Efficient entanglement purification based on noise guessing decoding

André Roque; Diogo Cruz; Francisco A. Monteiro; Bruno C. Coutinho

doi:10.22331/q-2024-09-19-1476

Efficient entanglement purification based on noise guessing decoding

André Roque¹, Diogo Cruz^1,2, Francisco A. Monteiro^2,3, and Bruno C. Coutinho²

¹Instituto Superior Técnico, Universidade de Lisboa, Portugal
²Instituto de Telecomunicações, Portugal
³ISCTE - Instituto Universitário de Lisboa, Portugal

Published:	2024-09-19, volume 8, page 1476
Eprint:	arXiv:2310.19914v4
Doi:	https://doi.org/10.22331/q-2024-09-19-1476
Citation:	Quantum 8, 1476 (2024).

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Abstract

In this paper, we propose a novel bipartite entanglement purification protocol built upon hashing and upon the guessing random additive noise decoding (GRAND) approach recently devised for classical error correction codes. Our protocol offers substantial advantages over existing hashing protocols, requiring fewer qubits for purification, achieving higher fidelities, and delivering better yields with reduced computational costs. We provide numerical and semi-analytical results to corroborate our findings and provide a detailed comparison with the hashing protocol of Bennet et al. Although that pioneering work devised performance bounds, it did not offer an explicit construction for implementation. The present work fills that gap, offering both an explicit and more efficient purification method. We demonstrate that our protocol is capable of purifying states with noise on the order of 10% per Bell pair even with a small ensemble of 16 pairs. The work explores a measurement-based implementation of the protocol to address practical setups with noise. This work opens the path to practical and efficient entanglement purification using hashing-based methods with feasible computational costs. Compared to the original hashing protocol, the proposed method can achieve some desired fidelity with a number of initial resources up to one hundred times smaller. Therefore, the proposed method seems well-fit for future quantum networks with a limited number of resources and entails a relatively low computational overhead.

Featured image: Quantum circuit for the PGRAND applied to n Bell pairs distributed between Alice (A) and Bob (B) in order to obtain k purified pairs. The error is described by a quantum channel C. In the circuit, the measurements performed by Alice are depicted changing the ancilla qubits, but in reality only a virtual syndrome update needs do be done. The same goes to the recovery procedure: if it consists only of Pauli strings, then R can be performed virtually (in software), without the need to apply extra gates.

► BibTeX data

@article{Roque2024efficient,
  doi = {10.22331/q-2024-09-19-1476},
  url = {https://doi.org/10.22331/q-2024-09-19-1476},
  title = {Efficient entanglement purification based on noise guessing decoding},
  author = {Roque, Andr{\'{e}} and Cruz, Diogo and Monteiro, Francisco A. and Coutinho, Bruno C.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1476},
  month = sep,
  year = {2024}
}

► References

[1] H. J. Kimble. ``The quantum internet''. Nature 453, 1023–1030 (2008).
https://doi.org/10.1038/nature07127

[2] Marcello Caleffi, Angela Sara Cacciapuoti, and Giuseppe Bianchi. ``Quantum internet: from communication to distributed computing!''. In Proceedings of the 5th ACM International Conference on Nanoscale Computing and Communication. Pages 1–4. Reykjavik Iceland (2018). ACM.
https://doi.org/10.1145/3233188.3233224

[3] Ang-Kun Wu, Liang Tian, Bruno Coelho Coutinho, Yasser Omar, and Yang-Yu Liu. ``Structural vulnerability of quantum networks''. Phys. Rev. A 101, 052315 (2020).
https://doi.org/10.1103/PhysRevA.101.052315

[4] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Reviews of Modern Physics 81, 865–942 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[5] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. Theoretical Computer Science 560, 7–11 (2014).
https://doi.org/10.1016/j.tcs.2014.05.025

[6] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. ``Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels''. Physical Review Letters 70, 1895–1899 (1993).
https://doi.org/10.1103/PhysRevLett.70.1895

