Optimized Entanglement Purification

Stefan Krastanov; Victor V. Albert; Liang Jiang

doi:10.22331/q-2019-02-18-123

Optimized Entanglement Purification

Stefan Krastanov^1,2, Victor V. Albert^1,2,3, and Liang Jiang^1,2

¹Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA
²Yale Quantum Institute, Yale University, New Haven, CT 06520, USA
³Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

Published:	2019-02-18, volume 3, page 123
Eprint:	arXiv:1712.09762v3
Doi:	https://doi.org/10.22331/q-2019-02-18-123
Citation:	Quantum 3, 123 (2019).

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

Abstract

We investigate novel protocols for entanglement purification of qubit Bell pairs. Employing genetic algorithms for the design of the purification circuit, we obtain shorter circuits achieving higher success rates and better final fidelities than what is currently available in the literature. We provide a software tool for analytical and numerical study of the generated purification circuits, under customizable error models. These new purification protocols pave the way to practical implementations of modular quantum computers and quantum repeaters. Our approach is particularly attentive to the effects of finite resources and imperfect local operations - phenomena neglected in the usual asymptotic approach to the problem. The choice of the building blocks permitted in the construction of the circuits is based on a thorough enumeration of the local Clifford operations that act as permutations on the basis of Bell states.

Featured image: A short optimized entanglement purification circuit for a 4-qubit register designed by a genetic algorithm.

► BibTeX data

@article{Krastanov2019optimized,
  doi = {10.22331/q-2019-02-18-123},
  url = {https://doi.org/10.22331/q-2019-02-18-123},
  title = {Optimized {E}ntanglement {P}urification},
  author = {Krastanov, Stefan and Albert, Victor V. and Jiang, Liang},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {123},
  month = feb,
  year = {2019}
}

► References

[1] Julia Cramer, Norbert Kalb, M Adriaan Rol, Bas Hensen, Machiel S Blok, Matthew Markham, Daniel J Twitchen, Ronald Hanson, and Tim H Taminiau. Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Nature communications, 7, 2016. 10.1038/ncomms11526.
https://doi.org/10.1038/ncomms11526

[2] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, SM Girvin, L Jiang, et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature, 536 (7617): 441–445, 2016. 10.1038/nature18949.
https://doi.org/10.1038/nature18949

[3] Daniel Gottesman and Isaac L Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390–393, 1999. 10.1038/46503.
https://doi.org/10.1038/46503

[4] C Monroe, R Raussendorf, A Ruthven, KR Brown, P Maunz, L-M Duan, and J Kim. Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Physical Review A, 89 (2): 022317, 2014. 10.1103/physreva.89.022317.
https://doi.org/10.1103/physreva.89.022317

[5] A Narla, S Shankar, M Hatridge, Z Leghtas, KM Sliwa, E Zalys-Geller, SO Mundhada, W Pfaff, L Frunzio, RJ Schoelkopf, et al. Robust concurrent remote entanglement between two superconducting qubits. Physical Review X, 6 (3): 031036, 2016. 10.1103/physrevx.6.031036.
https://doi.org/10.1103/physrevx.6.031036

[6] D Hucul, IV Inlek, G Vittorini, C Crocker, S Debnath, SM Clark, and Cl Monroe. Modular entanglement of atomic qubits using photons and phonons. Nature Physics, 11 (1): 37–42, 2015. 10.1038/nphys3150.
https://doi.org/10.1038/nphys3150

[7] Ramil Nigmatullin, Christopher J Ballance, Niel de Beaudrap, and Simon C Benjamin. Minimally complex ion traps as modules for quantum communication and computing. New Journal of Physics, 18 (10): 103028, 2016. 10.1088/1367-2630/18/10/103028.
https://doi.org/10.1088/1367-2630/18/10/103028

[8] Andreas Reiserer, Norbert Kalb, Machiel S Blok, Koen JM van Bemmelen, Tim H Taminiau, Ronald Hanson, Daniel J Twitchen, and Matthew Markham. Robust quantum-network memory using decoherence-protected subspaces of nuclear spins. Physical Review X, 6 (2): 021040, 2016. 10.1103/physrevx.6.021040.
https://doi.org/10.1103/physrevx.6.021040

