Parameter regimes for surpassing the PLOB bound with error-corrected qudit repeaters

Daniel Miller; Timo Holz; Hermann Kampermann; Dagmar Bruß

doi:10.22331/q-2019-12-16-216

Parameter regimes for surpassing the PLOB bound with error-corrected qudit repeaters

Daniel Miller, Timo Holz, Hermann Kampermann, and Dagmar Bruß

Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

Published:	2019-12-16, volume 3, page 216
Eprint:	arXiv:1906.05172v2
Doi:	https://doi.org/10.22331/q-2019-12-16-216
Citation:	Quantum 3, 216 (2019).

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Abstract

A potential quantum internet would open up the possibility of realizing numerous new applications, including provably secure communication. Since losses of photons limit long-distance, direct quantum communication and wide-spread quantum networks, quantum repeaters are needed. The so-called PLOB-repeaterless bound [Pirandola et al., Nat. Commun. 8, 15043 (2017)] is a fundamental limit on the quantum capacity of direct quantum communication. Here, we analytically derive the quantum-repeater gain for error-corrected, one-way quantum repeaters based on higher-dimensional qudits for two different physical encodings: Fock and multimode qudits. We identify parameter regimes in which such quantum repeaters can surpass the PLOB-repeaterless bound and systematically analyze how typical parameters manifest themselves in the quantum-repeater gain. This benchmarking provides a guideline for the implementation of error-corrected qudit repeaters.

► BibTeX data

@article{Miller2019parameterregimes,
  doi = {10.22331/q-2019-12-16-216},
  url = {https://doi.org/10.22331/q-2019-12-16-216},
  title = {Parameter regimes for surpassing the {PLOB} bound with error-corrected qudit repeaters},
  author = {Miller, Daniel and Holz, Timo and Kampermann, Hermann and Bru{\ss{}}, Dagmar},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {216},
  month = dec,
  year = {2019}
}

► References

[1] M. Riedel, D. Binosi, R. Thew, and T. Calarco, The European quantum technologies flagship programme, Quantum Sci. Technol. 2, 030501 (2017).
https://doi.org/10.1088/2058-9565/aa6aca

[2] A. Acín, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. Riedel, P. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. Wilhelm, The quantum technologies roadmap: a European community view, New J. Phys. 20, 080201 (2018).
https://doi.org/10.1088/1367-2630/aad1ea

[3] S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, 6412 (2018).
https://doi.org/10.1126/science.aam9288

[4] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced positioning and clock synchronization, Nature 412, 417 (2001).
https://doi.org/10.1038/35086525

[5] M. Christandl and S. Wehner, Quantum Anonymous Transmissions, ASIACRYPT 2005, 217-235 (2005).
https://doi.org/10.1007/11593447_12

[6] D. Gottesman, T. Jennewein, and S. Croke, Longer-Baseline Telescopes Using Quantum Repeaters, Phys. Rev. Lett. 109, 070503 (2012).
https://doi.org/10.1103/PhysRevLett.109.070503

[7] E. Khabiboulline, J. Borregaard, K. De Greve, and M. Lukin, Optical Interferometry with Quantum Networks, Phys. Rev. Lett. 123, 070504 (2019).
https://doi.org/10.1103/PhysRevLett.123.070504

[8] C. Bennett, G. Brassard, Public Key Distribution and Coin Tossing, Theor. Comput. Sci. 560, 7 (2014).
https://doi.org/10.1016/j.tcs.2014.05.025

[9] A. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett. 67, 661 (1991).
https://doi.org/10.1103/PhysRevLett.67.661

[10] D. Bruß, Optimal Eavesdropping in Quantum Cryptography with Six States, Phys. Rev. Lett. 81, 3018 (1998).
https://doi.org/10.1103/PhysRevLett.81.3018

[11] A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M. Li, D. Nolan, A. Martin, and H. Zbinden, Secure Quantum Key Distribution over 421 km of Optical Fiber, Phys. Rev. Lett. 121, 190502, (2018).
https://doi.org/10.1103/PhysRevLett.121.190502

[12] M. Takeoka, S. Guha, and M. Wilde, Fundamental rate-loss tradeoff for optical quantum key distribution, Nat. Comm. 5, 5235 (2014).
https://doi.org/10.1038/ncomms6235

