Blueprint For A Scalable Photonic Fault-Tolerant Quantum Computer

Abstract

Photonics is the platform of choice to build a modular, easy-to-network quantum computer operating at room temperature. However, no concrete architecture has been presented so far that exploits both the advantages of qubits encoded into states of light and the modern tools for their generation. Here we propose such a design for a scalable fault-tolerant photonic quantum computer informed by the latest developments in theory and technology. Central to our architecture is the generation and manipulation of three-dimensional resource states comprising both bosonic qubits and squeezed vacuum states. The proposal exploits state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states. Moreover, the architecture is based on two-dimensional integrated photonic chips used to produce a qubit cluster state in one temporal and two spatial dimensions. By reducing the experimental challenges as compared to existing architectures and by enabling room-temperature quantum computation, our design opens the door to scalable fabrication and operation, which may allow photonics to leap-frog other platforms on the path to a quantum computer with millions of qubits.

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Popular summary

The prototypical quantum computer ought to be universal, fault-tolerant, and scalable: ready to run any quantum algorithm, detect and correct the accruing errors, and accommodate scores of qubits. But there is a lot more to the design of a practical quantum computer, where one also looks for qualities like modularity, networkability, speed, and room-temperature operation. The photonic platform – a computer based on quantum states of light – gives perhaps the best hope to satisfy these criteria. In our paper we present the first detailed, comprehensive, top-down blueprint for such a computer. Our main theoretical innovation is to utilize a hybrid quantum state of light consisting of powerful but experimentally challenging checkerboard states, and more limited but easier-to-produce squeezed states. We lay out a complete mechanism for generating, processing, and measuring this state in the course of a fault-tolerant computation. The device we propose needs only planar, specialized, moderately-sized integrated photonic chips, a technology familiar to the telecommunications industry. The cryostats it needs – for the moment – are small and commercially available. And the quantum processor in our design establishes a brisk clock speed. These features, made possible by the flexibility of the photonic platform and by theoretical advancements in encoding and decoding that we detail, bring us closer to an operational quantum computer and its remarkable consequences.

► BibTeX data

@article{Bourassa2021blueprintscalable, doi = {10.22331/q-2021-02-04-392}, url = {https://doi.org/10.22331/q-2021-02-04-392}, title = {Blueprint for a {S}calable {P}hotonic {F}ault-{T}olerant {Q}uantum {C}omputer}, author = {Bourassa, J. Eli and Alexander, Rafael N. and Vasmer, Michael and Patil, Ashlesha and Tzitrin, Ilan and Matsuura, Takaya and Su, Daiqin and Baragiola, Ben Q. and Guha, Saikat and Dauphinais, Guillaume and Sabapathy, Krishna K. and Menicucci, Nicolas C. and Dhand, Ish}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {5}, pages = {392}, month = feb, year = {2021}<br /> }

► References

[1] Nicolas C. Menicucci. Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. Phys. Rev. Lett., 112: 120504, Mar 2014. 10.1103/PhysRevLett.112.120504.
https://doi.org/10.1103/PhysRevLett.112.120504

[2] Ulrik L Andersen, Jonas S Neergaard-Nielsen, Peter van Loock, and Akira Furusawa. Hybrid discrete- and continuous-variable quantum information. Nat. Phys., 11 (9): 713–719, 2015. 10.1038/nphys3410.
https://doi.org/10.1038/nphys3410

[3] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409 (6816): 46–52, 2001. 10.1038/35051009.
https://doi.org/10.1038/35051009

[4] Casey R. Myers and Timothy C. Ralph. Coherent state topological cluster state production. New J. Phys., 13 (11): 115015, 2011. 10.1088/1367-2630/13/11/115015.
https://doi.org/10.1088/1367-2630/13/11/115015

[5] Kosuke Fukui, Akihisa Tomita, Atsushi Okamoto, and Keisuke Fujii. High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction. Phys. Rev. X, 8 (2): 21054, 2018a. 10.1103/PhysRevX.8.021054.
https://doi.org/10.1103/PhysRevX.8.021054

[6] James M. Auger, Hussain Anwar, Mercedes Gimeno-Segovia, Thomas M. Stace, and Dan E. Browne. Fault-tolerant quantum computation with nondeterministic entangling gates. Phys. Rev. A, 97 (3): 5–9, 2018. 10.1103/PhysRevA.97.030301.
https://doi.org/10.1103/PhysRevA.97.030301

[7] Kyungjoo Noh and Christopher Chamberland. Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code. Phys. Rev. A, 101: 012316, Jan 2020. 10.1103/PhysRevA.101.012316.
https://doi.org/10.1103/PhysRevA.101.012316

[8] Nicolas C. Menicucci, Peter van Loock, Mile Gu, Christian Weedbrook, Timothy C. Ralph, and Michael A. Nielsen. Universal quantum computation with continuous-variable cluster states. Phys. Rev. Lett., 97: 110501, Sep 2006. 10.1103/PhysRevLett.97.110501.
https://doi.org/10.1103/PhysRevLett.97.110501

[9] Shota Yokoyama, Ryuji Ukai, Seiji C Armstrong, Chanond Sornphiphatphong, Toshiyuki Kaji, Shigenari Suzuki, Jun-ichi Yoshikawa, Hidehiro Yonezawa, Nicolas C. Menicucci, and Akira Furusawa. Ultra-large-scale continuous-variable cluster states multiplexed in the time domain. Nat. Photonics, 7 (12): 982–986, 2013. 10.1038/nphoton.2013.287.
https://doi.org/10.1038/nphoton.2013.287

[10] Moran Chen, Nicolas C. Menicucci, and Olivier Pfister. Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb. Phys. Rev. Lett., 112 (12): 120505, 2014. 10.1103/PhysRevLett.112.120505.
https://doi.org/10.1103/PhysRevLett.112.120505

[11] Jun-ichi Yoshikawa, Shota Yokoyama, Toshiyuki Kaji, Chanond Sornphiphatphong, Yu Shiozawa, Kenzo Makino, and Akira Furusawa. Invited article: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing. APL Photonics, 1 (6): 060801, 2016. 10.1063/1.4962732.
https://doi.org/10.1063/1.4962732

