Gauge invariant information concerning quantum channels
1Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77, D-50937, Cologne, Germany
2Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
3Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ulica Bałtycka 5, 44-100 Gliwice, Poland
4Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
Published: | 2018-04-11, volume 2, page 60 |
Eprint: | arXiv:1707.06926v2 |
Doi: | https://doi.org/10.22331/q-2018-04-11-60 |
Citation: | Quantum 2, 60 (2018). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.

Featured image: Exemplary ranges of eigenvalues for qubit quantum channels. Within the Gate Set Tomography, due to its gauge symmetry, the only meaningful characterization of a channel is provided by
the eigenvalues.
► BibTeX data
► References
[1] R. Blume-Kohout, et al., Robust, self-consistent,closed-form tomography of quantum logic gates on a trapped ion qubit, arXiv:1310.4492 (2013).
arXiv:1310.4492
[2] D. Kim, et al., Microwave-driven coherent operation of a semiconductor quantum dot charge qubit, Nat. Nanotechnology 10, 243 (2015).
https://doi.org/10.1038/nnano.2014.336
[3] D. Greenbaum, Introduction to Quantum Gate Set Tomography, arXiv:1509.02921 (2015).
arXiv:1509.02921
[4] 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 (1997).
https://doi.org/10.1103/PhysRevLett.78.390
[5] S. T. Merkel, et al., Self-consistent quantum process tomography, Phys. Rev. A 87, 062119 (2013).
https://doi.org/10.1103/PhysRevA.87.062119
[6] 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
[7] R. Blume-Kohout, et al., Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nat. Commun. 8, 14485 (2017).
[8] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
[9] A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and quantum computation, Vol. 47 (American Mathematical Society, 2002).
https://doi.org/10.1090/gsm/047
[10] J. Emerson, R. Alicki, K. Życzkowski, Scalable Noise Estimation with Random Unitary Operators, J. Opt. B: Quantum Semiclass. Opt. 7, S347 (2005).
https://doi.org/10.1088/1464-4266/7/10/021
[11] J. Emerson, et al., Symmetrized Characterization of Noisy Quantum Processes, Science 317, 1893 (2007).
https://doi.org/10.1126/science.1145699
[12] E. Knill, et al., Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
https://doi.org/10.1103/PhysRevA.77.012307
[13] J. J. Wallman and S. T. Flammia, Randomized benchmarking with confidence, New J. Phys. 16, 103032 (2014).
https://doi.org/10.1088/1367-2630/16/10/103032
[14] K. Życzkowski and I. Bengtsson, On duality between quantum maps and quantum states, Open Sys. & Information Dyn. 11, 3 (2004).
https://doi.org/10.1023/B:OPSY.0000024753.05661.c2
[15] W. Bruzda, V. Cappellini, H.-J. Sommers, and K. Życzkowski, Random Quantum Operations, Phys. Lett. A 373, 320 (2009).
https://doi.org/10.1016/j.physleta.2008.11.043
[16] D. Evans, R. Hoegh-Krohn, Spectral properties of positive maps on $C_*$–algebras, J. London Math. Soc. 17, 345 (1978).
https://doi.org/10.1112/jlms/s2-17.2.345
[17] A. Jamiołkowski, On applications of PI-algebras in the analysis of quantum channels, Int. J. Quantum Inf. 10, 1241007 (2012).
https://doi.org/10.1142/S0219749912410079
[18] D. R. Farenick, Irreducible positive linear maps on operator algebras, Proc. AMS 124, 3381, (1996).
https://doi.org/10.1090/S0002-9939-96-03441-7
[19] U. Groh, The peripheral point spectrum of Schwarz operators on $C_*$-algebras, Math. Z. 176, 311 (1981).
https://doi.org/10.1007/BF01214608
[20] M. M. Wolf and J. I. Cirac, Dividing Quantum Channels, Commun. Math. Phys. 279, 147 (2008).
