Entanglement-Free Parameter Estimation of Generalized Pauli Channels

Junaid ur Rehman and Hyundong Shin

Department of Electronics and Information Convergence Engineering, Kyung Hee University, Korea

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

Abstract

We propose a parameter estimation protocol for generalized Pauli channels acting on $d$-dimensional Hilbert space. The salient features of the proposed method include product probe states and measurements, the number of measurement configurations linear in $d$, minimal post-processing, and the scaling of the mean square error comparable to that of the entanglement-based parameter estimation scheme for generalized Pauli channels. We also show that while measuring generalized Pauli operators the errors caused by the Pauli noise can be modeled as measurement errors. This makes it possible to utilize the measurement error mitigation framework to mitigate the errors caused by the generalized Pauli channels. We use this result to mitigate noise on the probe states and recover the scaling of the noiseless probes, except with a noise strength-dependent constant factor. This method of modeling Pauli channel as measurement noise can also be of independent interest in other NISQ tasks, e.g., state tomography problems, variational quantum algorithms, and other channel estimation problems where Pauli measurements have the central role.

► BibTeX data

► References

[1] Marco Chiani and Lorenzo Valentini. Design of short codes for quantum channels with asymmetric Pauli errors. In International Conference on Computational Science – ICCS 2020, volume 12142, pages 638–649, Cham, June 2020. Springer International Publishing. ISBN 978-3-030-50433-5. 10.1007/​978-3-030-50433-5_49.
https:/​/​doi.org/​10.1007/​978-3-030-50433-5_49

[2] Andrew S. Fletcher, Peter W. Shor, and Moe Z. Win. Channel-adapted quantum error correction for the amplitude damping channel. IEEE Trans. Inf. Theory, 54 (12): 5705–5718, December 2008. 10.1109/​TIT.2008.2006458.
https:/​/​doi.org/​10.1109/​TIT.2008.2006458

[3] Yixuan Xie, J. Li, R. Malaney, and J. Yuan. Channel identification and its impact on quantum LDPC code performance. In 2012 Australian Communications Theory Workshop (AusCTW), pages 140–144, January 2012. 10.1109/​AusCTW.2012.6164921.
https:/​/​doi.org/​10.1109/​AusCTW.2012.6164921

[4] Isaac L. Chuang and M. A. Nielsen. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt., 44 (11-12): 2455–2467, November 1997. 10.1080/​09500349708231894.
https:/​/​doi.org/​10.1080/​09500349708231894

[5] J. F. Poyatos, J. I. Cirac, and P. Zoller. Complete characterization of a quantum process: The two-bit quantum gate. Phys. Rev. Lett., 78 (2): 390–393, January 1997. 10.1103/​PhysRevLett.78.390.
https:/​/​doi.org/​10.1103/​PhysRevLett.78.390

[6] J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White. Quantum process tomography of a controlled-NOT gate. Phys. Rev. Lett., 93 (8): 4, August 2004. 10.1103/​PhysRevLett.93.080502.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.080502

[7] G. M. D'Ariano and P. Lo Presti. Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation. Phys. Rev. Lett., 86 (19): 4195–4198, May 2001. 10.1103/​PhysRevLett.86.4195.
https:/​/​doi.org/​10.1103/​PhysRevLett.86.4195

[8] M. Mohseni and D. A. Lidar. Direct characterization of quantum dynamics: General theory. Phys. Rev. A, 75 (6): 15, June 2007. 10.1103/​PhysRevA.75.062331.
https:/​/​doi.org/​10.1103/​PhysRevA.75.062331

[9] G. Chiribella, G. M. D'Ariano, and M. F. Sacchi. Optimal estimation of group transformations using entanglement. Phys. Rev. A, 72 (4): 10, October 2005. 10.1103/​PhysRevA.72.042338.
https:/​/​doi.org/​10.1103/​PhysRevA.72.042338

[10] Indrani Chattopadhyay and Debasis Sarkar. Distinguishing quantum operations: LOCC versus separable operators. Int. J. Quantum Inf., 14 (06): 1640028, August 2016. 10.1142/​S0219749916400281.
https:/​/​doi.org/​10.1142/​S0219749916400281

