Physical Implementability of Linear Maps and Its Application in Error Mitigation

Jiaqing Jiang; Kun Wang; Xin Wang

doi:10.22331/q-2021-12-07-600

Physical Implementability of Linear Maps and Its Application in Error Mitigation

Jiaqing Jiang^1,2, Kun Wang¹, and Xin Wang¹

¹Institute for Quantum Computing, Baidu Research, Beijing 100193, China
²Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA USA

Published:	2021-12-07, volume 5, page 600
Eprint:	arXiv:2012.10959v2
Doi:	https://doi.org/10.22331/q-2021-12-07-600
Citation:	Quantum 5, 600 (2021).

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Abstract

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the $\textit{physical implementability}$ measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

Featured image: A linear map that is not physically implementable can be effectively realized or simulated by a combination of two physically implementable linear maps (i.e., quantum operations) with appropriate coefficients.

► BibTeX data

@article{Jiang2021physical,
  doi = {10.22331/q-2021-12-07-600},
  url = {https://doi.org/10.22331/q-2021-12-07-600},
  title = {Physical {I}mplementability of {L}inear {M}aps and {I}ts {A}pplication in {E}rror {M}itigation},
  author = {Jiang, Jiaqing and Wang, Kun and Wang, Xin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {600},
  month = dec,
  year = {2021}
}

► References

[1] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2011. 10.37686/qrl.v1i1.57.
https://doi.org/10.37686/qrl.v1i1.57

[2] Karl Kraus. States, effects, and Operations. Springer-Verlag, Berlin, 1983.

[3] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Separability of mixed states: necessary and sufficient conditions. Physics Letters A, 223 (1): 1 – 8, 1996. ISSN 0375-9601. 10.1016/S0375-9601(96)00706-2.
https://doi.org/10.1016/S0375-9601(96)00706-2

[4] Philip Pechukas. Reduced dynamics need not be completely positive. Physical Review Letters, 73 (8): 1060, 1994. 10.1103/PhysRevLett.73.1060.
https://doi.org/10.1103/PhysRevLett.73.1060

[5] Hilary A Carteret, Daniel R Terno, and Karol Życzkowski. Dynamics beyond completely positive maps: Some properties and applications. Physical Review A, 77 (4): 042113, 2008. 10.1103/PhysRevA.77.042113.
https://doi.org/10.1103/PhysRevA.77.042113

[6] Paweł Horodecki. From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation. Physical Review A, 68 (5): 052101, 2003. 10.1103/physreva.68.052101.
https://doi.org/10.1103/physreva.68.052101

[7] Paweł Horodecki and Artur Ekert. Method for direct detection of quantum entanglement. Physical Review Letters, 89 (12): 127902, 2002. 10.1103/physrevlett.89.127902.
https://doi.org/10.1103/physrevlett.89.127902

[8] Jaromír Fiurášek. Structural physical approximations of unphysical maps and generalized quantum measurements. Physical Review A, 66 (5): 052315, 2002. 10.1103/physreva.66.052315.
https://doi.org/10.1103/physreva.66.052315

[9] JK Korbicz, ML Almeida, Joonwoo Bae, M Lewenstein, and A Acin. Structural approximations to positive maps and entanglement-breaking channels. Physical Review A, 78 (6): 062105, 2008. 10.1103/physreva.78.062105.
https://doi.org/10.1103/physreva.78.062105

[10] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. Volume of the set of separable states. Physical Review A, 58 (2): 883, 1998. 10.1103/physreva.58.883.
https://doi.org/10.1103/physreva.58.883

[11] Leonid Gurvits and Howard Barnum. Largest separable balls around the maximally mixed bipartite quantum state. Physical Review A, 66 (6): 062311, 2002. 10.1103/PhysRevA.66.062311.
https://doi.org/10.1103/PhysRevA.66.062311

[12] Tanner Crowder. A linearization of quantum channels. Journal of Geometry and Physics, 92: 157–166, 2015. 10.1016/j.geomphys.2015.02.014.
https://doi.org/10.1016/j.geomphys.2015.02.014

[13] Kristan Temme, Sergey Bravyi, and Jay M Gambetta. Error mitigation for short-depth quantum circuits. Physical Review Letters, 119 (18): 180509, 2017. 10.1103/PhysRevLett.119.180509.
https://doi.org/10.1103/PhysRevLett.119.180509