[13] M. Christandl and A. Müller-Hermes, Relative Entropy Bounds on Quantum, Private and Repeater Capacities, Commun. Math. Phys. 353, 821 (2017).
https://doi.org/10.1007/s00220-017-2885-y

[14] S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Fundamental limits of repeaterless quantum communications, Nat. Commun. 8, 15043 (2017).
https://doi.org/10.1038/ncomms15043

[15] H. Briegel, W. Dür, J. Cirac, and P. Zoller, Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication, Phys. Rev. Lett. 81, 5932 (1998).
https://doi.org/10.1103/PhysRevLett.81.5932

[16] P. van Loock, T. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. Munro, and Y. Yamamoto, Hybrid Quantum Repeater Using Bright Coherent Light, Phys. Rev. Lett. 96, 240501 (2006).
https://doi.org/10.1103/PhysRevLett.96.240501

[17] L. Jiang, J. Taylor, K. Nemoto, W. Munro, R. Van Meter, and M. Lukin, Quantum repeater with encoding, Phys. Rev. A 79, 032325 (2009).
https://doi.org/10.1103/PhysRevA.79.032325

[18] A. Fowler, D. Wang, C. Hill, T. Ladd, R. Van Meter, and L. Hollenberg, Surface Code Quantum Communication, Phys. Rev. Lett. 104, 180503 (2010).
https://doi.org/10.1103/PhysRevLett.104.180503

[19] S. Muralidharan, J. Kim, N. Lütkenhaus, M Lukin, and L. Jiang, Ultrafast and Fault-Tolerant Quantum Communication across Long Distances, Phys. Rev. Lett. 112, 250501 (2014).
https://doi.org/10.1103/PhysRevLett.112.250501

[20] S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. Lukin and L. Jiang, Optimal architectures for long distance quantum communication, Sci Rep. 6, 20463 (2016).
https://doi.org/10.1038/srep20463

[21] D. Luong, L. Jiang, J. Kim, and N. Lütkenhaus, Overcoming lossy channel bounds using a single quantum repeater node, Appl. Phys. B 112: 96 (2016).
https://doi.org/10.1007/s00340-016-6373-4

[22] F. Rozpędek., K. Goodenough, J. Ribeiro, N. Kalb, V. Caprara Vivoli, A. Reiserer, R. Hanson, S. Wehner, and D. Elkouss, Parameter regimes for a single sequential quantum repeater, Quantum Sci. Technol. 3, 034002 (2018).
https://doi.org/10.1088/2058-9565/aab31b

[23] F. Rozpędek, R. Yehia, K. Goodenough, M. Ruf, P. Humphreys, R. Hanson, S. Wehner, and D. Elkouss, Near-term quantum-repeater experiments with nitrogen-vacancy centers: Overcoming the limitations of direct transmission, Phys. Rev. A 99, 052330 (2019).
https://doi.org/10.1103/PhysRevA.99.052330

[24] M. Lucamarini, Z. Yuan, J. Dynes, and A. Shields, Overcoming the rate-distance limit of quantum key distribution without quantum repeaters, Nature 557, 400 (2018).
https://doi.org/10.1038/s41586-018-0066-6

[25] M. Curty, K. Azuma, and H. Lo, Simple security proof of twin-field type quantum key distribution protocol, npj Quantum Inf. 5, 64 (2019).
https://doi.org/10.1038/s41534-019-0175-6

[26] F. Grasselli and M. Curty, Practical decoy-state method for twin-field quantum key distribution, New J. Phys. 21, 073001 (2019).
https://doi.org/10.1088/1367-2630/ab2b00

[27] M. Minder, M. Pittaluga, G. Roberts, M. Lucamarini, J. Dynes, Z. Yuan, and A. Shields, Experimental quantum key distribution beyond the repeaterless secret key capacity, Nat. Photonics 13, 334 (2019).
https://doi.org/10.1038/s41566-019-0377-7

[28] S. Wang, D. He, Z. Yin, F. Lu, C. Cui, W. Chen, Z. Zhou, G. Guo, and Z. Han, Beating the Fundamental Rate-Distance Limit in a Proof-of-Principle Quantum Key Distribution System, Phys. Rev. X 9, 021046 (2019).
https://doi.org/10.1103/PhysRevX.9.021046

