-9007. 10.1103/​PhysRevLett.106.180504. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.106.180504.

[29] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson. Characterizing quantum gates via randomized benchmarking. Physical Review A, 85 (4): 042311, April 2012a. ISSN 1050-2947. 10.1103/​PhysRevA.85.042311. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.85.042311.

[30] Easwar Magesan, Jay M. Gambetta, B. R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking. Physical Review Letters, 109 (8): 080505, August 2012b. ISSN 0031-9007. 10.1103/​PhysRevLett.109.080505. URL http:/​/​arxiv.org/​abs/​1203.4550.

[31] Michael A. Nielsen. A simple formula for the average gate fidelity of a quantum dynamical operation. Physics Letters A, 303 (4): 249, October 2002. ISSN 03759601. 10.1016/​S0375-9601(02)01272-0. URL http:/​/​linkinghub.elsevier.com/​retrieve/​pii/​S0375960102012720.

[32] D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai. Contractivity of positive and trace-preserving maps under L$_{p}$ norms. Journal of Mathematical Physics, 47 (8): 083506–083506, August 2006. 10.1063/​1.2218675.

[33] Timothy J. Proctor, Arnaud Carignan-Dugas, Kenneth Rudinger, Erik Nielsen, Robin Blume-Kohout, and Kevin Young. Direct randomized benchmarking for multi-qubit devices. arXiv e-prints, art. arXiv:1807.07975, July 2018. 10.1103/​PhysRevLett.123.030503.

[34] Mary Beth Ruskai, Stanislaw Szarek, and Elisabeth Werner. An analysis of completely-positive trace-preserving maps on m2. Linear Algebra and its Applications, 347 (1): 159 – 187, 2002. ISSN 0024-3795. https:/​/​doi.org/​10.1016/​S0024-3795(01)00547-X. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S002437950100547X.

[35] Sarah Sheldon, Lev S. Bishop, Easwar Magesan, Stefan Filipp, Jerry M. Chow, and Jay M. Gambetta. Characterizing errors on qubit operations via iterative randomized benchmarking. Phys. Rev. A, 93: 012301, Jan 2016. 10.1103/​PhysRevA.93.012301. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.93.012301.

[36] Joel J. Wallman, Marie Barnhill, and Joseph Emerson. Robust Characterization of Loss Rates. Phys. Rev. Lett., 115 (6): 060501, 2015a. ISSN 0031-9007. 10.1103/​PhysRevLett.115.060501. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.060501.

[37] Joel J. Wallman, Christopher Granade, Robin Harper, and Steven T. Flammia. Estimating the Coherence of Noise. New J. Phys., 17: 113020, 2015b. ISSN 1367-2630. 10.1088/​1367-2630/​17/​11/​113020. URL http:/​/​dx.doi.org/​10.1088/​1367-2630/​17/​11/​113020.

[38] Joel J Wallman, Marie Barnhill, and Joseph Emerson. Robust characterization of leakage errors. New Journal of Physics, 18 (4): 043021, apr 2016. 10.1088/​1367-2630/​18/​4/​043021. URL https:/​/​doi.org/​10.1088.

[39] Bo-Ying Wang and Ming-Peng Gong. Some eigenvalue inequalities for positive semidefinite matrix power products. Linear Algebra and its Applications, 184: 249 – 260, 1993. ISSN 0024-3795. https:/​/​doi.org/​10.1016/​0024-3795(93)90382-X. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​002437959390382X.

[40] C. H. Yang, K. W. Chan, R. Harper, W. Huang, T. Evans, J. C. C. Hwang, B. Hensen, A. Laucht, T. Tanttu, F. E. Hudson, S. T. Flammia, K. M. Itoh, A. Morello, S. D. Bartlett, and A. S. Dzurak. Silicon qubit fidelities approaching incoherent noise limits via pulse optimisation. Nature Electronics, April 2019. 10.1038/​s41928-019-0234-1.

Cited by

[1] J. Helsen, I. Roth, E. Onorati, A.H. Werner, and J. Eisert, "General Framework for Randomized Benchmarking", PRX Quantum 3 2, 020357 (2022).

[2] Alexander Hahn, Daniel Burgarth, and Kazuya Yuasa, "Unification of random dynamical decoupling and the quantum Zeno effect", New Journal of Physics 24 6, 063027 (2022).

[3] Yifeng Xiong, Daryus Chandra, Soon Xin Ng, and Lajos Hanzo, "Sampling Overhead Analysis of Quantum Error Mitigation: Uncoded vs. Coded Systems", IEEE Access 8, 228967 (2020).

[4] Jonas Helsen, "Quantum channels look simpler if you squint", Quantum Views 3, 22 (2019).

[5] Anthony M Polloreno and Kevin C Young, "Robustly decorrelating errors with mixed quantum gates", Quantum Science and Technology 7 2, 025004 (2022).

[6] Daniel Gottesman, "Maximally Sensitive Sets of States", arXiv:1907.05950, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2023-02-03 20:01:45) and SAO/NASA ADS (last updated successfully 2023-02-03 20:01:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “A polar decomposition for quantum channels (with applications to bounding error propagation in quantum circuits)

  1. Pingback: Perspective in Quantum Views by Jonas Helsen "Quantum channels look simpler if you squint"