[14] Mark Howard and Earl Campbell. Application of a resource theory for magic states to fault-tolerant quantum computing. Physical Review Letters, 118 (9): 090501, 2017. 10.1103/PhysRevLett.118.090501.
https://doi.org/10.1103/PhysRevLett.118.090501

[15] Suguru Endo, Simon C Benjamin, and Ying Li. Practical quantum error mitigation for near-future applications. Physical Review X, 8 (3): 031027, 2018. 10.1103/PhysRevX.8.031027.
https://doi.org/10.1103/PhysRevX.8.031027

[16] Ryuji Takagi. Optimal resource cost for error mitigation. arXiv preprint arXiv:2006.12509, 3, 2020. ISSN 2643-1564. 10.1103/physrevresearch.3.033178.
https://doi.org/10.1103/physrevresearch.3.033178
arXiv:2006.12509

[17] Lieven Vandenberghe and Stephen Boyd. Semidefinite Programming. SIAM Review, 38 (1): 49–95, mar 1996. ISSN 0036-1445. 10.1137/1038003.
https://doi.org/10.1137/1038003

[18] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear algebra and its applications, 10 (3): 285–290, 1975. 10.1016/0024-3795(75)90075-0.
https://doi.org/10.1016/0024-3795(75)90075-0

[19] Andrzej Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3 (4): 275–278, 1972. 10.1016/0034-4877(72)90011-0.
https://doi.org/10.1016/0034-4877(72)90011-0

[20] John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. 10.1017/9781316848142.
https://doi.org/10.1017/9781316848142

[21] Ashwin Nayak and Pranab Sen. Invertible quantum operations and perfect encryption of quantum states. arXiv preprint quant-ph/0605041, 2006.
arXiv:quant-ph/0605041

[22] Maksim E Shirokov. Reversibility conditions for quantum channels and their applications. Sbornik: Mathematics, 204 (8): 1215, 2013. 10.1070/sm2013v204n08abeh004337.
https://doi.org/10.1070/sm2013v204n08abeh004337

[23] Eugene Paul Wigner and U Fano. Group theory and its application to the quantum mechanics of atomic spectra. AmJPh, 28 (4): 408–409, 1960. 10.1119/1.1935822.
https://doi.org/10.1119/1.1935822

[24] Tom Cooney, Milán Mosonyi, and Mark M Wilde. Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication. Communications in Mathematical Physics, 344 (3): 797–829, 2016. 10.1007/s00220-016-2645-4.
https://doi.org/10.1007/s00220-016-2645-4

[25] Mark M. Wilde, Mario Berta, Christoph Hirche, and Eneet Kaur. Amortized channel divergence for asymptotic quantum channel discrimination. Letters in Mathematical Physics, 110 (8): 2277–2336, aug 2020. ISSN 0377-9017. 10.1007/s11005-020-01297-7.
https://doi.org/10.1007/s11005-020-01297-7

[26] Kun Fang, Xin Wang, Marco Tomamichel, and Mario Berta. Quantum Channel Simulation and the Channel's Smooth Max-Information. IEEE Transactions on Information Theory, 66 (4): 2129–2140, apr 2020. ISSN 0018-9448. 10.1109/TIT.2019.2943858.
https://doi.org/10.1109/TIT.2019.2943858

[27] Xin Wang, Kun Fang, and Marco Tomamichel. On Converse Bounds for Classical Communication Over Quantum Channels. IEEE Transactions on Information Theory, 65 (7): 4609–4619, jul 2019a. ISSN 0018-9448. 10.1109/TIT.2019.2898656.
https://doi.org/10.1109/TIT.2019.2898656

[28] Ryuji Takagi, Kun Wang, and Masahito Hayashi. Application of the resource theory of channels to communication scenarios. Physical Review Letters, 124 (12): 120502, 2020. 10.1103/physrevlett.124.120502.
https://doi.org/10.1103/physrevlett.124.120502

[29] Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. 10.1017/cbo9780511804441.
https://doi.org/10.1017/cbo9780511804441

[30] Eric Chitambar and Gilad Gour. Quantum resource theories. Reviews of Modern Physics, 91 (2): 025001, 2019. 10.1103/revmodphys.91.025001.
https://doi.org/10.1103/revmodphys.91.025001