[12] Rafael N. Alexander, Pei Wang, Niranjan Sridhar, Moran Chen, Olivier Pfister, and Nicolas C. Menicucci. One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator. Phys. Rev. A, 94: 032327, Sep 2016. 10.1103/PhysRevA.94.032327.
https://doi.org/10.1103/PhysRevA.94.032327

[13] Mikkel V. Larsen, Xueshi Guo, Casper R. Breum, Jonas S. Neergaard-Nielsen, and Ulrik L. Andersen. Deterministic generation of a two-dimensional cluster state. Science, 366 (6463): 369–372, Oct 2019. 10.1126/science.aay4354.
https://doi.org/10.1126/science.aay4354

[14] Mikkel V. Larsen, Jonas S. Neergaard-Nielsen, and Ulrik L. Andersen. Architecture and noise analysis of continuous-variable quantum gates using two-dimensional cluster states. Phys. Rev. A, 102: 042608, Oct 2020. 10.1103/PhysRevA.102.042608.
https://doi.org/10.1103/PhysRevA.102.042608

[15] Rafael N. Alexander, Shota Yokoyama, Akira Furusawa, and Nicolas C. Menicucci. Universal quantum computation with temporal-mode bilayer square lattices. Phys. Rev. A, 97: 032302, Mar 2018. 10.1103/PhysRevA.97.032302.
https://doi.org/10.1103/PhysRevA.97.032302

[16] Warit Asavanant, Yu Shiozawa, Shota Yokoyama, Baramee Charoensombutamon, Hiroki Emura, Rafael N. Alexander, Shuntaro Takeda, Jun-ichi Yoshikawa, Nicolas C. Menicucci, Hidehiro Yonezawa, and et al. Generation of time-domain-multiplexed two-dimensional cluster state. Science, 366 (6463): 373–376, Oct 2019. 10.1126/science.aay2645.
https://doi.org/10.1126/science.aay2645

[17] Pei Wang, Moran Chen, Nicolas C. Menicucci, and Olivier Pfister. Weaving quantum optical frequency combs into continuous-variable hypercubic cluster states. Phys. Rev. A, 90 (3): 032325, 2014. 10.1103/PhysRevA.90.032325.
https://doi.org/10.1103/PhysRevA.90.032325

[18] Bo-Han Wu, Rafael N Alexander, Shuai Liu, and Zheshen Zhang. Quantum computing with multidimensional continuous-variable cluster states in a scalable photonic platform. Phys. Rev. Res., 2 (2): 023138, 2020. 10.1103/PhysRevResearch.2.023138.
https://doi.org/10.1103/PhysRevResearch.2.023138

[19] Kosuke Fukui, Warit Asavanant, and Akira Furusawa. Temporal-mode continuous-variable three-dimensional cluster state for topologically protected measurement-based quantum computation. Phys. Rev. A, 102: 032614, Sep 2020. 10.1103/PhysRevA.102.032614.
https://doi.org/10.1103/PhysRevA.102.032614

[20] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64: 012310, Jun 2001. 10.1103/PhysRevA.64.012310.
https://doi.org/10.1103/PhysRevA.64.012310

[21] Austin P. Lund, Timothy C. Ralph, and Henry L. Haselgrove. Fault-tolerant linear optical quantum computing with small-amplitude coherent states. Phys. Rev. Lett., 100: 030503, Jan 2008. 10.1103/PhysRevLett.100.030503.
https://doi.org/10.1103/PhysRevLett.100.030503

[22] Kosuke Fukui, Akihisa Tomita, Atsushi Okamoto, and Keisuke Fujii. High-threshold fault-tolerant quantum computation with analog quantum error correction. Phys. Rev. X, 8: 021054, May 2018b. 10.1103/PhysRevX.8.021054.
https://doi.org/10.1103/PhysRevX.8.021054

[23] Kosuke Fukui. High-threshold fault-tolerant quantum computation with the GKP qubit and realistically noisy devices. arXiv preprint arXiv:1906.09767, 2019.
arXiv:1906.09767

[24] Terry Rudolph. Why I am optimistic about the silicon-photonic route to quantum computing. APL Photonics, 2 (3): 030901, 2017. 10.1063/1.4976737.
https://doi.org/10.1063/1.4976737

[25] M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch. Generation of arbitrary quantum states of traveling fields. Phys. Rev. A, 59: 1658–1661, Feb 1999. 10.1103/PhysRevA.59.1658.
https://doi.org/10.1103/PhysRevA.59.1658

[26] Jaromír Fiurášek, Raúl García-Patrón, and Nicolas J. Cerf. Conditional generation of arbitrary single-mode quantum states of light by repeated photon subtractions. Phys. Rev. A, 72: 033822, Sep 2005. 10.1103/PhysRevA.72.033822.
https://doi.org/10.1103/PhysRevA.72.033822

[27] Krishna Kumar Sabapathy, Haoyu Qi, Josh Izaac, and Christian Weedbrook. Production of photonic universal quantum gates enhanced by machine learning. Phys. Rev. A, 100: 012326, Jul 2019. 10.1103/PhysRevA.100.012326.
https://doi.org/10.1103/PhysRevA.100.012326

[28] Daiqin Su, Casey R. Myers, and Krishna Kumar Sabapathy. Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors. Phys. Rev. A, 100: 052301, Nov 2019. 10.1103/PhysRevA.100.052301.
https://doi.org/10.1103/PhysRevA.100.052301

[29] N. Quesada, L. G. Helt, J. Izaac, J. M. Arrazola, R. Shahrokhshahi, C. R. Myers, and K. K. Sabapathy. Simulating realistic non-Gaussian state preparation. Phys. Rev. A, 100: 022341, Aug 2019. 10.1103/PhysRevA.100.022341.
https://doi.org/10.1103/PhysRevA.100.022341

[30] Ilan Tzitrin, J. Eli Bourassa, Nicolas C. Menicucci, and Krishna Kumar Sabapathy. Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes. Phys. Rev. A, 101: 032315, Mar 2020. 10.1103/PhysRevA.101.032315.
https://doi.org/10.1103/PhysRevA.101.032315

[31] Hilma M Vasconcelos, Liliana Sanz, and Scott Glancy. All-optical generation of states for “Encoding a qubit in an oscillator”. Opt. Lett., 35 (19): 3261–3263, 2010. 10.1364/OL.35.003261.
https://doi.org/10.1364/OL.35.003261