[11] M. Mohseni, A. T. Rezakhani, and D. A. Lidar. Quantum-process tomography: Resource analysis of different strategies. Phys. Rev. A, 77 (3): 15, March 2008. 10.1103/​PhysRevA.77.032322.
https:/​/​doi.org/​10.1103/​PhysRevA.77.032322

[12] Andrey V. Rodionov, Andrzej Veitia, R. Barends, J. Kelly, Daniel Sank, J. Wenner, John M. Martinis, Robert L. Kosut, and Alexander N. Korotkov. Compressed sensing quantum process tomography for superconducting quantum gates. Phys. Rev. B, 90 (14): 16, October 2014. 10.1103/​PhysRevB.90.144504.
https:/​/​doi.org/​10.1103/​PhysRevB.90.144504

[13] Michael Frey, David Collins, and Karl Gerlach. Probing the qudit depolarizing channel. J. Phys. A Math. Theor., 44 (20): 205306, April 2011. 10.1088/​1751-8113/​44/​20/​205306.
https:/​/​doi.org/​10.1088/​1751-8113/​44/​20/​205306

[14] Z. Ji, G. Wang, R. Duan, Y. Feng, and M. Ying. Parameter estimation of quantum channels. IEEE Trans. Inf. Theory, 54 (11): 5172–5185, November 2008. 10.1109/​TIT.2008.929940.
https:/​/​doi.org/​10.1109/​TIT.2008.929940

[15] Akio Fujiwara. Estimation of a generalized amplitude-damping channel. Phys. Rev. A, 70 (1): 8, July 2004. 10.1103/​PhysRevA.70.012317.
https:/​/​doi.org/​10.1103/​PhysRevA.70.012317

[16] H. Ohno and D. Petz. Generalizations of Pauli channels. Acta Math. Hung., 124 (1): 165–177, July 2009. 10.1007/​s10474-009-8171-5.
https:/​/​doi.org/​10.1007/​s10474-009-8171-5

[17] Katarzyna Siudzińska and Dariusz Chruściński. Quantum channels irreducibly covariant with respect to the finite group generated by the Weyl operators. J. Math. Phys., 59 (3): 033508, March 2018. 10.1063/​1.5013604.
https:/​/​doi.org/​10.1063/​1.5013604

[18] Zbigniew Puchała, Łukasz Rudnicki, and Karol Życzkowski. Pauli semigroups and unistochastic quantum channels. Phys. Lett. A, 383 (20): 2376–2381, July 2019. 10.1016/​j.physleta.2019.04.057.
https:/​/​doi.org/​10.1016/​j.physleta.2019.04.057

[19] E. Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39–44, March 2005. 10.1038/​nature03350.
https:/​/​doi.org/​10.1038/​nature03350

[20] Robin Harper, Steven T. Flammia, and Joel J. Wallman. Efficient learning of quantum noise. Nat. Phys., 16 (12): 1184–1188, December 2020. 10.1038/​s41567-020-0992-8.
https:/​/​doi.org/​10.1038/​s41567-020-0992-8

[21] Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme. Symmetrized characterization of noisy quantum processes. Science, 317 (5846): 1893–1896, September 2007. 10.1126/​science.1145699.
https:/​/​doi.org/​10.1126/​science.1145699

[22] Michael R. Geller and Zhongyuan Zhou. Efficient error models for fault-tolerant architectures and the Pauli twirling approximation. Phys. Rev. A, 88: 012314, July 2013. 10.1103/​PhysRevA.88.012314.
https:/​/​doi.org/​10.1103/​PhysRevA.88.012314

[23] Joel J. Wallman and Joseph Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Phys. Rev. A, 94 (5): 9, November 2016. 10.1103/​PhysRevA.94.052325.
https:/​/​doi.org/​10.1103/​PhysRevA.94.052325

[24] Zhenyu Cai, Xiaosi Xu, and Simon C. Benjamin. Mitigating coherent noise using Pauli conjugation. npj Quantum Inform., 6 (1): 17, February 2020. 10.1038/​s41534-019-0233-0.
https:/​/​doi.org/​10.1038/​s41534-019-0233-0