[9] 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. 10.1038/ncomms2773.
https://doi.org/10.1038/ncomms2773

[10] W Dür, H-J Briegel, JI Cirac, and P Zoller. Quantum repeaters based on entanglement purification. Physical Review A, 59 (1): 169, 1999. 10.1103/physreva.59.169.
https://doi.org/10.1103/physreva.59.169

[11] Jian-Wei Pan, Christoph Simon, Časlav Brukner, and Anton Zeilinger. Entanglement purification for quantum communication. Nature, 410 (6832): 1067–1070, 2001. 10.1038/35074041.
https://doi.org/10.1038/35074041

[12] L Childress, JM Taylor, Anders Søndberg Sørensen, and Mikhail D Lukin. Fault-tolerant quantum repeaters with minimal physical resources and implementations based on single-photon emitters. Physical Review A, 72 (5): 052330, 2005. 10.1103/physreva.72.052330.
https://doi.org/10.1103/physreva.72.052330

[13] L Jiang, JM Taylor, and MD Lukin. Fast and robust approach to long-distance quantum communication with atomic ensembles. Physical Review A, 76 (1): 012301, 2007. 10.1103/physreva.76.012301.
https://doi.org/10.1103/physreva.76.012301

[14] Naomi H Nickerson, Joseph F Fitzsimons, and Simon C Benjamin. Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links. Physical Review X, 4 (4): 041041, 2014. 10.1103/physrevx.4.041041.
https://doi.org/10.1103/physrevx.4.041041

[15] DL Moehring, P Maunz, S Olmschenk, KC Younge, DN Matsukevich, L-M Duan, and C Monroe. Entanglement of single-atom quantum bits at a distance. Nature, 449 (7158): 68–71, 2007. 10.1038/nature06118.
https://doi.org/10.1038/nature06118

[16] Wolfgang Pfaff, Tim H Taminiau, Lucio Robledo, Hannes Bernien, Matthew Markham, Daniel J Twitchen, and Ronald Hanson. Demonstration of entanglement-by-measurement of solid-state qubits. Nature Physics, 9 (1): 29–33, 2013. 10.1038/nphys2444.
https://doi.org/10.1038/nphys2444

[17] Bas Hensen, Hannes Bernien, Anaïs E Dréau, Andreas Reiserer, Norbert Kalb, Machiel S Blok, Just Ruitenberg, Raymond FL Vermeulen, Raymond N Schouten, Carlos Abellán, et al. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526 (7575): 682–686, 2015. 10.1038/nature15759.
https://doi.org/10.1038/nature15759

[18] Stephan Ritter, Christian Nölleke, Carolin Hahn, Andreas Reiserer, Andreas Neuzner, Manuel Uphoff, Martin Mücke, Eden Figueroa, Joerg Bochmann, and Gerhard Rempe. An elementary quantum network of single atoms in optical cavities. Nature, 484 (7393): 195–200, 2012. 10.1364/icqi.2011.qwb2.
https://doi.org/10.1364/icqi.2011.qwb2

[19] 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 (13): 2818, 1996. 10.1103/physrevlett.77.2818.
https://doi.org/10.1103/physrevlett.77.2818

[20] 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 (5): 722, 1996a. 10.1103/physrevlett.76.722.
https://doi.org/10.1103/physrevlett.76.722

[21] Wolfgang Dür and Hans J Briegel. Entanglement purification and quantum error correction. Reports on Progress in Physics, 70 (8): 1381, 2007. 10.1088/0034-4885/70/8/r03.
https://doi.org/10.1088/0034-4885/70/8/r03

[22] Keisuke Fujii and Katsuji Yamamoto. Entanglement purification with double selection. Physical Review A, 80 (4): 042308, 2009. 10.1103/physreva.80.042308.
https://doi.org/10.1103/physreva.80.042308

[23] Hans Aschauer. Quantum communication in noisy environments. PhD thesis, lmu, 2005.