[32] Daniel J. Weigand and Barbara M. Terhal. Generating grid states from Schrödinger-cat states without postselection. Phys. Rev. A, 97: 022341, Feb 2018. 10.1103/PhysRevA.97.022341.
https://doi.org/10.1103/PhysRevA.97.022341

[33] Miller Eaton, Rajveer Nehra, and Olivier Pfister. Non-Gaussian and Gottesman–Kitaev–Preskill state preparation by photon catalysis. New J. Phys., 21 (11): 113034, nov 2019. 10.1088/1367-2630/ab5330.
https://doi.org/10.1088/1367-2630/ab5330

[34] Keith R. Motes, Ben Q. Baragiola, Alexei Gilchrist, and Nicolas C. Menicucci. Encoding qubits into oscillators with atomic ensembles and squeezed light. Phys. Rev. A, 95: 053819, May 2017. 10.1103/PhysRevA.95.053819.
https://doi.org/10.1103/PhysRevA.95.053819

[35] Stefano Pirandola, Stefano Mancini, David Vitali, and Paolo Tombesi. Generating continuous variable quantum codewords in the near-field atomic lithography. J. Phys. B, 39 (4): 997, 2006. 10.1088/0953-4075/39/4/023.
https://doi.org/10.1088/0953-4075/39/4/023

[36] Adriana E. Lita, Aaron J. Miller, and Sae Woo Nam. Counting near-infrared single-photons with 95% efficiency. Opt. Express, 16 (5): 3032, 2008. 10.1364/oe.16.003032.
https://doi.org/10.1364/oe.16.003032

[37] J. Sperling, D. S. Phillips, J. F. F Bulmer, G. S. Thekkadath, A. Eckstein, T. A. W. Wolterink, J. Lugani, S. W. Nam, A. Lita, T. Gerrits, W. Vogel, G. S. Agarwal, C. Silberhorn, and I. A. Walmsley. Detector-agnostic phase-space distributions. Phys. Rev. Lett., 124: 013605, Jan 2020. 10.1103/PhysRevLett.124.013605.
https://doi.org/10.1103/PhysRevLett.124.013605

[38] G. S. Thekkadath, D. S. Phillips, J. F. F. Bulmer, W. R. Clements, A. Eckstein, B. A. Bell, J. Lugani, T. A. W. Wolterink, A. Lita, S. W. Nam, T. Gerrits, C. G. Wade, and I. A. Walmsley. Tuning between photon-number and quadrature measurements with weak-field homodyne detection. Phys. Rev. A, 101: 031801, Mar 2020. 10.1103/PhysRevA.101.031801.
https://doi.org/10.1103/PhysRevA.101.031801

[39] V. D. Vaidya, B. Morrison, L. G. Helt, R. Shahrokshahi, D. H. Mahler, M. J. Collins, K. Tan, J. Lavoie, A. Repingon, M. Menotti, N. Quesada, R. C. Pooser, A. E. Lita, T. Gerrits, S. W. Nam, and Z. Vernon. Broadband quadrature-squeezed vacuum and nonclassical photon number correlations from a nanophotonic device. Sci. Adv., 6 (39): eaba9186, September 2020. 10.1126/sciadv.aba9186.
https://doi.org/10.1126/sciadv.aba9186

[40] Robert Raussendorf, Sergey Bravyi, and Jim Harrington. Long-range quantum entanglement in noisy cluster states. Phys. Rev. A, 71: 062313, Jun 2005. 10.1103/PhysRevA.71.062313.
https://doi.org/10.1103/PhysRevA.71.062313

[41] Robert Raussendorf, Jim Harrington, and Kovid Goyal. A fault-tolerant one-way quantum computer. Ann. Phys. (N. Y.), 321 (9): 2242–2270, 2006. 10.1016/j.aop.2006.01.012.
https://doi.org/10.1016/j.aop.2006.01.012

[42] Robert Raussendorf, Jim Harrington, and Kovid Goyal. Topological fault-tolerance in cluster state quantum computation. New J. Phys., 9 (6): 199, 2007. 10.1088/1367-2630/9/6/199.
https://doi.org/10.1088/1367-2630/9/6/199

[43] Giacomo Pantaleoni, Ben Q. Baragiola, and Nicolas C. Menicucci. Modular bosonic subsystem codes. Phys. Rev. Lett., 125 (4): 040501, 2020. 10.1103/PhysRevLett.125.040501.
https://doi.org/10.1103/PhysRevLett.125.040501

[44] Stephen D Bartlett, Barry C Sanders, Samuel L Braunstein, and Kae Nemoto. Efficient classical simulation of continuous variable quantum information processes. Phys. Rev. Lett., 88 (9): 097904, 2002. 10.1103/PhysRevLett.88.097904.
https://doi.org/10.1103/PhysRevLett.88.097904

[45] Thomas M Stace, Sean D Barrett, and Andrew C Doherty. Thresholds for topological codes in the presence of loss. Phys. Rev. Lett., 102 (20): 200501, 2009. 10.1103/PhysRevLett.102.200501.
https://doi.org/10.1103/PhysRevLett.102.200501

[46] Sean D Barrett and Thomas M Stace. Fault tolerant quantum computation with very high threshold for loss errors. Phys. Rev. Lett., 105 (20): 200502, 2010. 10.1103/PhysRevLett.105.200502.
https://doi.org/10.1103/PhysRevLett.105.200502

[47] Adam C. Whiteside and Austin G. Fowler. Upper bound for loss in practical topological-cluster-state quantum computing. Phys. Rev. A, 90: 052316, Nov 2014. 10.1103/PhysRevA.90.052316.
https://doi.org/10.1103/PhysRevA.90.052316

[48] Henning Vahlbruch, Moritz Mehmet, Karsten Danzmann, and Roman Schnabel. Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency. Phys. Rev. Lett., 117: 110801, Sep 2016. 10.1103/PhysRevLett.117.110801.
https://doi.org/10.1103/PhysRevLett.117.110801

[49] M.J. Collins, C. Xiong, I.H. Rey, T.D. Vo, J. He, S. Shahnia, C. Reardon, T.F. Krauss, M.J. Steel, A.S. Clark, and B.J. Eggleton. Integrated spatial multiplexing of heralded single-photon sources. Nat. Commun., 4 (1), Oct 2013. 10.1038/ncomms3582.
https://doi.org/10.1038/ncomms3582