[29] X. Zhong, J. Hu, M. Curty, L. Qian, and H. Lo, Proof-of-Principle Experimental Demonstration of Twin-Field Type Quantum Key Distribution, Phys. Rev. Lett. 123, 100506 (2019).
https://doi.org/10.1103/PhysRevLett.123.100506

[30] F. Ewert and P. van Loock, Ultrafast fault-tolerant long-distance quantum communication with static linear optics, Phys. Rev. A 95, 012327 (2017).
https://doi.org/10.1103/PhysRevA.95.012327

[31] F. Schmidt and P. van Loock, Efficiencies of logical Bell measurements on Calderbank-Shor-Steane codes with static linear optics, Phys. Rev. A 99, 062308 (2019).
https://doi.org/10.1103/PhysRevA.99.062308

[32] S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise, Phys. Rev. X 9, 041042 (2019).
https://doi.org/10.1103/PhysRevX.9.041042

[33] S. Muralidharan, C. Zou, L. Li, J. Wen, and L. Jiang, Overcoming erasure errors with multilevel systems, New J. Phys. 19, 013026 (2017).
https://doi.org/10.1088/1367-2630/aa573a

[34] S. Muralidharan, C. Zou, L. Li, and L. Jiang, One-way quantum repeaters with quantum Reed-Solomon codes, Phys. Rev. A 97, 052316 (2018).
https://doi.org/10.1103/PhysRevA.97.052316

[35] D. Miller, T. Holz, H. Kampermann, and D. Bruß, Propagation of generalized Pauli errors in qudit Clifford circuits, Phys. Rev. A 98, 052316 (2018).
https://doi.org/10.1103/PhysRevA.98.052316

[36] M. Bergmann and P. van Loock, Hybrid quantum repeater for qudits, Phys. Rev. A 99, 032349 (2019).
https://doi.org/10.1103/PhysRevA.99.032349

[37] S. Abruzzo, S. Bratzik, N. Bernardes, H. Kampermann, P. van Loock, and D. Bruß, Quantum repeaters and quantum key distribution: Analysis of secret-key rates, Phys. Rev. A 87, 052315 (2013).
https://doi.org/10.1103/PhysRevA.87.052315

[38] S. Bratzik, H. Kampermann, and D. Bruß, Secret key rates for an encoded quantum repeater, Phys. Rev. A 89, 032335 (2014).
https://doi.org/10.1103/PhysRevA.89.032335

[39] M. Epping, H. Kampermann, and D. Bruß, On the error analysis of quantum repeaters with encoding, Appl. Phys. B 122: 54 (2016).
https://doi.org/10.1007/s00340-015-6314-7

[40] M. Epping, H. Kampermann, and D. Bruß, Large-scale quantum networks based on graphs, New J. Phys. 18, 53036 (2016).
https://doi.org/10.1088/1367-2630/18/5/053036

[41] M. Epping, H. Kampermann, and D. Bruß, Robust entanglement distribution via quantum network coding, New J. Phys. 18, 103052 (2016).
https://doi.org/10.1088/1367-2630/18/10/103052

[42] K. Vollbrecht and M. Wolf, Efficient distillation beyond qubits, Phys. Rev. A 67, 012303 (2003).
https://doi.org/10.1103/PhysRevA.67.012303

[43] S. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
https://doi.org/10.1103/RevModPhys.77.513

[44] C. Weedbrook, S. Pirandola, R. García-Patrón, N. Cerf, T. Ralph, J. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
https://doi.org/10.1103/RevModPhys.84.621

[45] J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication, Phys. Rev. Lett. 82, 2594 (1999).
https://doi.org/10.1103/PhysRevLett.82.2594

[46] H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, Creating high dimensional time-bin entanglement using mode-locked lasers, Quant. Inf. Comp. 2, 425 (2002).
https://doi.org/10.26421/QIC2.6