[24] John Preskill. Quantum computing in the nisq era and beyond. arXiv preprint arXiv:1801.00862, 2018. 10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79
arXiv:1801.00862

[25] Charles H Bennett, David P DiVincenzo, John A Smolin, and William K Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54 (5): 3824, 1996b. 10.1103/physreva.54.3824.
https://doi.org/10.1103/physreva.54.3824

[26] Jeroen Dehaene, Maarten Van den Nest, Bart De Moor, and Frank Verstraete. Local permutations of products of bell states and entanglement distillation. Physical Review A, 67 (2): 022310, 2003. 10.1103/physreva.67.022310.
https://doi.org/10.1103/physreva.67.022310

[27] H Bombin and MA Martin-Delgado. Entanglement distillation protocols and number theory. Physical Review A, 72 (3): 032313, 2005. 10.1103/physreva.72.032313.
https://doi.org/10.1103/physreva.72.032313

[28] Maris Ozols. Clifford group. 2008. 10.1117/12.266854.
https://doi.org/10.1117/12.266854

[29] A Robert Calderbank, Eric M Rains, PM Shor, and Neil JA Sloane. Quantum error correction via codes over gf (4). IEEE Transactions on Information Theory, 44 (4): 1369–1387, 1998. 10.1109/isit.1997.613213.
https://doi.org/10.1109/isit.1997.613213

[30] Thomas Geijtenbeek, Michiel van de Panne, and A Frank van der Stappen. Flexible muscle-based locomotion for bipedal creatures. ACM Transactions on Graphics (TOG), 32 (6): 206, 2013. 10.1145/2508363.2508399.
https://doi.org/10.1145/2508363.2508399

[31] CA Weidner, Hoon Yu, Ronnie Kosloff, and Dana Z Anderson. Atom interferometry using a shaken optical lattice. Physical Review A, 95 (4): 043624, 2017. 10.1103/physreva.95.043624.
https://doi.org/10.1103/physreva.95.043624

[32] PA Knott. A search algorithm for quantum state engineering and metrology. New Journal of Physics, 18 (7): 073033, 2016. 10.1088/1367-2630/18/7/073033.
https://doi.org/10.1088/1367-2630/18/7/073033

[33] Robin Harper, Robert J Chapman, Christopher Ferrie, Christopher Granade, Richard Kueng, Daniel Naoumenko, Steven T Flammia, and Alberto Peruzzo. Explaining quantum correlations through evolution of causal models. Physical Review A, 95 (4): 042120, 2017. 10.1103/physreva.95.042120.
https://doi.org/10.1103/physreva.95.042120

[34] Rami Barends, Julian Kelly, Anthony Megrant, Andrzej Veitia, Daniel Sank, Evan Jeffrey, Ted C White, Josh Mutus, Austin G Fowler, Brooks Campbell, et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508 (7497): 500–503, 2014. 10.1038/nature13171.
https://doi.org/10.1038/nature13171

[35] CJ Ballance, TP Harty, NM Linke, MA Sepiol, and DM Lucas. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Physical review letters, 117 (6): 060504, 2016. 10.1103/physrevlett.117.060504.
https://doi.org/10.1103/physrevlett.117.060504

[36] Joseph M Renes, David Sutter, Frédéric Dupuis, and Renato Renner. Efficient quantum polar codes requiring no preshared entanglement. IEEE Transactions on Information Theory, 61 (11): 6395–6414, 2015. 10.1109/isit.2013.6620247.
https://doi.org/10.1109/isit.2013.6620247

[37] Lan Zhou and Yu-Bo Sheng. Purification of logic-qubit entanglement. Scientific reports, 6: 28813, 2016. 10.1038/srep28813.
https://doi.org/10.1038/srep28813

[38] Lan Zhou and Yu-Bo Sheng. Polarization entanglement purification for concatenated greenberger–horne–zeilinger state. Annals of Physics, 385: 10–35, 2017. 10.1016/j.aop.2017.07.012.
https://doi.org/10.1016/j.aop.2017.07.012

