Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise

Pedro Figueroa-Romero; Kavan Modi; Min-Hsiu Hsieh

doi:10.22331/q-2022-12-01-868

Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise

Pedro Figueroa-Romero¹, Kavan Modi^2,3, and Min-Hsiu Hsieh¹

¹Hon Hai Quantum Computing Research Center, Taipei, Taiwan
²School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia
³Centre for Quantum Technology, Transport for New South Wales, Sydney, NSW 2000, Australia

Published:	2022-12-01, volume 6, page 868
Eprint:	arXiv:2202.11338v4
Doi:	https://doi.org/10.22331/q-2022-12-01-868
Citation:	Quantum 6, 868 (2022).

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

Abstract

The rapid progress in the development of quantum devices is in large part due to the availability of a wide range of characterization techniques allowing to probe, test and adjust them. Nevertheless, these methods often make use of approximations that hold in rather simplistic circumstances. In particular, assuming that error mechanisms stay constant in time and have no dependence in the past, is something that will be impossible to do as quantum processors continue scaling up in depth and size. We establish a theoretical framework for the Randomized Benchmarking protocol encompassing temporally-correlated, so-called non-Markovian noise, at the gate level, for any gate set belonging to a wide class of finite groups. We obtain a general expression for the Average Sequence Fidelity (ASF) and propose a way to obtain average gate fidelities of full non-Markovian noise processes. Moreover, we obtain conditions that are fulfilled when an ASF displays authentic non-Markovian deviations. Finally, we show that even though gate-dependence does not translate into a perturbative term within the ASF, as in the Markovian case, the non-Markovian sequence fidelity nevertheless remains stable under small gate-dependent perturbations.

► BibTeX data

@article{FigueroaRomero2022towardsgeneral,
  doi = {10.22331/q-2022-12-01-868},
  url = {https://doi.org/10.22331/q-2022-12-01-868},
  title = {Towards a general framework of {R}andomized {B}enchmarking incorporating non-{M}arkovian {N}oise},
  author = {Figueroa-Romero, Pedro and Modi, Kavan and Hsieh, Min-Hsiu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {868},
  month = dec,
  year = {2022}
}

► References

[1] J. Emerson, R. Alicki, and K. Życzkowski, ``Scalable noise estimation with random unitary operators,'' J. Opt. B-Quantum S.O. 7, S347 (2005).
https://doi.org/10.1088/1464-4266/7/10/021

[2] B. Lévi, C. C. López, J. Emerson, and D. G. Cory, ``Efficient error characterization in quantum information processing,'' Phys. Rev. A 75, 022314 (2007).
https://doi.org/10.1103/PhysRevA.75.022314

[3] 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,'' Phys. Rev. A 77, 012307 (2008).
https://doi.org/10.1103/PhysRevA.77.012307

[4] E. Magesan, J. M. Gambetta, and J. Emerson, ``Scalable and robust randomized benchmarking of quantum processes,'' Phys. Rev. Lett. 106, 180504 (2011).
https://doi.org/10.1103/PhysRevLett.106.180504

[5] E. Magesan, J. M. Gambetta, and J. Emerson, ``Characterizing quantum gates via randomized benchmarking,'' Phys. Rev. A 85, 042311 (2012).
https://doi.org/10.1103/PhysRevA.85.042311

[6] I. L. Chuang and M. A. Nielsen, ``Prescription for experimental determination of the dynamics of a quantum black box,'' J. Mod. Optic 44, 2455–2467 (1997).
https://doi.org/10.1080/09500349708231894

[7] E. Nielsen, J. K. Gamble, K. Rudinger, T. Scholten, K. Young, and R. Blume-Kohout, ``Gate Set Tomography,'' Quantum 5, 557 (2021).
https://doi.org/10.22331/q-2021-10-05-557

[8] J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eisert, ``General framework for randomized benchmarking,'' PRX Quantum 3, 020357 (2022).
https://doi.org/10.1103/PRXQuantum.3.020357

[9] S. J. van Enk and R. Blume-Kohout, ``When quantum tomography goes wrong: drift of quantum sources and other errors,'' New J. Phys. 15, 025024 (2013).
https://doi.org/10.1088/1367-2630/15/2/025024

[10] M. A. Fogarty, M. Veldhorst, R. Harper, C. H. Yang, S. D. Bartlett, S. T. Flammia, and A. S. Dzurak, ``Nonexponential fidelity decay in randomized benchmarking with low-frequency noise,'' Phys. Rev. A 92, 022326 (2015).
https://doi.org/10.1103/PhysRevA.92.022326

