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).

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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

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[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).

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[15] 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).

[16] Xuanxuan Xin, Shiwen He, Yongxing Li, and Chong Li, "Deterministic joint remote state preparation via a non-maximally entangled channel", Physica Scripta 98 6, 065103 (2023).

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[19] Fang-Fang Du, Gang Fan, Yi-Ming Wu, and Bao-Cang Ren, "Faithful and efficient hyperentanglement purification for spatial-polarization-time-bin photon system", Chinese Physics B 32 6, 060304 (2023).

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

[21] 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).

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

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