[39] M Murao, MB Plenio, Sandu Popescu, V Vedral, and PL Knight. Multiparticle entanglement purification protocols. Physical Review A, 57 (6): R4075, 1998. 10.1103/physreva.57.r4075.
https://doi.org/10.1103/physreva.57.r4075

[40] F Fröwis and W Dür. Stable macroscopic quantum superpositions. Physical review letters, 106 (11): 110402, 2011. 10.1103/physrevlett.106.110402.
https://doi.org/10.1103/physrevlett.106.110402

[41] Florian Fröwis and Wolfgang Dür. Stability of encoded macroscopic quantum superpositions. Physical Review A, 85 (5): 052329, 2012. 10.1103/physreva.85.052329.
https://doi.org/10.1103/physreva.85.052329

Cited by

[1] Stefan Krastanov, Alexander Sanchez de la Cerda, and Prineha Narang, "Heterogeneous multipartite entanglement purification for size-constrained quantum devices", Physical Review Research 3 3, 033164 (2021).

[2] Francesco Preti, Tommaso Calarco, Juan Mauricio Torres, and József Zsolt Bernád, "Optimal two-qubit gates in recurrence protocols of entanglement purification", Physical Review A 106 2, 022422 (2022).

[3] Cheng-Chen Luo, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Multipartite entanglement purification using time-bin entanglement ", Laser Physics Letters 18 6, 065205 (2021).

[4] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Feasible time-bin entanglement purification based on sum-frequency generation", Optics Express 29 2, 571 (2021).

[5] F. Riera-Sàbat, P. Sekatski, A. Pirker, and W. Dür, "Entanglement purification by counting and locating errors with entangling measurements", Physical Review A 104 1, 012419 (2021).

[6] Cheng-Chen Luo, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Purification for hybrid logical qubit entanglement", Quantum Information Processing 21 8, 300 (2022).

[7] Kun Fang and Zi-Wen Liu, "No-Go Theorems for Quantum Resource Purification", Physical Review Letters 125 6, 060405 (2020).

[8] Si-Yi Chen, Gang Xu, Xiu-Bo Chen, Tao Shang, and Yi-Xian Yang, "Quantum Cooperative Multicast in a Quantum Hybrid Topology Network", Frontiers in Physics 10, 842035 (2022).

[9] Gui‐Long Jiang, Hai‐Rui Wei, Guo‐Zhu Song, and Ming Hua, "Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled‐Unitary Gates", Annalen der Physik 534 11, 2200317 (2022).

[10] Derek S. Wang, Tomáš Neuman, and Prineha Narang, "Dipole-coupled emitters as deterministic entangled photon-pair sources", Physical Review Research 2 4, 043328 (2020).

[11] Jude Alnas, Muneer Alshowkan, Nageswara S. V. Rao, Nicholas A. Peters, and Joseph M. Lukens, "Optimal resource allocation for flexible-grid entanglement distribution networks", Optics Express 30 14, 24375 (2022).

[12] Lu‐Cong Lu, Bao‐Cang Ren, Xin Wang, Mei Zhang, and Fu‐Guo Deng, "General Quantum Entanglement Purification Protocol using a Controlled‐Phase‐Flip Gate", Annalen der Physik 532 4, 2000011 (2020).

[13] Haoxiong Yan, Youpeng Zhong, Hung-Shen Chang, Audrey Bienfait, Ming-Han Chou, Christopher R. Conner, Étienne Dumur, Joel Grebel, Rhys G. Povey, and Andrew N. Cleland, "Entanglement Purification and Protection in a Superconducting Quantum Network", Physical Review Letters 128 8, 080504 (2022).

[14] Xuanqiang Zhao, Benchi Zhao, Zihe Wang, Zhixin Song, and Xin Wang, "Practical distributed quantum information processing with LOCCNet", npj Quantum Information 7 1, 159 (2021).

[15] Lan Zhou, Ze-Kai Liu, Zi-Xuan Xu, Yi-Lun Cui, Hai-Jiang Ran, and Yu-Bo Sheng, "Economical multi-photon polarization entanglement purification with Bell state", Quantum Information Processing 20 8, 257 (2021).