[7] J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello. ``Distributed quantum computation over noisy channels''. Physical Review A 59, 4249–4254 (1999).
https://doi.org/10.1103/PhysRevA.59.4249

[8] W. Dür and H.-J. Briegel. ``Entanglement Purification for Quantum Computation''. Physical Review Letters 90, 067901 (2003).
https://doi.org/10.1103/PhysRevLett.90.067901

[9] P. Sekatski, M. Skotiniotis, and W. Dür. ``Improved sensing with a single qubit''. Physical Review Letters 118, 170801 (2017).
https://doi.org/10.1103/PhysRevLett.118.170801

[10] V. Krutyanskiy, M. Galli, V. Krcmarsky, S. Baier, D. A. Fioretto, Y. Pu, A. Mazloom, P. Sekatski, M. Canteri, M. Teller, J. Schupp, J. Bate, M. Meraner, N. Sangouard, B. P. Lanyon, and T. E. Northup. ``Entanglement of Trapped-Ion Qubits Separated by 230 Meters''. Physical Review Letters 130, 050803 (2023).
https://doi.org/10.1103/PhysRevLett.130.050803

[11] C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov, and C. Monroe. ``Parallel entangling operations on a universal ion-trap quantum computer''. Nature 572, 368–372 (2019).
https://doi.org/10.1038/s41586-019-1427-5

[12] Colin D. Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M. Sage. ``Trapped-ion quantum computing: Progress and challenges''. Applied Physics Reviews 6, 021314 (2019).
https://doi.org/10.1063/1.5088164

[13] Daniel Riedel, Immo Söllner, Brendan J. Shields, Sebastian Starosielec, Patrick Appel, Elke Neu, Patrick Maletinsky, and Richard J. Warburton. ``Deterministic enhancement of coherent photon generation from a nitrogen-vacancy center in ultrapure diamond''. Physical Review X 7, 031040 (2017).
https://doi.org/10.1103/PhysRevX.7.031040

[14] Maximilian Ruf, Mark IJspeert, Suzanne Van Dam, Nick De Jong, Hans Van Den Berg, Guus Evers, and Ronald Hanson. ``Optically coherent nitrogen-vacancy centers in micrometer-thin etched diamond membranes''. Nano Letters 19, 3987–3992 (2019).
https://doi.org/10.1021/acs.nanolett.9b01316

[15] A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, and C. A. Regal. ``Entangling two transportable neutral atoms via local spin exchange''. Nature 527, 208–211 (2015).
https://doi.org/10.1038/nature16073

[16] Ivaylo S. Madjarov, Jacob P. Covey, Adam L. Shaw, Joonhee Choi, Anant Kale, Alexandre Cooper, Hannes Pichler, Vladimir Schkolnik, Jason R. Williams, and Manuel Endres. ``High-fidelity entanglement and detection of alkaline-earth Rydberg atoms''. Nature Physics 16, 857–861 (2020).
https://doi.org/10.1038/s41567-020-0903-z

[17] N. Leung, Y. Lu, S. Chakram, R. K. Naik, N. Earnest, R. Ma, K. Jacobs, A. N. Cleland, and D. I. Schuster. ``Deterministic bidirectional communication and remote entanglement generation between superconducting qubits''. npj Quantum Information 5, 1–5 (2019).
https://doi.org/10.1038/s41534-019-0128-0

[18] E. Flurin, N. Roch, J. D. Pillet, F. Mallet, and B. Huard. ``Superconducting Quantum Node for Entanglement and Storage of Microwave Radiation''. Physical Review Letters 114, 090503 (2015).
https://doi.org/10.1103/PhysRevLett.114.090503

[19] Joschka Roffe. ``Quantum error correction: an introductory guide''. Contemporary Physics 60, 226–245 (2019).
https://doi.org/10.1080/00107514.2019.1667078

[20] Simon J. Devitt, William J. Munro, and Kae Nemoto. ``Quantum error correction for beginners''. Reports on Progress in Physics 76, 076001 (2013).
https://doi.org/10.1088/0034-4885/76/7/076001