[50] Seth Lloyd and Samuel L Braunstein. Quantum computation over continuous variables. Phys. Rev. Lett., 82 (8): 1784, 1999. 10.1103/PhysRevLett.82.1784.
https://doi.org/10.1103/PhysRevLett.82.1784

[51] Daniel Gottesman. Stabilizer codes and quantum error correction. arXiv preprint quant-ph/9705052, 1997.
arXiv:quant-ph/9705052

[52] Emanuel Knill. Non-binary unitary error bases and quantum codes. arXiv preprint quant-ph/9608048, 1996.
arXiv:quant-ph/9608048

[53] Avanti Ketkar, Andreas Klappenecker, Santosh Kumar, and Pradeep Kiran Sarvepalli. Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory, 52 (11): 4892–4914, 2006. 10.1109/TIT.2006.883612.
https://doi.org/10.1109/TIT.2006.883612

[54] Isaac L Chuang and Yoshihisa Yamamoto. Simple quantum computer. Phys. Rev. A, 52 (5): 3489, 1995. 10.1103/PhysRevA.52.3489.
https://doi.org/10.1103/PhysRevA.52.3489

[55] P. T. Cochrane, G. J. Milburn, and W. J. Munro. Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping. Phys. Rev. A, 59: 2631–2634, Apr 1999. 10.1103/PhysRevA.59.2631.
https://doi.org/10.1103/PhysRevA.59.2631

[56] Timothy C. Ralph, Alexei Gilchrist, Gerard J Milburn, William J Munro, and Scott Glancy. Quantum computation with optical coherent states. Phys. Rev. A, 68 (4): 042319, 2003. 10.1103/PhysRevA.68.042319.
https://doi.org/10.1103/PhysRevA.68.042319

[57] Zaki Leghtas, Gerhard Kirchmair, Brian Vlastakis, Robert J. Schoelkopf, Michel H. Devoret, and Mazyar Mirrahimi. Hardware-efficient autonomous quantum memory protection. Phys. Rev. Lett., 111: 120501, Sep 2013. 10.1103/PhysRevLett.111.120501.
https://doi.org/10.1103/PhysRevLett.111.120501

[58] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. Dynamically protected cat-qubits: a new paradigm for universal quantum computation. New J. Phys., 16 (4): 045014, April 2014. 10.1088/1367-2630/16/4/045014.
https://doi.org/10.1088/1367-2630/16/4/045014

[59] Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, and S. M. Girvin. New class of quantum error-correcting codes for a bosonic mode. Phys. Rev. X, 6: 031006, Jul 2016. 10.1103/PhysRevX.6.031006.
https://doi.org/10.1103/PhysRevX.6.031006

[60] Arne L. Grimsmo, Joshua Combes, and Ben Q. Baragiola. Quantum computing with rotation-symmetric bosonic codes. Phys. Rev. X, 10: 011058, Mar 2020. 10.1103/PhysRevX.10.011058.
https://doi.org/10.1103/PhysRevX.10.011058

[61] A. I. Lvovsky and M. G. Raymer. Continuous-variable optical quantum-state tomography. Rev. Mod. Phys., 81: 299–332, Mar 2009. 10.1103/RevModPhys.81.299.
https://doi.org/10.1103/RevModPhys.81.299

[62] Kazunori Miyata, Hisashi Ogawa, Petr Marek, Radim Filip, Hidehiro Yonezawa, Jun-ichi Yoshikawa, and Akira Furusawa. Implementation of a quantum cubic gate by an adaptive non-Gaussian measurement. Phys. Rev. A, 93 (2): 022301, 2016. 10.1103/PhysRevA.93.022301.
https://doi.org/10.1103/PhysRevA.93.022301

[63] Jacob Hastrup, Mikkel V Larsen, Jonas S Neergaard-Nielsen, Nicolas C Meniccuci, and Ulrik L Andersen. Cubic phase gates are not suitable for non-Clifford operations on GKP states. arXiv preprint arXiv:2009.05309, 2020.
arXiv:2009.05309

[64] Victor V. Albert, Kyungjoo Noh, Kasper Duivenvoorden, Dylan J. Young, R. T. Brierley, Philip Reinhold, Christophe Vuillot, Linshu Li, Chao Shen, S. M. Girvin, Barbara M. Terhal, and Liang Jiang. Performance and structure of single-mode bosonic codes. Phys. Rev. A, 97: 032346, Mar 2018. 10.1103/PhysRevA.97.032346.
https://doi.org/10.1103/PhysRevA.97.032346

[65] Kyungjoo Noh, Victor V Albert, and Liang Jiang. Quantum capacity bounds of Gaussian thermal loss channels and achievable rates with Gottesman-Kitaev-Preskill codes. IEEE Trans. Inf. Theory, 65 (4): 2563–2582, 2018. 10.1109/TIT.2018.2873764.
https://doi.org/10.1109/TIT.2018.2873764

[66] Takaya Matsuura, Hayata Yamasaki, and Masato Koashi. Equivalence of approximate Gottesman-Kitaev-Preskill codes. Phys. Rev. A, 102: 032408, Sep 2020. 10.1103/PhysRevA.102.032408.
https://doi.org/10.1103/PhysRevA.102.032408

[67] S. Glancy and E. Knill. Error analysis for encoding a qubit in an oscillator. Phys. Rev. A, 73: 012325, Jan 2006. 10.1103/PhysRevA.73.012325.
https://doi.org/10.1103/PhysRevA.73.012325

[68] Hans J Briegel and Robert Raussendorf. Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett., 86 (5): 910, 2001. 10.1103/PhysRevLett.86.910.
https://doi.org/10.1103/PhysRevLett.86.910

[69] Robert Raussendorf and Hans J Briegel. A one-way quantum computer. Phys. Rev. Lett., 86 (22): 5188, 2001. 10.1103/PhysRevLett.86.5188.
https://doi.org/10.1103/PhysRevLett.86.5188

[70] Mile Gu, Christian Weedbrook, Nicolas C. Menicucci, Timothy C. Ralph, and Peter van Loock. Quantum computing with continuous-variable clusters. Phys. Rev. A, 79: 062318, Jun 2009. 10.1103/PhysRevA.79.062318.
https://doi.org/10.1103/PhysRevA.79.062318