[25] Nilanjana Datta and Mary Beth Ruskai. Maximal output purity and capacity for asymmetric unital qudit channels. J. Phys. A Math. Gen., 38 (45): 9785, October 2005. 10.1088/​0305-4470/​38/​45/​005.
https:/​/​doi.org/​10.1088/​0305-4470/​38/​45/​005

[26] Akio Fujiwara and Hiroshi Imai. Quantum parameter estimation of a generalized Pauli channel. J. Phys. A Math. Gen., 36 (29): 8093, July 2003. 10.1088/​0305-4470/​36/​29/​314.
https:/​/​doi.org/​10.1088/​0305-4470/​36/​29/​314

[27] Masahito Hayashi. Quantum channel estimation and asymptotic bound. J. Phys. Conf. Ser, 233 (1): 012016, July 2010. 10.1088/​1742-6596/​233/​1/​012016.
https:/​/​doi.org/​10.1088/​1742-6596/​233/​1/​012016

[28] Masahito Hayashi. Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation. Commun. Math. Phys., 304 (3): 689–709, June 2011. 10.1007/​s00220-011-1239-4.
https:/​/​doi.org/​10.1007/​s00220-011-1239-4

[29] G. Ballo, K. M. Hangos, and D. Petz. Convex optimization-based parameter estimation and experiment design for Pauli channels. IEEE Trans. Autom. Control, 57 (8): 2056–2061, August 2012. 10.1109/​TAC.2012.2195835.
https:/​/​doi.org/​10.1109/​TAC.2012.2195835

[30] László Ruppert, Dániel Virosztek, and Katalin Hangos. Optimal parameter estimation of Pauli channels. J. Phys. A Math. Theor., 45 (26): 265305, June 2012. 10.1088/​1751-8113/​45/​26/​265305.
https:/​/​doi.org/​10.1088/​1751-8113/​45/​26/​265305

[31] Dániel Virosztek, László Ruppert, and Katalin Hangos. Pauli channel tomography with unknown channel directions. arXiv preprint arXiv:1304.4492, April 2013.
arXiv:1304.4492

[32] David Collins. Mixed-state Pauli-channel parameter estimation. Phys. Rev. A, 87 (3): 15, March 2013. 10.1103/​PhysRevA.87.032301.
https:/​/​doi.org/​10.1103/​PhysRevA.87.032301

[33] David Collins and Jaimie Stephens. Depolarizing-channel parameter estimation using noisy initial states. Phys. Rev. A, 92 (3): 14, September 2015. 10.1103/​PhysRevA.92.032324.
https:/​/​doi.org/​10.1103/​PhysRevA.92.032324

[34] Steven T. Flammia and Joel J. Wallman. Efficient estimation of Pauli channels. ACM Trans. Quant. Comput., 1 (1): 3, December 2020. 10.1145/​3408039.
https:/​/​doi.org/​10.1145/​3408039

[35] Robin Harper, Wenjun Yu, and Steven T. Flammia. Fast estimation of sparse quantum noise. PRX Quantum, 2: 010322, February 2021. 10.1103/​PRXQuantum.2.010322.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010322

[36] A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni. Experimental realization of optimal noise estimation for a general Pauli channel. Phys. Rev. Lett., 107 (25): 5, December 2011. 10.1103/​PhysRevLett.107.253602.
https:/​/​doi.org/​10.1103/​PhysRevLett.107.253602

[37] C. E. López, G. Romero, F. Lastra, E. Solano, and J. C. Retamal. Sudden birth versus sudden death of entanglement in multipartite systems. Phys. Rev. Lett., 101 (8): 080503, August 2008. 10.1103/​PhysRevLett.101.080503.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.080503

[38] Giacomo Sorelli, Nina Leonhard, Vyacheslav N Shatokhin, Claudia Reinlein, and Andreas Buchleitner. Entanglement protection of high-dimensional states by adaptive optics. New J. Phys., 21 (2): 023003, February 2019. 10.1088/​1367-2630/​ab006e.
https:/​/​doi.org/​10.1088/​1367-2630/​ab006e