[21] Michelle Victora, Spyros Tserkis, Stefan Krastanov, Alexander Sanchez de la Cerda, Steven Willis, and Prineha Narang. ``Entanglement purification on quantum networks''. Phys. Rev. Res. 5, 033171 (2023).
https://doi.org/10.1103/PhysRevResearch.5.033171

[22] William J. Munro, Koji Azuma, Kiyoshi Tamaki, and Kae Nemoto. ``Inside Quantum Repeaters''. IEEE Journal of Selected Topics in Quantum Electronics 21, 78–90 (2015).
https://doi.org/10.1109/JSTQE.2015.2392076

[23] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller. ``Long-distance quantum communication with atomic ensembles and linear optics''. Nature 414, 413–418 (2001).
https://doi.org/10.1038/35106500

[24] Sara Santos, Francisco A. Monteiro, Bruno C. Coutinho, and Yasser Omar. ``Shortest Path Finding in Quantum Networks With Quasi-Linear Complexity''. IEEE Access 11, 7180–7194 (2023).
https://doi.org/10.1109/ACCESS.2023.3237997

[25] Luís Bugalho, Bruno C. Coutinho, Francisco A. Monteiro, and Yasser Omar. ``Distributing Multipartite Entanglement over Noisy Quantum Networks''. Quantum 7, 920 (2023).
https://doi.org/10.22331/q-2023-02-09-920

[26] Bruno C. Coutinho, Raul Monteiro, Luís Bugalho, and Francisco A. Monteiro. ``Entanglement routing based on fidelity curves'' (2024). arXiv:2303.12864.
arXiv:2303.12864

[27] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. ``Mixed-state entanglement and quantum error correction''. Physical Review A 54, 3824–3851 (1996).
https://doi.org/10.1103/PhysRevA.54.3824

[28] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters. ``Purification of noisy entanglement and faithful teleportation via noisy channels''. Physical Review Letters 76, 722–725 (1996).
https://doi.org/10.1103/PhysRevLett.76.722

[29] David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu, and Anna Sanpera. ``Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels''. Physical Review Letters 77, 2818–2821 (1996).
https://doi.org/10.1103/PhysRevLett.77.2818

[30] H. F. Chau and K. H. Ho. ``Practical entanglement distillation scheme using recurrence method and quantum low density parity check codes''. Quantum Information Processing 10, 213–229 (2011).
https://doi.org/10.1007/s11128-010-0190-1

[31] Stefan Krastanov, Victor V. Albert, and Liang Jiang. ``Optimized Entanglement Purification''. Quantum 3, 123 (2019).
https://doi.org/10.22331/q-2019-02-18-123

[32] J. Miguel-Ramiro and W. Dür. ``Efficient entanglement purification protocols for d-level systems''. Physical Review A 98, 042309 (2018).
https://doi.org/10.1103/PhysRevA.98.042309

[33] W. Dür and H. J. Briegel. ``Entanglement purification and quantum error correction''. Reports on Progress in Physics 70, 1381 (2007).
https://doi.org/10.1088/0034-4885/70/8/R03

[34] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng. ``Advances in quantum entanglement purification''. Science China Physics, Mechanics & Astronomy 66, 250301 (2023).
https://doi.org/10.1007/s11433-022-2065-x

[35] H. Aschauer, W. Dür, and H.-J. Briegel. ``Multiparticle entanglement purification for two-colorable graph states''. Physical Review A 71, 012319 (2005).
https://doi.org/10.1103/PhysRevA.71.012319

[36] Diogo Cruz, Francisco A. Monteiro, and Bruno C. Coutinho. ``Quantum error correction via noise guessing decoding''. IEEE Access 11 (2023).
https://doi.org/10.1109/ACCESS.2023.3327214