[71] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52 (4): 2493–2496, 1995. 10.1103/PhysRevA.52.R2493.
https://doi.org/10.1103/PhysRevA.52.R2493

[72] A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77 (5): 793–797, 1996a. 10.1103/PhysRevLett.77.793.
https://doi.org/10.1103/PhysRevLett.77.793

[73] Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error rate. SIAM J. Comput., 38 (4): 1207–1282, 2008. 10.1137/S0097539799359385.
https://doi.org/10.1137/S0097539799359385

[74] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. Resilient quantum computation. Science, 279 (5349): 342–345, 1998. 10.1126/science.279.5349.342.
https://doi.org/10.1126/science.279.5349.342

[75] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549 (7671): 172–179, Sep 2017. 10.1038/nature23460.
https://doi.org/10.1038/nature23460

[76] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys. (N. Y.), 303 (1): 2 – 30, 2003. 10.1016/S0003-4916(02)00018-0.
https://doi.org/10.1016/S0003-4916(02)00018-0

[77] Sergey B Bravyi and A Yu Kitaev. Quantum codes on a lattice with boundary. arXiv preprint quant-ph/9811052, 1998.
arXiv:quant-ph/9811052

[78] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett., 98 (19), May 2007. 10.1103/PhysRevLett.98.190504.
https://doi.org/10.1103/PhysRevLett.98.190504

[79] Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski. High-threshold universal quantum computation on the surface code. Phys. Rev. A, 80: 052312, Nov 2009. 10.1103/PhysRevA.80.052312.
https://doi.org/10.1103/PhysRevA.80.052312

[80] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86 (3), Sep 2012. 10.1103/PhysRevA.86.032324.
https://doi.org/10.1103/PhysRevA.86.032324

[81] A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54 (2): 1098–1105, aug 1996. 10.1103/PhysRevA.54.1098.
https://doi.org/10.1103/PhysRevA.54.1098

[82] Andrew Steane. Multiple-particle interference and quantum error correction. Proc. R. Soc. A, 452 (1954): 2551–2577, 1996b. 10.1098/rspa.1996.0136.
https://doi.org/10.1098/rspa.1996.0136

[83] A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace. Foliated Quantum Error-Correcting Codes. Phys. Rev. Lett., 117 (7), 2016. 10.1103/PhysRevLett.117.070501.
https://doi.org/10.1103/PhysRevLett.117.070501

[84] Austin G Fowler and Kovid Goyal. Topological cluster state quantum computing. Quantum Inf. Comput., 9 (9-10): 0721–0738, 2009. 10.26421/QIC9.9-10.
https://doi.org/10.26421/QIC9.9-10

[85] Naomi Nickerson and Héctor Bombín. Measurement based fault tolerance beyond foliation. arXiv preprint arXiv:1810.09621, 2018.
arXiv:1810.09621

[86] Hector Bombin. 2D quantum computation with 3D topological codes. arXiv preprint arXiv:1810.09571, 2018a.
arXiv:1810.09571

[87] Hector Bombin. Transversal gates and error propagation in 3D topological codes. arXiv preprint arXiv:1810.09575, 2018b.
arXiv:1810.09575

[88] Benjamin J. Brown and Sam Roberts. Universal fault-tolerant measurement-based quantum computation. Phys. Rev. Research, 2: 033305, Aug 2020. 10.1103/PhysRevResearch.2.033305.
https://doi.org/10.1103/PhysRevResearch.2.033305

[89] Christophe Vuillot, Hamed Asasi, Yang Wang, Leonid P. Pryadko, and Barbara M. Terhal. Quantum error correction with the toric Gottesman-Kitaev-Preskill code. Phys. Rev. A, 99: 032344, Mar 2019. 10.1103/PhysRevA.99.032344.
https://doi.org/10.1103/PhysRevA.99.032344

[90] Lisa Hänggli, Margret Heinze, and Robert König. Enhanced noise resilience of the surface–gottesman-kitaev-preskill code via designed bias. Phys. Rev. A, 102: 052408, Nov 2020. 10.1103/PhysRevA.102.052408.
https://doi.org/10.1103/PhysRevA.102.052408

[91] B M Terhal, J Conrad, and C Vuillot. Towards scalable bosonic quantum error correction. Quantum Science and Technology, 5 (4): 043001, Jul 2020. 10.1088/2058-9565/ab98a5.
https://doi.org/10.1088/2058-9565/ab98a5

[92] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi. Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code. arXiv preprint arXiv:2006.05416, 2020.
arXiv:2006.05416

[93] Damien Bonneau, Gabriel J Mendoza, Jeremy L O’Brien, and Mark G Thompson. Effect of loss on multiplexed single-photon sources. New J. Phys., 17 (4): 043057, April 2015. 10.1088/1367-2630/17/4/043057.
https://doi.org/10.1088/1367-2630/17/4/043057

[94] Francesco Lenzini, Ben Haylock, Juan C. Loredo, Raphael A. Abrahão, Nor A. Zakaria, Sachin Kasture, Isabelle Sagnes, Aristide Lemaitre, Hoang-Phuong Phan, Dzung Viet Dao, Pascale Senellart, Marcelo P. Almeida, Andrew G. White, and Mirko Lobino. Active demultiplexing of single photons from a solid-state source. Laser & Photonics Reviews, 11 (3): 1600297, April 2017. 10.1002/lpor.201600297.
https://doi.org/10.1002/lpor.201600297

[95] Nicolas C. Menicucci. Temporal-mode continuous-variable cluster states using linear optics. Phys. Rev. A, 83: 062314, Jun 2011. 10.1103/PhysRevA.83.062314.
https://doi.org/10.1103/PhysRevA.83.062314

[96] Ish Dhand, Melanie. Engelkemeier, Linda. Sansoni, Sonja Barkhofen, Christine Silberhorn, and Martin B. Plenio. Proposal for quantum simulation via all-optically-generated tensor network states. Phys. Rev. Lett., 120: 130501, 2018. 10.1103/PhysRevLett.120.130501.
https://doi.org/10.1103/PhysRevLett.120.130501