[47] T. Zhong, H. Zhou, R. Horansky, C. Lee, V. Verma, A. Lita, A. Restelli, J. Bienfang, R. Mirin, T. Gerrits, S. Nam, F. Marsili, M. Shaw, Z. Zhang, L. Wang, D. Englund, G. Wornell, J. Shapiro, and F. Wong, Photon-efficient quantum key distribution using time-energy entanglement with high-dimensional encoding, New J. Phys. 17, 022002 (2015).
https://doi.org/10.1088/1367-2630/17/2/022002

[48] N. Montaut, O. Magaña-Loaiza, T. Bartley, V. Verma, S. Nam, R. Mirin, C. Silberhorn, and T. Gerrits, Compressive characterization of telecom photon pairs in the spatial and spectral degrees of freedom, Optica 5, 1418 (2018).
https://doi.org/10.1364/OPTICA.5.001418

[49] B. Brecht, D. Reddy, C. Silberhorn and M. Raymer, Photon Temporal Modes: A Complete Framework for Quantum Information Science, Phys. Rev. X 5, 041017 (2015).
https://doi.org/10.1103/PhysRevX.5.041017

[50] V. Ansari, J. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, Tomography and Purification of the Temporal-Mode Structure of Quantum Light, Phys. Rev. Lett. 120, 213601 (2018).
https://doi.org/10.1103/PhysRevLett.120.213601

[51] L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 45, 8185 (1992).
https://doi.org/10.1103/PhysRevA.45.8185

[52] G. Calvo, A. Picón, and E. Bagan, Quantum field theory of photons with orbital angular momentum, Phys. Rev. A 73, 013805 (2006).
https://doi.org/10.1103/PhysRevA.73.013805

[53] W. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, Quantum orbital angular momentum of elliptically symmetric light, Phys. Rev. A 87, (2013).
https://doi.org/10.1103/PhysRevA.87.033806

[54] M. Krenn, M. Malik, M. Erhard, and A. Zeilinger, Orbital angular momentum of photons and the entanglement of Laguerre–Gaussian modes, Phil. Trans. R. Soc. A 375: 20150442 (2017).
https://doi.org/10.1098/rsta.2015.0442

[55] E. Knill, Group Representations, Error Bases and Quantum Codes, Technical Report LAUR-96-2807, Los Alamos National Laboratory, arXiv:9608049 [quant-ph] (1996).
arXiv:quant-ph/9608049

[56] D. Gottesman, Fault-Tolerant Quantum Computation with Higher-Dimensional Systems, Chaos Solitons Fractals 10, 1749-1758 (1999).
https://doi.org/10.1016/S0960-0779(98)00218-5

[57] D. Lidar and T. Brun, Quantum Error Correction, Cambridge University Press (2013).
https://doi.org/10.1017/CBO9781139034807

[58] R. Cleve, D. Gottesman and H. Lo, How to share a quantum secret, Phys. Rev. Lett. 83, 648 (1999).
https://doi.org/10.1103/PhysRevLett.83.648

[59] D. Aharonov and M. Ben-Or, Fault-Tolerant Quantum Computation with Constant Error Rate, SIAM J. Comput. 38(4), 1207 (2008).
https://doi.org/10.1137/S0097539799359385

[60] A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52(11), 4892 (2006).
https://doi.org/10.1109/TIT.2006.883612

[61] A. Cross, Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes, MIT 1721.1/44407 (2008).
http://hdl.handle.net/1721.1/44407

[62] N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).
https://doi.org/10.1103/RevModPhys.74.145

[63] R. Hadfield, Single-photon detectors for optical quantum information applications, Nat. Photonics 3, 696 (2009).
https://doi.org/10.1038/nphoton.2009.230

[64] B. Hacker, S. Welte, G. Rempe, and S. Ritter, A photon–photon quantum gate based on a single atom in an optical resonator Nature 536, 193 (2016).
https://doi.org/10.1038/nature18592

[65] G. Alber, A. Delgado, N. Gisin, and I. Jex, Efficient bipartite quantum state purification in arbitrary dimensional Hilbert spaces, J. Phys. A: Math. Gen. 34 8821 (2001).
https://doi.org/10.1088/0305-4470/34/42/307

[66] R. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. Albert, L. Frunzio, L. Jiang, and R. Schoelkopf, Cavity State Manipulation Using Photon-Number Selective Phase Gates, Phys. Rev. Lett. 115, 137002 (2015).
https://doi.org/10.1103/PhysRevLett.115.137002