https://doi.org/10.1007/s00220-008-0411-y
[21] M. M. Wolf and D. Perez-Garcia, The inverse eigenvalue problems for quantum channels, preprint arXiv:1005.4545, (2010).
arXiv:1005.4545
[22] M. Rahaman, Multiplicative Properties of Quantum Channels, J. Phys. A: Math. Theor. 50, 345302 (2017).
https://doi.org/10.1088/1751-8121/aa7b57
[23] M. Sanz, D. Perez-Garcia, M. M. Wolf, and J. I. Cirac, A quantum version of Wielandt's inequality, IEEE Trans. Inf. Theory 56, 4668 (2010).
https://doi.org/10.1109/TIT.2010.2054552
[24] W. Bruzda, M. Smaczyński, V. Cappellini, H.-J. Sommers, K. Życzkowski, Universality of spectra for interacting quantum chaotic systems, Phys. Rev. E 81, 066209 (2010).
https://doi.org/10.1103/PhysRevE.81.066209
[25] I. Bengtsson and K. Życzkowski, Geometry of Quantum States, II edition, Cambridge University Press, Cambridge, 2017.
[26] A. Fujiwara and P. Algoet, One-to-one parametrization of quantum channels, Phys. Rev. A 59, 3290 (1999).
https://doi.org/10.1103/PhysRevA.59.3290
[27] M. B. Ruskai, S. Szarek, and E. Werner, An analysis of completely positive trace-preserving maps on $\mathscr{M}_{2}$, Linear Algebra Appl. 347, 159 (2002).
https://doi.org/10.1016/S0024-3795(01)00547-X
[28] D. Braun, O. Giraud, I. Nechita, C. Pellegrini, and M. Znidaric, A universal set of qubit quantum channels, J. Phys. A 47, 135302 (2014).
https://doi.org/10.1088/1751-8113/47/13/135302
[29] J. Watrous, Simpler semidefinite programs for completely bounded norms, Chicago J. of Th. Comp. Sci. 2013, 8 (2013).
http://cjtcs.cs.uchicago.edu/articles/2013/8/contents.html
[30] I. Nechita, Z. Puchała, Ł. Pawela, and K. Życzkowski Almost all quantum channels are equidistant, arXiv:1612.00401 (2016).
arXiv:1612.00401
[31] M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002).
https://doi.org/10.1016/S0375-9601(02)01272-0
[32] M. D. Bowdrey, D. K. L. Oi, A. J. Short, K. Banaszek, and J. A. Jones, Fidelity of single qubit maps, Phys. Lett. A. 294, 258 (2002).
https://doi.org/10.1016/S0375-9601(02)00069-5
[33] M. Horodecki, P. Horodecki, and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A 60, 1888 (1999).
https://doi.org/10.1103/PhysRevA.60.1888
[34] E. Magesan, R. Blume-Kohout, and J. Emerson, Gate fidelity fluctuations and quantum process invariants, Phys. Rev. A 84, 012309 (2011).
https://doi.org/10.1103/PhysRevA.84.012309
[35] L. H. Pedersen, N. M. Møller, and K. Mølmer, The distribution of quantum fidelities, Phys. Lett. A 372, 7028 (2011).
https://doi.org/10.1016/j.physleta.2008.10.034
[36] N. Johnston and D. W. Kribs, Quantum gate fidelity in terms of Choi matrices, J. Phys. A 44, 495303 (2011).
https://doi.org/10.1088/1751-8113/44/49/495303
[37] J. Wallman, C. Granade, R. Harper, and S. T. Flammia, Estimating the Coherence of Noise, New J. Phys. 17, 113020 (2015).
https://doi.org/10.1088/1367-2630/17/11/113020
[38] 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
[39] J. Wallman, Bounding experimental quantum error rates relative to fault-tolerant thresholds, arXiv:1511.00727v2 (2016).
arXiv:1511.00727
Cited by
[1] H. R. Sumathi and C. Vidya Raj, "Quantum Multipartite correlation in Optical channel", International Journal of Scientific Research in Computer Science, Engineering and Information Technology 168 (2022).