[31] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. Transforming quantum operations: Quantum supermaps. EPL (Europhysics Letters), 83 (3): 30004, 2008. 10.1209/0295-5075/83/30004.
https://doi.org/10.1209/0295-5075/83/30004

[32] Xin Wang and Mark M. Wilde. Exact entanglement cost of quantum states and channels under PPT-preserving operations. arXiv:1809.09592, sep 2018. URL http://arxiv.org/abs/1809.09592.
arXiv:1809.09592

[33] María García Díaz, Kun Fang, Xin Wang, Matteo Rosati, Michalis Skotiniotis, John Calsamiglia, and Andreas Winter. Using and reusing coherence to realize quantum processes. Quantum, 2: 100, oct 2018. ISSN 2521-327X. 10.22331/q-2018-10-19-100.
https://doi.org/10.22331/q-2018-10-19-100

[34] Xin Wang, Mark M Wilde, and Yuan Su. Quantifying the magic of quantum channels. New Journal of Physics, 21 (10): 103002, oct 2019b. ISSN 1367-2630. 10.1088/1367-2630/ab451d.
https://doi.org/10.1088/1367-2630/ab451d

[35] Xiao Yuan, Yunchao Liu, Qi Zhao, Bartosz Regula, Jayne Thompson, and Mile Gu. Universal and operational benchmarking of quantum memories. npj Quantum Information, 7 (1): 108, dec 2021. ISSN 2056-6387. 10.1038/s41534-021-00444-9.
https://doi.org/10.1038/s41534-021-00444-9

[36] Kun Fang and Hamza Fawzi. Geometric Rényi Divergence and its Applications in Quantum Channel Capacities. Communications in Mathematical Physics, 384 (3): 1615–1677, jun 2021. ISSN 0010-3616. 10.1007/s00220-021-04064-4.
https://doi.org/10.1007/s00220-021-04064-4

[37] Xin Wang and Mark M. Wilde. Resource theory of asymmetric distinguishability for quantum channels. Physical Review Research, 1 (3): 033169, dec 2019. ISSN 2643-1564. 10.1103/PhysRevResearch.1.033169.
https://doi.org/10.1103/PhysRevResearch.1.033169

[38] Luca Chirolli and Guido Burkard. Decoherence in solid-state qubits. Advances in Physics, 57 (3): 225–285, 2008. 10.1080/00018730802218067.
https://doi.org/10.1080/00018730802218067

[39] Sumeet Khatri, Kunal Sharma, and Mark M. Wilde. Information-theoretic aspects of the generalized amplitude-damping channel. Physical Review A, 102 (1): 012401, jul 2020. ISSN 2469-9926. 10.1103/PhysRevA.102.012401.
https://doi.org/10.1103/PhysRevA.102.012401

[40] Xin Wang. Pursuing the fundamental limits for quantum communication. arXiv:1912.00931, 67 (7): 4524–4532, 2019. ISSN 0018-9448. 10.1109/tit.2021.3068818.
https://doi.org/10.1109/tit.2021.3068818
arXiv:1912.00931

[41] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, 2018. 10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[42] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien. Low-cost error mitigation by symmetry verification. Physical Review A, 98 (6): 062339, dec 2018. ISSN 2469-9926. 10.1103/PhysRevA.98.062339.
https://doi.org/10.1103/PhysRevA.98.062339

[43] Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation. Journal of the Physical Society of Japan, 90 (3): 032001, mar 2021. ISSN 0031-9015. 10.7566/JPSJ.90.032001.
https://doi.org/10.7566/JPSJ.90.032001

[44] Kun Wang, Yu-Ao Chen, and Xin Wang. Measurement Error Mitigation via Truncated Neumann Series. arXiv preprint arXiv:2103.13856, (2): 1–14, mar 2021. URL http://arxiv.org/abs/2103.13856.
arXiv:2103.13856

[45] Zhenyu Cai. Multi-exponential error extrapolation and combining error mitigation techniques for NISQ applications. npj Quantum Information, 7 (1), 2021. ISSN 20566387. 10.1038/s41534-021-00404-3.
https://doi.org/10.1038/s41534-021-00404-3

[46] Aram W Harrow and Michael A Nielsen. Robustness of quantum gates in the presence of noise. Physical Review A, 68 (1): 012308, 2003. 10.1103/PhysRevA.68.012308.
https://doi.org/10.1103/PhysRevA.68.012308