[97] Michael Lubasch, Antonio A. Valido, Jelmer J. Renema, W. Steven Kolthammer, Dieter Jaksch, M. S. Kim, Ian Walmsley, and Raúl García-Patrón. Tensor network states in time-bin quantum optics. Phys. Rev. A, 97: 062304, Jun 2018. 10.1103/PhysRevA.97.062304.
https://doi.org/10.1103/PhysRevA.97.062304

[98] Michael JW Hall. Gaussian noise and quantum-optical communication. Phys. Rev. A, 50 (4): 3295, 1994. 10.1103/PhysRevA.50.3295.
https://doi.org/10.1103/PhysRevA.50.3295

[99] Ben Q. Baragiola, Giacomo Pantaleoni, Rafael N. Alexander, Angela Karanjai, and Nicolas C. Menicucci. All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code. Phys. Rev. Lett., 123: 200502, Nov 2019. 10.1103/PhysRevLett.123.200502.
https://doi.org/10.1103/PhysRevLett.123.200502

[100] Christopher Bishop. Pattern recognition and machine learning. Springer, New York, 2006. ISBN 0-387-31073-8.

[101] Christoph Buchheim, Ruth Hübner, and Anita Schöbel. Ellipsoid bounds for convex quadratic integer programming. SIAM J. Optim., 25 (2): 741–769, January 2015. 10.1137/130929187.
https://doi.org/10.1137/130929187

[102] Jaehyun Park and Stephen Boyd. A semidefinite programming method for integer convex quadratic minimization. Optim. Lett., 12 (3): 499–518, March 2017. 10.1007/s11590-017-1132-y.
https://doi.org/10.1007/s11590-017-1132-y

[103] Jack Edmonds. Optimum branchings. J. Res. Natl. Bur. Stand. Sect. B Math. Math. Phys., 71B (4): 233, oct 1967. 10.6028/jres.071B.032.
https://doi.org/10.6028/jres.071B.032

[104] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. J. Math. Phys., 43 (9): 4452–4505, Sep 2002. 10.1063/1.1499754.
https://doi.org/10.1063/1.1499754

[105] David S. Wang, Austin G. Fowler, and Lloyd C. L. Hollenberg. Surface code quantum computing with error rates over 1%. Phys. Rev. A, 83: 020302, Feb 2011. 10.1103/PhysRevA.83.020302.
https://doi.org/10.1103/PhysRevA.83.020302

[106] Nicolas Delfosse and Naomi H Nickerson. Almost-linear time decoding algorithm for topological codes. arXiv preprint arXiv:1709.06218, 2017.
arXiv:1709.06218

[107] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A, 90 (3): 032326, sep 2014. 10.1103/PhysRevA.90.032326.
https://doi.org/10.1103/PhysRevA.90.032326

[108] Guillaume Duclos-Cianci and David Poulin. Fast decoders for topological quantum codes. Phys. Rev. Lett., 104 (5): 1–5, 2010. 10.1103/PhysRevLett.104.050504.
https://doi.org/10.1103/PhysRevLett.104.050504

[109] Guillaume Duclos-Cianci and David Poulin. Fault-tolerant renormalization group decoder for Abelian topological codes. Quantum Inf. Comput., 14 (9-10): 721–740, 2014. 10.5555/2638670.2638671.
https://doi.org/10.5555/2638670.2638671

[110] Sergey Bravyi and Jeongwan Haah. Quantum self-correction in the 3D cubic code model. Phys. Rev. Lett., 111 (20), dec 2013. 10.1103/PhysRevLett.111.200501.
https://doi.org/10.1103/PhysRevLett.111.200501

[111] Hussain Anwar, Benjamin J. Brown, Earl T. Campbell, and Dan E. Browne. Fast decoders for qudit topological codes. New J. Phys., 16: 1–20, 2014. 10.1088/1367-2630/16/6/063038.
https://doi.org/10.1088/1367-2630/16/6/063038

[112] James R. Wootton and Daniel Loss. High Threshold Error Correction for the Surface Code. Phys. Rev. Lett., 109 (16), 2012. 10.1103/PhysRevLett.109.160503.
https://doi.org/10.1103/PhysRevLett.109.160503

[113] Adrian Hutter, James R. Wootton, and Daniel Loss. Efficient Markov chain Monte Carlo algorithm for the surface code. Phys. Rev. A, 89 (2): 1–10, 2014. 10.1103/PhysRevA.89.022326.
https://doi.org/10.1103/PhysRevA.89.022326

[114] Pavel Panteleev and Gleb Kalachev. Degenerate quantum LDPC codes with good finite length performance. arXiv preprint arXiv:1904.02703, 2019.
arXiv:1904.02703

[115] Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. Decoding across the quantum low-density parity-check code landscape. Phys. Rev. Research, 2: 043423, Dec 2020. 10.1103/PhysRevResearch.2.043423.
https://doi.org/10.1103/PhysRevResearch.2.043423

[116] Austin G. Fowler. Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time. Quantum Inf. Comput., 15 (1-2): 145–158, 2014. 10.5555/2685188.2685197.
https://doi.org/10.5555/2685188.2685197

[117] Giacomo Torlai and Roger G. Melko. Neural Decoder for Topological Codes. Phys. Rev. Lett., 119 (3): 1–6, 2017. 10.1103/PhysRevLett.119.030501.
https://doi.org/10.1103/PhysRevLett.119.030501

[118] Savvas Varsamopoulos, Ben Criger, and Koen Bertels. Decoding small surface codes with feedforward neural networks. Quantum Sci. Technol., 3 (1): 1–13, 2018. 10.1088/2058-9565/aa955a.
https://doi.org/10.1088/2058-9565/aa955a

[119] David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. Ultrahigh Error Threshold for Surface Codes with Biased Noise. Phys. Rev. Lett., 120 (5): 1–6, 2018. 10.1103/PhysRevLett.120.050505.
https://doi.org/10.1103/PhysRevLett.120.050505

[120] Michael Herold, Earl T. Campbell, Jens Eisert, and Michael J. Kastoryano. Cellular-automaton decoders for topological quantum memories. npj Quantum Inf., 1 (1): 15010, dec 2015. 10.1038/npjqi.2015.10.
https://doi.org/10.1038/npjqi.2015.10

[121] Michael Herold, Michael J. Kastoryano, Earl T. Campbell, and Jens Eisert. Cellular automaton decoders of topological quantum memories in the fault tolerant setting. New J. Phys., 19 (6), 2017. 10.1088/1367-2630/aa7099.
https://doi.org/10.1088/1367-2630/aa7099

[122] J Harrington. Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes. PhD thesis, Caltech, 2004.