[37] Daryus Chandra, Zeynep B. Kaykac Egilmez, Yifeng Xiong, Soon Xin Ng, Robert G. Maunder, and Lajos Hanzo. ``Universal decoding of quantum stabilizer codes via classical guesswork''. IEEE Access 11, 19059–19072 (2023).
https://doi.org/10.1109/ACCESS.2023.3247966

[38] M. Zwerger, H. J. Briegel, and W. Dür. ``Robustness of hashing protocols for entanglement purification''. Physical Review A 90, 012314 (2014).
https://doi.org/10.1103/PhysRevA.90.012314

[39] M. Zwerger, H. J. Briegel, and W. Dür. ``Measurement-based quantum communication''. Applied Physics B 122, 50 (2016).
https://doi.org/10.1007/s00340-015-6285-8

[40] Robert Raussendorf. ``Measurement-based quantum computation with cluster states''. International Journal of Quantum Information 07, 1053–1203 (2009).
https://doi.org/10.1142/S0219749909005699

[41] Robert Raussendorf and Hans J. Briegel. ``A one-way quantum computer''. Physical Review Letters 86, 5188–5191 (2001).
https://doi.org/10.1103/PhysRevLett.86.5188

[42] Ken R. Duffy, Jiange Li, and Muriel Médard. ``Capacity-Achieving Guessing Random Additive Noise Decoding''. IEEE Transactions on Information Theory 65, 4023–4040 (2019).
https://doi.org/10.1109/TIT.2019.2896110

[43] Ioannis Chatzigeorgiou and Francisco A Monteiro. ``Symbol-level GRAND for high-order modulation over block fading channels''. IEEE Communications Letters 27, 447–451 (2023).
https://doi.org/10.1109/LCOMM.2022.3227593

[44] Hans Aschauer. ``Quantum communication in noisy environments''. PhD thesis. Ludwig-Maximilians-Universität München. (2005).
https://doi.org/10.5282/edoc.3588

[45] Ryutaroh Matsumoto. ``Conversion of a general quantum stabilizer code to an entanglement distillation protocol''. Journal of Physics A: Mathematical and General 36, 8113–8127 (2003).
https://doi.org/10.1088/0305-4470/36/29/316

[46] Reinhard F. Werner. ``Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model''. Physical Review A 40, 4277–4281 (1989).
https://doi.org/10.1103/PhysRevA.40.4277

[47] W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel. ``Standard forms of noisy quantum operations via depolarization''. Physical Review A 72, 052326 (2005).
https://doi.org/10.1103/PhysRevA.72.052326

[48] Robert Raussendorf and Hans Briegel. ``Computational model underlying the one-way quantum computer'' (2002). arxiv:quant-ph/0108067.
arXiv:quant-ph/0108067

[49] Michael A. Nielsen and Christopher M. Dawson. ``Fault-tolerant quantum computation with cluster states''. Physical Review A 71, 042323 (2005).
https://doi.org/10.1103/PhysRevA.71.042323

[50] Christopher M. Dawson, Henry L. Haselgrove, and Michael A. Nielsen. ``Noise Thresholds for Optical Quantum Computers''. Physical Review Letters 96, 020501 (2006).
https://doi.org/10.1103/PhysRevLett.96.020501

[51] Benjamin J. Brown and Sam Roberts. ``Universal fault-tolerant measurement-based quantum computation''. Physical Review Research 2, 033305 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033305

[52] Mercedes Gimeno-Segovia, Terry Rudolph, and Sophia E. Economou. ``Deterministic generation of large-scale entangled photonic cluster state from interacting solid state emitters''. Physical Review Letters 123, 070501 (2019).
https://doi.org/10.1103/PhysRevLett.123.070501

[53] W. Dür, H. Aschauer, and H.-J. Briegel. ``Multiparticle Entanglement Purification for Graph States''. Physical Review Letters 91, 107903 (2003).
https://doi.org/10.1103/PhysRevLett.91.107903

[54] Daniel Gottesman. ``The Heisenberg representation of quantum computers'' (1998). arxiv:quant-ph/9807006.
arXiv:quant-ph/9807006