[47] Guifré Vidal and Rolf Tarrach. Robustness of entanglement. Physical Review A, 59 (1): 141, 1999. 10.1103/PhysRevA.59.141.
https://doi.org/10.1103/PhysRevA.59.141

[48] Michael Steiner. Generalized robustness of entanglement. Phys. Rev. A, 67: 054305, May 2003. 10.1103/PhysRevA.67.054305.
https://doi.org/10.1103/PhysRevA.67.054305

[49] Fernando GSL Brandao. Entanglement activation and the robustness of quantum correlations. Physical Review A, 76 (3): 030301, 2007. 10.1103/PhysRevA.76.030301.
https://doi.org/10.1103/PhysRevA.76.030301

[50] Mafalda L Almeida, Stefano Pironio, Jonathan Barrett, Géza Tóth, and Antonio Acín. Noise robustness of the nonlocality of entangled quantum states. Physical Review Letters, 99 (4): 040403, 2007. 10.1103/physrevlett.99.040403.
https://doi.org/10.1103/physrevlett.99.040403

[51] Ryuji Takagi, Bartosz Regula, Kaifeng Bu, Zi-Wen Liu, and Gerardo Adesso. Operational advantage of quantum resources in subchannel discrimination. Physical Review Letters, 122 (14): 140402, 2019. 10.1103/PhysRevLett.122.140402.
https://doi.org/10.1103/PhysRevLett.122.140402

[52] Marco Piani and John Watrous. Necessary and sufficient quantum information characterization of einstein-podolsky-rosen steering. Physical Review Letters, 114 (6): 060404, 2015. 10.1103/physrevlett.114.060404.
https://doi.org/10.1103/physrevlett.114.060404

[53] Carmine Napoli, Thomas R Bromley, Marco Cianciaruso, Marco Piani, Nathaniel Johnston, and Gerardo Adesso. Robustness of coherence: an operational and observable measure of quantum coherence. Physical Review Letters, 116 (15): 150502, 2016. 10.1103/physrevlett.116.150502.
https://doi.org/10.1103/physrevlett.116.150502

[54] Marco Piani, Marco Cianciaruso, Thomas R Bromley, Carmine Napoli, Nathaniel Johnston, and Gerardo Adesso. Robustness of asymmetry and coherence of quantum states. Physical Review A, 93 (4): 042107, 2016. 10.1103/physreva.93.042107.
https://doi.org/10.1103/physreva.93.042107

[55] Namit Anand and Todd A Brun. Quantifying non-markovianity: a quantum resource-theoretic approach. arXiv preprint arXiv:1903.03880, 2019.
arXiv:1903.03880

[56] Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: operational characterization via discrimination tasks. Physical Review X, 9 (3): 031053, 2019. 10.1103/PhysRevX.9.031053.
https://doi.org/10.1103/PhysRevX.9.031053

[57] Zi-Wen Liu and Andreas Winter. Resource theories of quantum channels and the universal role of resource erasure. arXiv preprint arXiv:1904.04201, 2019.
arXiv:1904.04201

[58] Joonwoo Bae, Dariusz Chruściński, and Marco Piani. More entanglement implies higher performance in channel discrimination tasks. Physical Review Letters, 122 (14): 140404, 2019. 10.1103/PhysRevLett.122.140404.
https://doi.org/10.1103/PhysRevLett.122.140404

[59] Paul Skrzypczyk and Noah Linden. Robustness of measurement, discrimination games, and accessible information. Physical Review Letters, 122 (14): 140403, 2019. 10.1103/PhysRevLett.122.140403.
https://doi.org/10.1103/PhysRevLett.122.140403

[60] John Watrous. Semidefinite programming in quantum information (winter 2017). https://cs.uwaterloo.ca/ watrous/CS867.Winter2017/, 2017.
https://cs.uwaterloo.ca/~watrous/CS867.Winter2017/

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[17] Maurits S. J. Tepaske and David J. Luitz, "Compressed quantum error mitigation", Physical Review B 107 20, L201114 (2023).

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[21] Kento Tsubouchi, Takahiro Sagawa, and Nobuyuki Yoshioka, "Universal Cost Bound of Quantum Error Mitigation Based on Quantum Estimation Theory", Physical Review Letters 131 21, 210601 (2023).

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