[11] T. Proctor, M. Revelle, E. Nielsen, K. Rudinger, D. Lobser, P. Maunz, R. Blume-Kohout, and K. Young, ``Detecting and tracking drift in quantum information processors,'' Nat. Commun. 11, 5396 (2020).
https://doi.org/10.1038/s41467-020-19074-4

[12] J. M. Gambetta, A. D. Córcoles, S. T. Merkel, B. R. Johnson, J. A. Smolin, J. M. Chow, C. A. Ryan, C. Rigetti, S. Poletto, T. A. Ohki, M. B. Ketchen, and M. Steffen, ``Characterization of addressability by simultaneous randomized benchmarking,'' Phys. Rev. Lett. 109, 240504 (2012).
https://doi.org/10.1103/PhysRevLett.109.240504

[13] M. Sarovar, T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, ``Detecting crosstalk errors in quantum information processors,'' Quantum 4, 321 (2020).
https://doi.org/10.22331/q-2020-09-11-321

[14] P. Parrado-Rodríguez, C. Ryan-Anderson, A. Bermudez, and M. Müller, ``Crosstalk Suppression for Fault-tolerant Quantum Error Correction with Trapped Ions,'' Quantum 5, 487 (2021).
https://doi.org/10.22331/q-2021-06-29-487

[15] C. J. Wood and J. M. Gambetta, ``Quantification and characterization of leakage errors,'' Phys. Rev. A 97, 032306 (2018).
https://doi.org/10.1103/PhysRevA.97.032306

[16] J. J. Wallman, M. Barnhill, and J. Emerson, ``Robust characterization of leakage errors,'' New J. Phys. 18, 043021 (2016).
https://doi.org/10.1088/1367-2630/18/4/043021

[17] T. Chasseur and F. K. Wilhelm, ``Complete randomized benchmarking protocol accounting for leakage errors,'' Phys. Rev. A 92, 042333 (2015).
https://doi.org/10.1103/PhysRevA.92.042333

[18] K. Young, S. Bartlett, R. J. Blume-Kohout, J. K. Gamble, D. Lobser, P. Maunz, E. Nielsen, T. J. Proctor, M. Revelle, and K. M. Rudinger, Diagnosing and Destroying Non-Markovian Noise, Tech. Rep. (U.S. Department of Energy, Office of Scientific and Technical Information, 2020).
https://doi.org/10.2172/1671379

[19] C. A. Ryan, M. Laforest, and R. Laflamme, ``Randomized benchmarking of single- and multi-qubit control in liquid-state NMR quantum information processing,'' New J. Phys. 11, 013034 (2009).
https://doi.org/10.1088/1367-2630/11/1/013034

[20] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, ``Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,'' Nat. Phys. 7, 565 (2011).
https://doi.org/10.1038/nphys1994

[21] C. Müller, J. Lisenfeld, A. Shnirman, and S. Poletto, ``Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits,'' Phys. Rev. B 92, 035442 (2015).
https://doi.org/10.1103/PhysRevB.92.035442

[22] K. W. Chan, W. Huang, C. H. Yang, J. C. C. Hwang, B. Hensen, T. Tanttu, F. E. Hudson, K. M. Itoh, A. Laucht, A. Morello, and A. S. Dzurak, ``Assessment of a silicon quantum dot spin qubit environment via noise spectroscopy,'' Phys. Rev. Applied 10, 044017 (2018).
https://doi.org/10.1103/PhysRevApplied.10.044017

[23] S. M. Meißner, A. Seiler, J. Lisenfeld, A. V. Ustinov, and G. Weiss, ``Probing individual tunneling fluctuators with coherently controlled tunneling systems,'' Phys. Rev. B 97, 180505 (2018).
https://doi.org/10.1103/PhysRevB.97.180505

[24] J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, ``Decoherence benchmarking of superconducting qubits,'' npj Quantum Inf. 5 (2019), 10.1038/s41534-019-0168-5.
https://doi.org/10.1038/s41534-019-0168-5

[25] B. H. Fong and S. T. Merkel, ``Randomized benchmarking, correlated noise, and ising models,'' (2017), arXiv:1703.09747 [quant-ph].
arXiv:1703.09747

[26] 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 Inf. 4, 7 (2018).
https://doi.org/10.1038/s41534-017-0052-0

[27] H. Ball, T. M. Stace, S. T. Flammia, and M. J. Biercuk, ``Effect of noise correlations on randomized benchmarking,'' Phys. Rev. A 93, 022303 (2016).
https://doi.org/10.1103/PhysRevA.93.022303