[39] Junaid ur Rehman, Youngmin Jeong, Jeong San Kim, and Hyundong Shin. Holevo capacity of discrete Weyl channels. Sci. Rep., 8 (1): 17457, November 2018. 10.1038/​s41598-018-35777-7.
https:/​/​doi.org/​10.1038/​s41598-018-35777-7

[40] Junaid ur Rehman, Youngmin Jeong, and Hyundong Shin. Directly estimating the Holevo capacity of discrete Weyl channels. Phys. Rev. A, 99 (4): 8, April 2019. 10.1103/​PhysRevA.99.042312.
https:/​/​doi.org/​10.1103/​PhysRevA.99.042312

[41] Filip B. Maciejewski, Zoltán Zimborás, and Michał Oszmaniec. Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography. Quantum, 4: 257, April 2020. 10.22331/​q-2020-04-24-257.
https:/​/​doi.org/​10.22331/​q-2020-04-24-257

[42] R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro. Qudit quantum-state tomography. Phys. Rev. A, 66 (1): 6, July 2002. 10.1103/​PhysRevA.66.012303.
https:/​/​doi.org/​10.1103/​PhysRevA.66.012303

[43] David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. Quantum state tomography via compressed sensing. Phys. Rev. Lett., 105 (15): 4, October 2010. 10.1103/​PhysRevLett.105.150401.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.150401

[44] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun., 5 (1): 4213, July 2014. 10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213

[45] Steven T. Flammia and Yi-Kai Liu. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett., 106 (23): 4, June 2011. 10.1103/​PhysRevLett.106.230501.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.230501

[46] H. Shin and M. Z. Win. MIMO diversity in the presence of double scattering. IEEE Trans. Inf. Theory, 54 (7): 2976–2996, July 2008. 10.1109/​TIT.2008.924672.
https:/​/​doi.org/​10.1109/​TIT.2008.924672

[47] Katarzyna Siudzińska. Classical capacity of generalized Pauli channels. J. Phys. A Math. Theor., 53 (44): 445301, October 2020. 10.1088/​1751-8121/​abb276.
https:/​/​doi.org/​10.1088/​1751-8121/​abb276

[48] A Yu Kitaev. Quantum computations: Algorithms and error correction. Russ. Math. Surv., 52 (6): 1191–1249, December 1997. 10.1070/​rm1997v052n06abeh002155.
https:/​/​doi.org/​10.1070/​rm1997v052n06abeh002155

[49] Giuliano Benenti and Giuliano Strini. Computing the distance between quantum channels: Usefulness of the Fano representation. Journal of Physics B: Atomic Molecular and Optical Physics, 43 (21): 215508, October 2010. 10.1088/​0953-4075/​43/​21/​215508.
https:/​/​doi.org/​10.1088/​0953-4075/​43/​21/​215508

[50] John Watrous. Semidefinite programs for completely bounded norms. Theory of Computing, 5 (11): 217–238, November 2009. 10.4086/​toc.2009.v005a011.
https:/​/​doi.org/​10.4086/​toc.2009.v005a011

[51] Avraham Ben-Aroya and Amnon Ta-Shma. On the complexity of approximating the diamond norm. Quantum Info. Comput., 10 (1): 77–86, January 2010. 10.5555/​2011438.2011444.
https:/​/​doi.org/​10.5555/​2011438.2011444

[52] Massimiliano F. Sacchi. Optimal discrimination of quantum operations. Phys. Rev. A, 71 (6): 4, June 2005. 10.1103/​PhysRevA.71.062340.
https:/​/​doi.org/​10.1103/​PhysRevA.71.062340

[53] Yanjun Han, Jiantao Jiao, and Tsachy Weissman. Minimax estimation of discrete distributions under $\ell _{1}$ loss. IEEE Trans. Inf. Theory, 61: 6343–6354, November 2015. 10.1109/​TIT.2015.2478816.
https:/​/​doi.org/​10.1109/​TIT.2015.2478816

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2021-08-04 16:30:50). On SAO/NASA ADS no data on citing works was found (last attempt 2021-08-04 16:30:51).