[2] Junan Lin, Brandon Buonacorsi, Raymond Laflamme, and Joel J Wallman, "On the freedom in representing quantum operations", New Journal of Physics 21 2, 023006 (2019).
[3] Zbigniew Puchała, Łukasz Rudnicki, and Karol Życzkowski, "Pauli semigroups and unistochastic quantum channels", Physics Letters A 383 20, 2376 (2019).
[4] Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, and Robin Blume-Kohout, "Gate Set Tomography", Quantum 5, 557 (2021).
[5] Olivia Di Matteo, John Gamble, Chris Granade, Kenneth Rudinger, and Nathan Wiebe, "Operational, gauge-free quantum tomography", Quantum 4, 364 (2020).
[6] S. Mavadia, C. L. Edmunds, C. Hempel, H. Ball, F. Roy, T. M. Stace, and M. J. Biercuk, "Experimental quantum verification in the presence of temporally correlated noise", npj Quantum Information 4 1, 7 (2018).
[7] Kenneth Rudinger, Craig W. Hogle, Ravi K. Naik, Akel Hashim, Daniel Lobser, David I. Santiago, Matthew D. Grace, Erik Nielsen, Timothy Proctor, Stefan Seritan, Susan M. Clark, Robin Blume-Kohout, Irfan Siddiqi, and Kevin C. Young, "Experimental Characterization of Crosstalk Errors with Simultaneous Gate Set Tomography", PRX Quantum 2 4, 040338 (2021).
[8] Marco Cattaneo, Matteo A.C. Rossi, Guillermo García-Pérez, Roberta Zambrini, and Sabrina Maniscalco, "Quantum Simulation of Dissipative Collective Effects on Noisy Quantum Computers", PRX Quantum 4 1, 010324 (2023).
[9] David Davalos, Mario Ziman, and Carlos Pineda, "Divisibility of qubit channels and dynamical maps", Quantum 3, 144 (2019).
[10] Yu Luo, Yongming Li, and Zhengjun Xi, "Coherence-breaking superchannels", Quantum Information Processing 21 5, 176 (2022).
[11] B I Bantysh and Yu I Bogdanov, "Quantum tomography of noisy ion-based qudits", Laser Physics Letters 18 1, 015203 (2021).
[12] Ruyu Yang and Ying Li, "Perturbative tomography of small errors in quantum gates", Physical Review A 103 3, 032421 (2021).
[13] Jonas Helsen, Francesco Battistel, and Barbara M. Terhal, "Spectral quantum tomography", npj Quantum Information 5 1, 74 (2019).
[14] Raphael Brieger, Ingo Roth, and Martin Kliesch, "Compressive Gate Set Tomography", PRX Quantum 4 1, 010325 (2023).
[15] Ahmed Abid Moueddene, Nader Khammassi, Sebastian Feld, and Said Hamdioui, "A context-aware gate set tomography characterization of superconducting qubits", arXiv:2103.09922, (2021).
[16] Andrzej Veitia, Marcus P. da Silva, Robin Blume-Kohout, and Steven J. van Enk, "Macroscopic instructions vs microscopic operations in quantum circuits", arXiv:1708.08173, (2017).
[17] A. Veitia, M. P. da Silva, R. Blume-Kohout, and S. J. van Enk, "Macroscopic instructions vs microscopic operations in quantum circuits", Physics Letters A 384, 126131 (2020).
[18] John Gamble, Chris Granade, and Nathan Wiebe, "Bayesian ACRONYM Tuning", arXiv:1902.05940, (2019).
The above citations are from Crossref's cited-by service (last updated successfully 2023-09-28 01:41:20) and SAO/NASA ADS (last updated successfully 2023-09-28 01:41:20). 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.