[55] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Physical Review A 70, 052328 (2004).
https://doi.org/10.1103/PhysRevA.70.052328

[56] Alexandru Paler, Simon Devitt, Kae Nemoto, and Ilia Polian. ``Software-based Pauli tracking in fault-tolerant quantum circuits''. In 2014 Design, Automation & Test in Europe Conference & Exhibition (DATE). Pages 1–4. (2014).
https://doi.org/10.7873/DATE.2014.137

[57] Artur Ekert and Chiara Macchiavello. ``Quantum error correction for communication''. Physical Review Letters 77, 2585–2588 (1996).
https://doi.org/10.1103/PhysRevLett.77.2585

[58] David P. DiVincenzo, Peter W. Shor, and John A. Smolin. ``Quantum-channel capacity of very noisy channels''. Physical Review A 57, 830–839 (1998).
https://doi.org/10.1103/PhysRevA.57.830

[59] A.S. Holevo. ``The capacity of the quantum channel with general signal states''. IEEE Transactions on Information Theory 44, 269–273 (1998).
https://doi.org/10.1109/18.651037

[60] C. King. ``The capacity of the quantum depolarizing channel''. IEEE Transactions on Information Theory 49, 221–229 (2003).
https://doi.org/10.1109/TIT.2002.806153

[61] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. PhD thesis. California Institute of Technology. (1997).
https://doi.org/10.48550/arXiv.quant-ph/9705052
arXiv:quant-ph/9705052

[62] M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür. ``Long-range big quantum-data transmission''. Physical Review Letters 120, 030503 (2018).
https://doi.org/10.1103/PhysRevLett.120.030503

[63] M. Zwerger, W. Dür, and H. J. Briegel. ``Measurement-based quantum repeaters''. Physical Review A 85, 062326 (2012).
https://doi.org/10.1103/PhysRevA.85.062326

[64] M. Zwerger, H. J. Briegel, and W. Dür. ``Universal and optimal error thresholds for measurement-based entanglement purification''. Physical Review Letters 110, 260503 (2013).
https://doi.org/10.1103/PhysRevLett.110.260503

[65] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng. ``Feasible measurement-based entanglement purification in linear optics''. Optics Express 29, 9363 (2021).
https://doi.org/10.1364/OE.420348

[66] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng. ``Measurement-based entanglement purification for entangled coherent states''. Frontiers of Physics 17, 21501 (2021).
https://doi.org/10.1007/s11467-021-1103-8

[67] M. Hein, J. Eisert, and H. J. Briegel. ``Multiparty entanglement in graph states''. Physical Review A 69, 062311 (2004).
https://doi.org/10.1103/PhysRevA.69.062311

[68] J. Wallnöfer and W. Dür. ``Measurement-based quantum communication with resource states generated by entanglement purification''. Physical Review A 95, 012303 (2017).
https://doi.org/10.1103/PhysRevA.95.012303

[69] Diogo Cruz, Francisco A. Monteiro, André Roque, and Bruno C. Coutinho. ``Fault-tolerant noise guessing decoding of quantum random codes'' (2024). arXiv:2407.01658.
arXiv:2407.01658

[70] Narayanan Rengaswamy, Ankur Raina, Nithin Raveendran, and Bane Vasić. ``Distilling GHZ states using stabilizer codes'' (2022). arxiv:2109.06248.
arXiv:2109.06248

[71] Chiara Macchiavello. ``On the analytical convergence of the QPA procedure''. Physics Letters A 246, 385–388 (1998).
https://doi.org/10.1016/S0375-9601(98)00550-7

[72] Keisuke Fujii and Katsuji Yamamoto. ``Entanglement purification with double selection''. Physical Review A 80, 042308 (2009).
https://doi.org/10.1103/PhysRevA.80.042308

[73] Naomi H. Nickerson, Ying Li, and Simon C. Benjamin. ``Topological quantum computing with a very noisy network and local error rates approaching one percent''. Nature Communications 4, 1756 (2013).
https://doi.org/10.1038/ncomms2773

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