[28] J. Qi and H. K. Ng, ``Randomized benchmarking in the presence of time-correlated dephasing noise,'' Phys. Rev. A 103, 022607 (2021).
https://doi.org/10.1103/PhysRevA.103.022607

[29] P. Figueroa-Romero, K. Modi, R. J. Harris, T. M. Stace, and M.-H. Hsieh, ``Randomized benchmarking for non-Markovian noise,'' PRX Quantum 2, 040351 (2021).
https://doi.org/10.1103/PRXQuantum.2.040351

[30] M. A. Graydon, J. Skanes-Norman, and J. J. Wallman, ``Clifford groups are not always 2-designs,'' (2021), arXiv:2108.04200 [quant-ph].
arXiv:2108.04200

[31] D. S. França and A. K. Hashagen, ``Approximate randomized benchmarking for finite groups,'' J. Phys. A: Math. Theor. 51, 395302 (2018).
https://doi.org/10.1088/1751-8121/aad6fa

[32] J. Helsen, X. Xue, L. M. K. Vandersypen, and S. Wehner, ``A new class of efficient randomized benchmarking protocols,'' npj Quantum Inf. 5, 71 (2019).
https://doi.org/10.1038/s41534-019-0182-7

[33] J. J. Wallman, ``Randomized benchmarking with gate-dependent noise,'' Quantum 2, 47 (2018).
https://doi.org/10.22331/q-2018-01-29-47

[34] T. Proctor, K. Rudinger, K. Young, M. Sarovar, and R. Blume-Kohout, ``What randomized benchmarking actually measures,'' Phys. Rev. Lett. 119, 130502 (2017).
https://doi.org/10.1103/PhysRevLett.119.130502

[35] A. W. Cross, E. Magesan, L. S. Bishop, J. A. Smolin, and J. M. Gambetta, ``Scalable randomised benchmarking of non-Clifford gates,'' npj Quantum Inf. 2 (2016).
https://doi.org/10.1038/npjqi.2016.12

[36] W. G. Brown and B. Eastin, ``Randomized benchmarking with restricted gate sets,'' Phys. Rev. A 97, 062323 (2018).
https://doi.org/10.1103/PhysRevA.97.062323

[37] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman, ``Real randomized benchmarking,'' Quantum 2, 85 (2018).
https://doi.org/10.22331/q-2018-08-22-85

[38] A. Carignan-Dugas, J. J. Wallman, and J. Emerson, ``Characterizing universal gate sets via dihedral benchmarking,'' Phys. Rev. A 92, 060302 (2015).
https://doi.org/10.1103/PhysRevA.92.060302

[39] 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

[40] S. T. Merkel, E. J. Pritchett, and B. H. Fong, ``Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors,'' Quantum 5, 581 (2021).
https://doi.org/10.22331/q-2021-11-16-581

[41] A. Carignan-Dugas, K. Boone, J. J. Wallman, and J. Emerson, ``From randomized benchmarking experiments to gate-set circuit fidelity: how to interpret randomized benchmarking decay parameters,'' New J. Phys. 20, 092001 (2018).
https://doi.org/10.1088/1367-2630/aadcc7

[42] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum circuit architecture,'' Phys. Rev. Lett. 101, 060401 (2008).
https://doi.org/10.1103/PhysRevLett.101.060401

[43] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Phys. Rev. A 80, 022339 (2009).
https://doi.org/10.1103/PhysRevA.80.022339

[44] C. Portmann, C. Matt, U. Maurer, R. Renner, and B. Tackmann, ``Causal boxes: Quantum information-processing systems closed under composition,'' IEEE Trans. Inf. Theory , 1–1 (2017).
https://doi.org/10.1109/tit.2017.2676805

[45] H. I. Nurdin and J. Gough, ``From the heisenberg to the schrödinger picture: Quantum stochastic processes and process tensors,'' 2021 60th IEEE Conference on Decision and Control (CDC) (2021), 10.1109/cdc45484.2021.9683765.
https://doi.org/10.1109/cdc45484.2021.9683765

[46] F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New J. Phys. 18, 063032 (2016).
https://doi.org/10.1088/1367-2630/18/6/063032

[47] D. Kretschmann and R. F. Werner, ``Quantum channels with memory,'' Phys. Rev. A 72, 062323 (2005).
https://doi.org/10.1103/PhysRevA.72.062323