[123] Guillaume Dauphinais and David Poulin. Fault-tolerant quantum error correction for non-abelian anyons. Commun. Math. Phys., 355 (2): 519–560, Jul 2017. 10.1007/s00220-017-2923-9.
https://doi.org/10.1007/s00220-017-2923-9

[124] E. Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39–44, Mar 2005. 10.1038/nature03350.
https://doi.org/10.1038/nature03350

[125] Clare Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. Surface code quantum computing by lattice surgery. New J. Phys., 14 (12): 123011, dec 2012. 10.1088/1367-2630/14/12/123011.
https://doi.org/10.1088/1367-2630/14/12/123011

[126] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton. Poking holes and cutting corners to achieve Clifford gates with the surface code. Phys. Rev. X, 7: 021029, May 2017. 10.1103/PhysRevX.7.021029.
https://doi.org/10.1103/PhysRevX.7.021029

[127] Daniel Litinski and Felix von Oppen. Lattice surgery with a twist: Simplifying Clifford gates of surface codes. Quantum, 2: 62, May 2018. 10.22331/q-2018-05-04-62.
https://doi.org/10.22331/q-2018-05-04-62

[128] Daniel Litinski. A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery. Quantum, 3: 128, March 2019. 10.22331/q-2019-03-05-128.
https://doi.org/10.22331/q-2019-03-05-128

[129] Daniel Herr, Alexandru Paler, Simon J Devitt, and Franco Nori. Lattice surgery on the Raussendorf lattice. Quantum Sci. Technol., 3 (3): 035011, jun 2018. 10.1088/2058-9565/aac450.
https://doi.org/10.1088/2058-9565/aac450

[130] Daniel Herr, Franco Nori, and Simon J. Devitt. Optimization of lattice surgery is NP-hard. npj Quantum Inf., 3 (1), Sep 2017. 10.1038/s41534-017-0035-1.
https://doi.org/10.1038/s41534-017-0035-1

[131] H. Bombin. Topological order with a twist: Ising anyons from an Abelian model. Phys. Rev. Lett., 105 (3), Jul 2010. 10.1103/PhysRevLett.105.030403.
https://doi.org/10.1103/PhysRevLett.105.030403

[132] Ying Li. A magic state’s fidelity can be superior to the operations that created it. New J. Phys., 17 (2): 023037, feb 2015. 10.1088/1367-2630/17/2/023037.
https://doi.org/10.1088/1367-2630/17/2/023037

[133] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A, 71: 022316, Feb 2005. 10.1103/PhysRevA.71.022316.
https://doi.org/10.1103/PhysRevA.71.022316

[134] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86 (5), Nov 2012. 10.1103/PhysRevA.86.052329.
https://doi.org/10.1103/PhysRevA.86.052329

[135] Jeongwan Haah and Matthew B. Hastings. Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates. Quantum, 2: 71, June 2018. 10.22331/q-2018-06-07-71.
https://doi.org/10.22331/q-2018-06-07-71

[136] Jeongwan Haah, Matthew B. Hastings, D. Poulin, and D. Wecker. Magic state distillation with low space overhead and optimal asymptotic input count. Quantum, 1: 31, October 2017. 10.22331/q-2017-10-03-31.
https://doi.org/10.22331/q-2017-10-03-31

[137] Cody Jones. Multilevel distillation of magic states for quantum computing. Phys. Rev. A, 87: 042305, Apr 2013. 10.1103/PhysRevA.87.042305.
https://doi.org/10.1103/PhysRevA.87.042305

[138] Matthew B. Hastings and Jeongwan Haah. Distillation with sublogarithmic overhead. Phys. Rev. Lett., 120 (5), Jan 2018. 10.1103/PhysRevLett.120.050504.
https://doi.org/10.1103/PhysRevLett.120.050504

[139] Alastair Kay. Tutorial on the quantikz package. arXiv preprint arXiv:1809.03842, 2018. 10.17637/rh.7000520.
https://doi.org/10.17637/rh.7000520
arXiv:1809.03842

[140] Daniel Litinski and Felix von Oppen. Quantum computing with Majorana fermion codes. Phys. Rev. B, 97: 205404, May 2018. 10.1103/PhysRevB.97.205404.
https://doi.org/10.1103/PhysRevB.97.205404

[141] Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and $T$ gates. Quantum Info. Comput., 13 (7–8): 607–630, July 2013a. 10.5555/2535649.2535653.
https://doi.org/10.5555/2535649.2535653

[142] Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Asymptotically optimal approximation of single qubit unitaries by Clifford and $T$ circuits using a constant number of ancillary qubits. Phys. Rev. Lett., 110: 190502, May 2013b. 10.1103/PhysRevLett.110.190502.
https://doi.org/10.1103/PhysRevLett.110.190502

[143] David Gosset, Vadym Kliuchnikov, Michele Mosca, and Vincent Russo. An algorithm for the t-count. arXiv preprint arXiv:1308.4134, 2013.
arXiv:1308.4134

[144] Luke E Heyfron and Earl T Campbell. An efficient quantum compiler that reduces $T$ count. Quantum Sci. Technol., 4 (1): 015004, sep 2018. 10.1088/2058-9565/aad604.
https://doi.org/10.1088/2058-9565/aad604

[145] M. Amy, D. Maslov, M. Mosca, and M. Roetteler. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 32 (6): 818–830, 2013. 10.1109/TCAD.2013.2244643.
https://doi.org/10.1109/TCAD.2013.2244643

[146] Peter Selinger. Quantum circuits of $T$-depth one. Phys. Rev. A, 87: 042302, Apr 2013. 10.1103/PhysRevA.87.042302.
https://doi.org/10.1103/PhysRevA.87.042302

[147] M. Amy, D. Maslov, and M. Mosca. Polynomial-time $T$-depth optimization of Clifford+$T$ circuits via matroid partitioning. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 33 (10): 1476–1489, 2014. 10.1109/TCAD.2014.2341953.
https://doi.org/10.1109/TCAD.2014.2341953