[16] Francisco Ferreira da Silva, Ariana Torres-Knoop, Tim Coopmans, David Maier, and Stephanie Wehner, "Optimizing entanglement generation and distribution using genetic algorithms", Quantum Science and Technology 6 3, 035007 (2021).

[17] Li Ding and Lee Spector, "Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits", Entropy 25 1, 93 (2023).

[18] Sebastian de Bone, Runsheng Ouyang, Kenneth Goodenough, and David Elkouss, "Protocols for Creating and Distilling Multipartite GHZ States With Bell Pairs", IEEE Transactions on Quantum Engineering 1, 1 (2020).

[19] Stefan Krastanov, Hamza Raniwala, Jeffrey Holzgrafe, Kurt Jacobs, Marko Lončar, Matthew J. Reagor, and Dirk R. Englund, "Optically Heralded Entanglement of Superconducting Systems in Quantum Networks", Physical Review Letters 127 4, 040503 (2021).

[20] Lan Zhou and Yu-Bo Sheng, "One-step device-independent quantum secure direct communication", Science China Physics, Mechanics & Astronomy 65 5, 250311 (2022).

[21] Gui-Long Jiang, Wen-Qiang Liu, and Hai-Rui Wei, "Heralded and high-efficient entanglement concentrations based on linear optics assisted by time-delay degree of freedom", Optics Express 30 26, 47836 (2022).

[22] Hamza Jnane, Brennan Undseth, Zhenyu Cai, Simon C. Benjamin, and Bálint Koczor, "Multicore Quantum Computing", Physical Review Applied 18 4, 044064 (2022).

[23] Benjamin Desef and Martin B. Plenio, "Optimizing quantum codes with an application to the loss channel with partial erasure information", Quantum 6, 667 (2022).

[24] F. Riera-Sàbat, P. Sekatski, A. Pirker, and W. Dür, "Entanglement-Assisted Entanglement Purification", Physical Review Letters 127 4, 040502 (2021).

[25] Cen-Xiao Huang, Xiao-Min Hu, Bi-Heng Liu, Lan Zhou, Yu-Bo Sheng, Chuan-Feng Li, and Guang-Can Guo, "Experimental one-step deterministic polarization entanglement purification", Science Bulletin 67 6, 593 (2022).

[26] Michele Amoretti, Mattia Pizzoni, and Stefano Carretta, "Enhancing distributed functional monitoring with quantum protocols", Quantum Information Processing 18 12, 371 (2019).

[27] Sumeet Khatri, Corey T. Matyas, Aliza U. Siddiqui, and Jonathan P. Dowling, "Practical figures of merit and thresholds for entanglement distribution in quantum networks", Physical Review Research 1 2, 023032 (2019).

[28] Sarah Jansen, Kenneth Goodenough, Sébastian de Bone, Dion Gijswijt, and David Elkouss, "Enumerating all bilocal Clifford distillation protocols through symmetry reduction", Quantum 6, 715 (2022).

[29] Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Purification of the residual entanglement", Optics Express 28 2, 2291 (2020).

[30] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Feasible measurement-based entanglement purification in linear optics", Optics Express 29 6, 9363 (2021).

[31] Narayanan Rengaswamy, Nithin Raveendran, Ankur Raina, and Bane Vasić, "Entanglement Purification with Quantum LDPC Codes and Iterative Decoding", arXiv:2210.14143, (2022).

[32] Jibing Yuan, Shiqing Tang, Xinwen Wang, and Dengyu Zhang, "One-step distillation of local-unitary-equivalent GHZ-type states", Quantum Information Processing 17 10, 259 (2018).

[33] Narayanan Rengaswamy, Ankur Raina, Nithin Raveendran, and Bane Vasić, "Distilling GHZ States using Stabilizer Codes", arXiv:2109.06248, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2023-03-23 05:15:22) and SAO/NASA ADS (last updated successfully 2023-03-23 05:15:23). 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.