[48] G. Gutoski and J. Watrous, ``Toward a general theory of quantum games,'' STOC '07, 565–574 (2007).
https://doi.org/10.1145/1250790.1250873

[49] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Non-Markovian quantum processes: Complete framework and efficient characterization,'' Phys. Rev. A 97, 012127 (2018a).
https://doi.org/10.1103/PhysRevA.97.012127

[50] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Operational Markov condition for quantum processes,'' Phys. Rev. Lett. 120, 040405 (2018b).
https://doi.org/10.1103/PhysRevLett.120.040405

[51] S. Milz, F. A. Pollock, and K. Modi, ``An introduction to operational quantum dynamics,'' Open Syst. Inf. Dyn. 24, 1740016 (2017).
https://doi.org/10.1142/s1230161217400169

[52] S. Milz, F. Sakuldee, F. A. Pollock, and K. Modi, ``Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories,'' Quantum 4, 255 (2020a).
https://doi.org/10.22331/q-2020-04-20-255

[53] P. Taranto, F. A. Pollock, and K. Modi, ``Non-Markovian memory strength bounds quantum process recoverability,'' npj Quantum Inf. 7 (2021), 10.1038/s41534-021-00481-4.
https://doi.org/10.1038/s41534-021-00481-4

[54] P. Figueroa-Romero, ``Equilibration and typicality in quantum processes,'' (2021), arXiv:2102.02289 [quant-ph].
arXiv:2102.02289

[55] S. Milz and K. Modi, ``Quantum stochastic processes and quantum non-Markovian phenomena,'' PRX Quantum 2, 030201 (2021).
https://doi.org/10.1103/PRXQuantum.2.030201

[56] S. Milz, M. S. Kim, F. A. Pollock, and K. Modi, ``Completely positive divisibility does not mean Markovianity,'' Phys. Rev. Lett. 123, 040401 (2019).
https://doi.org/10.1103/PhysRevLett.123.040401

[57] P. Figueroa-Romero, K. Modi, and F. A. Pollock, ``Almost Markovian processes from closed dynamics,'' Quantum 3, 136 (2019).
https://doi.org/10.22331/q-2019-04-30-136

[58] P. Figueroa-Romero, K. Modi, and F. A. Pollock, ``Equilibration on average in quantum processes with finite temporal resolution,'' Phys. Rev. E 102, 032144 (2020).
https://doi.org/10.1103/PhysRevE.102.032144

[59] S. Milz, C. Spee, Z.-P. Xu, F. A. Pollock, K. Modi, and O. Gühne, ``Genuine Multipartite Entanglement in Time,'' SciPost Phys. 10, 141 (2021).
https://doi.org/10.21468/SciPostPhys.10.6.141

[60] S. Milz, D. Egloff, P. Taranto, T. Theurer, M. B. Plenio, A. Smirne, and S. F. Huelga, ``When is a non-Markovian quantum process classical?'' Phys. Rev. X 10, 041049 (2020b).
https://doi.org/10.1103/PhysRevX.10.041049

[61] S. Milz, J. Bavaresco, and G. Chiribella, ``Resource theory of causal connection,'' Quantum 6, 788 (2022).
https://doi.org/10.22331/q-2022-08-25-788

[62] G. A. L. White, C. D. Hill, F. A. Pollock, L. C. L. Hollenberg, and K. Modi, ``Demonstration of non-Markovian process characterisation and control on a quantum processor,'' Nat. Commun. 11 (2020), 10.1038/s41467-020-20113-3.
https://doi.org/10.1038/s41467-020-20113-3

[63] C. Guo, K. Modi, and D. Poletti, ``Tensor-network-based machine learning of non-Markovian quantum processes,'' Phys. Rev. A 102, 062414 (2020).
https://doi.org/10.1103/PhysRevA.102.062414

[64] G. A. L. White, F. A. Pollock, L. C. L. Hollenberg, K. Modi, and C. D. Hill, ``Non-Markovian quantum process tomography,'' (2021), arXiv:2106.11722 [quant-ph].
arXiv:2106.11722

[65] G. D. Berk, S. Milz, F. A. Pollock, and K. Modi, ``Extracting quantum dynamical resources: Consumption of non-Markovianity for noise reduction,'' (2021), arXiv:2110.02613 [quant-ph].
arXiv:2110.02613

[66] D. Greenbaum, ``Introduction to quantum gate set tomography,'' (2015), arXiv:1509.02921 [quant-ph].
arXiv:1509.02921

[67] J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018).
https://doi.org/10.1017/9781316848142