[67] Y. Cho, G. Campbell, J. Everett, J. Bernu, D. Higginbottom, M. Cao, J. Geng, N. Robins, P. Lam, and B. Buchler, Highly efficient optical quantum memory with long coherence time in cold atoms, Optica 3, 100 (2016).
https://doi.org/10.1364/OPTICA.3.000100

[68] P. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Yao, S. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. Twitchen, J. Cirac, and M. Lukin, Room-Temperature Quantum Bit Memory Exceeding One Second, Sci Rep. 336, 6086 1283 (2012).
https://doi.org/10.1126/science.1220513

[69] M. Abobeih, J. Cramer, M. Bakker, N. Kalb, M. Markham, D. Twitchen, and T. Taminiau, One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment, Nat. Comm. 9, 2552 (2018).
https://doi.org/10.1038/s41467-018-04916-z

[70] C. Bradley, J. Randall, M. Abobeih, R. Berrevoets, M. Degen, M. Bakker, M. Markham, D. Twitchen, T. Taminiau, A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute, Phys. Rev. X 9, 031045 (2019).
https://doi.org/10.1103/PhysRevX.9.031045

[71] Y. Wang, M. Um, J. Zhang, S. An, M. Lyu, J. Zhang, L. Duan, D. Yum, and K. Kim, Single-qubit quantum memory exceeding ten-minute coherence time, Nat. Photonics 11, 646 (2017).
https://doi.org/10.1038/s41566-017-0007-1

[72] M. Körber, O. Morin, S. Langenfeld, A. Neuzner, S. Ritter, and G. Rempe, Decoherence-protected memory for a single-photon qubit, Nat. Photonics 12, 18 (2018).
https://doi.org/10.1038/s41566-017-0050-y

[73] K. Brown, K. Dani, D. Stamper-Kurn, and K. Whaley, Deterministic optical Fock-state generation, Phys. Rev. A 67, 043818 (2003).
https://doi.org/10.1103/PhysRevA.67.043818

[74] E. Waks, E. Diamanti, and Y. Yamamoto, Generation of photon number states, New J. Phys. 8, 4 (2006).
https://doi.org/10.1088/1367-2630/8/1/004

[75] J. Tiedau, T. Bartley, G. Harder, A. Lita, S. Nam, T. Gerrits, and C. Silberhorn, On the scalability of parametric down-conversion for generating higher-order Fock states, Phys. Rev. A 100, 041802(R) (2019).
https://doi.org/10.1103/PhysRevA.100.041802

[76] N. Brown, M. Newman, and K. Brown, Handling Leakage with Subsystem Codes, New J. Phys. 21 073055 (2019).
https://doi.org/10.1088/1367-2630/ab3372

[77] P. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R) (1995).
https://doi.org/10.1103/PhysRevA.52.R2493

[78] D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A 73, 012340 (2006).
https://doi.org/10.1103/PhysRevA.73.012340

[79] P. Aliferis and A. Cross, Subsystem Fault Tolerance with the Bacon-Shor Code, Phys. Rev. Lett. 98, 220502 (2007).
https://doi.org/10.1103/PhysRevLett.98.220502

[80] S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, Subsystem surface codes with three-qubit check operators, Quant. Inf. Comp. 13, 963 (2013).
https://doi.org/10.26421/QIC13.11-12

[81] M. Li, D. Miller, M. Newman, Y. Wu, and K. Brown, 2D Compass Codes, Phys. Rev. X 9, 021041 (2019).
https://doi.org/10.1103/PhysRevX.9.021041

[82] T. Yoder, Optimal quantum subsystem codes in 2-dimensions, Phys. Rev. A 99, 052333 (2019).
https://doi.org/10.1103/PhysRevA.99.052333

[83] D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
https://doi.org/10.1103/PhysRevA.64.012310

[84] C. Vuillot, H. Asasi, Y. Wang, L. Pryadko, and B. Terhal, Quantum error correction with the toric Gottesman-Kitaev-Preskill code, Phys. Rev. A 99, 032344 (2019).
https://doi.org/10.1103/PhysRevA.99.032344

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