[68] J. Claes, E. Rieffel, and Z. Wang, ``Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking,'' PRX Quantum 2, 010351 (2021).
https://doi.org/10.1103/PRXQuantum.2.010351

[69] P. Taranto, F. A. Pollock, S. Milz, M. Tomamichel, and K. Modi, ``Quantum Markov order,'' Phys. Rev. Lett. 122, 140401 (2019a).
https://doi.org/10.1103/PhysRevLett.122.140401

[70] P. Taranto, S. Milz, F. A. Pollock, and K. Modi, ``Structure of quantum stochastic processes with finite Markov order,'' Phys. Rev. A 99, 042108 (2019b).
https://doi.org/10.1103/PhysRevA.99.042108

[71] J. Helsen, M. Ioannou, I. Roth, J. Kitzinger, E. Onorati, A. H. Werner, and J. Eisert, ``Estimating gate-set properties from random sequences,'' (2021), arXiv:2110.13178 [quant-ph].
arXiv:2110.13178

[72] S. T. Flammia, ``Averaged circuit eigenvalue sampling,'' (2021), arXiv:2108.05803 [quant-ph].
arXiv:2108.05803

[73] T. Heinosaari, M. A. Jivulescu, and I. Nechita, ``Random positive operator valued measures,'' J. Math. Phys. 61, 042202 (2020).
https://doi.org/10.1063/1.5131028

[74] A. E. Rastegin, ``Relations for certain symmetric norms and anti-norms before and after partial trace,'' J. Stat. Phys. 148, 1040–1053 (2012).
https://doi.org/10.1007/s10955-012-0569-8

[75] D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai, ``Contractivity of positive and trace-preserving maps under lp norms,'' J. Math. Phys. 47, 083506 (2006).
https://doi.org/10.1063/1.2218675

[76] I. de Vega and D. Alonso, ``Dynamics of non-markovian open quantum systems,'' Rev. Mod. Phys. 89, 015001 (2017).
https://doi.org/10.1103/RevModPhys.89.015001

[77] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, ``Colloquium: Non-markovian dynamics in open quantum systems,'' Rev. Mod. Phys. 88, 021002 (2016).
https://doi.org/10.1103/RevModPhys.88.021002

[78] M. Tinkham, Group Theory and Quantum Mechanics, Dover Books on Chemistry and Earth Sciences (Dover Publications, 2003).
https://books.google.com.tw/books?id=r4GIU2wJCAEC

[79] W. Harris, W. Fulton, and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics (Springer New York, 1991).
https://books.google.com.tw/books?id=6GUH8ARxhp8C

[80] M. Horodecki and P. Horodecki, ``Reduction criterion of separability and limits for a class of protocols of entanglement distillation,'' (1997), arXiv:quant-ph/9708015 [quant-ph].
arXiv:quant-ph/9708015

[81] D. Chruściński and A. Kossakowski, ``Multipartite invariant states. I. Unitary symmetry,'' Phys. Rev. A 73, 062314 (2006).
https://doi.org/10.1103/PhysRevA.73.062314

Cited by

[1] J. Helsen, M. Ioannou, J. Kitzinger, E. Onorati, A. H. Werner, J. Eisert, and I. Roth, "Shadow estimation of gate-set properties from random sequences", Nature Communications 14 1, 5039 (2023).

[2] P Figueroa-Romero, M Papič, A Auer, M-H Hsieh, K Modi, and I de Vega, "Operational Markovianization in randomized benchmarking", Quantum Science and Technology 9 3, 035020 (2024).

[3] Philip Taranto, Thomas J. Elliott, and Simon Milz, "Hidden Quantum Memory: Is Memory There When Somebody Looks?", Quantum 7, 991 (2023).

[4] Tatsuki Odake, Philip Taranto, Nobuyuki Yoshioka, Toshinari Itoko, Kunal Sharma, Antonio Mezzacapo, and Mio Murao, "Robust Error Accumulation Suppression", arXiv:2401.16884, (2024).

[5] Markus Heinrich, Martin Kliesch, and Ingo Roth, "Randomized benchmarking with random quantum circuits", arXiv:2212.06181, (2022).

[6] Shih-Xian Yang, Pedro Figueroa-Romero, and Min-Hsiu Hsieh, "Machine Learning of Average Non-Markovianity from Randomized Benchmarking", arXiv:2207.01542, (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-05 22:30:05) and SAO/NASA ADS (last updated successfully 2024-05-05 22:30:06). 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.