[148] John Preskill. Reliable Quantum Computers. Proc. R. Soc. Lond. A, 454: 385–410, 1997. 10.1098/rspa.1998.0167.
https://doi.org/10.1098/rspa.1998.0167

[149] A Yu Kitaev. Quantum computations: algorithms and error correction. Russ. Math. Surv., 52 (6): 1191–1249, dec 1997. 10.1070/RM1997v052n06ABEH002155.
https://doi.org/10.1070/RM1997v052n06ABEH002155

[150] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput., 6 (2): 097–165, 2006. 10.26421/QIC6.2.
https://doi.org/10.26421/QIC6.2

[151] David P. Divincenzo and Panos Aliferis. Effective fault-tolerant quantum computation with slow measurements. Phys. Rev. Lett., 98 (2): 020501, 2007. 10.1103/PhysRevLett.98.020501.
https://doi.org/10.1103/PhysRevLett.98.020501

[152] Panos Aliferis and John Preskill. Fault-tolerant quantum computation against biased noise. Phys. Rev. A, 78 (5): 052331, 2008. 10.1103/PhysRevA.78.052331.
https://doi.org/10.1103/PhysRevA.78.052331

[153] Panos Aliferis, Daniel Gottesman, and John Preskill. Accuracy threshold for postselected quantum computation. Quantum Inf. Comput., 8 (3-4): 181–244, mar 2008. 10.26421/QIC8.3-4.
https://doi.org/10.26421/QIC8.3-4

[154] Alexey A. Kovalev and Leonid P. Pryadko. Fault tolerance of quantum low-density parity check codes with sublinear distance scaling. Phys. Rev. A, 87 (2): 020304, feb 2013. 10.1103/PhysRevA.87.020304.
https://doi.org/10.1103/PhysRevA.87.020304

[155] Daniel Gottesman. Fault-tolerant quantum computation with constant overhead. Quantum Inf. Comput., 14 (15-16): 1338–1372, oct 2014. 10.26421/QIC14.15-16.
https://doi.org/10.26421/QIC14.15-16

[156] Omar Fawzi, Antoine Grospellier, and Anthony Leverrier. Constant overhead quantum fault-tolerance with quantum expander codes. In Proceedings – Annual IEEE Symposium on Foundations of Computer Science, FOCS, volume 2018-Octob, pages 743–754. IEEE, oct 2018. 10.1109/FOCS.2018.00076.
https://doi.org/10.1109/FOCS.2018.00076

[157] Barbara M. Terhal. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307–346, Apr 2015. 10.1103/RevModPhys.87.307.
https://doi.org/10.1103/RevModPhys.87.307

[158] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70: 052328, Nov 2004. 10.1103/PhysRevA.70.052328.
https://doi.org/10.1103/PhysRevA.70.052328

[159] D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C.L. Hollenberg. Threshold error rates for the toric and planar codes. Quantum Inf. Comput., 10 (5-6): 456–469, may 2010. 10.5555/2011362.2011368.
https://doi.org/10.5555/2011362.2011368

[160] E. W. Dijkstra. A note on two problems in connexion with graphs. Numer. Math., 1 (1): 269–271, December 1959. 10.1007/bf01386390.
https://doi.org/10.1007/bf01386390

[161] Christian D. Lorenz and Robert M. Ziff. Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices. Phys. Rev. E, 57: 230–236, Jan 1998. 10.1103/PhysRevE.57.230.
https://doi.org/10.1103/PhysRevE.57.230

[162] Blayney W. Walshe, Ben Q. Baragiola, Rafael N. Alexander, and Nicolas C. Menicucci. Continuous-variable gate teleportation and bosonic-code error correction. Phys. Rev. A, 102: 062411, Dec 2020. 10.1103/PhysRevA.102.062411.
https://doi.org/10.1103/PhysRevA.102.062411

[163] Daiqin Su, Krishna Kumar Sabapathy, Casey R. Myers, Haoyu Qi, Christian Weedbrook, and Kamil Brádler. Implementing quantum algorithms on temporal photonic cluster states. Phys. Rev. A, 98: 032316, Sep 2018. 10.1103/PhysRevA.98.032316.
https://doi.org/10.1103/PhysRevA.98.032316

[164] Yunong Shi, Christopher Chamberland, and Andrew Cross. Fault-tolerant preparation of approximate GKP states. New J. Phys., 21 (9): 093007, 2019. 10.1088/1367-2630/ab3a62.
https://doi.org/10.1088/1367-2630/ab3a62

[165] Yong Zhang, M Menotti, K Tan, VD Vaidya, DH Mahler, L Zatti, M Liscidini, B Morrison, and Z Vernon. Single-mode quadrature squeezing using dual-pump four-wave mixing in an integrated nanophotonic device. arXiv preprint arXiv:2001.09474, 2020.
arXiv:2001.09474

[166] Matthew Collins and Zachary Vernon. On the road to room temperature quantum computation. https://bit.ly/2NR5xvm, 2020.
https://bit.ly/2NR5xvm

[167] Adrian Cho. IBM promises 1000-qubit quantum computer—a milestone—by 2023. Science, September 2020. 10.1126/science.abe8122.
https://doi.org/10.1126/science.abe8122

[168] Radim Filip, Petr Marek, and Ulrik L. Andersen. Measurement-induced continuous-variable quantum interactions. Phys. Rev. A, 71: 042308, Apr 2005. 10.1103/PhysRevA.71.042308.
https://doi.org/10.1103/PhysRevA.71.042308

[169] Ryuji Ukai, Shota Yokoyama, Jun-ichi Yoshikawa, Peter van Loock, and Akira Furusawa. Demonstration of a controlled-phase gate for continuous-variable one-way quantum computation. Phys. Rev. Lett., 107: 250501, Dec 2011. 10.1103/PhysRevLett.107.250501.
https://doi.org/10.1103/PhysRevLett.107.250501

[170] Jun-ichi Yoshikawa, Yoshichika Miwa, Alexander Huck, Ulrik L. Andersen, Peter van Loock, and Akira Furusawa. Demonstration of a quantum nondemolition sum gate. Phys. Rev. Lett., 101: 250501, Dec 2008. 10.1103/PhysRevLett.101.250501.
https://doi.org/10.1103/PhysRevLett.101.250501

[171] Filippo Caruso, Vittorio Giovannetti, and Alexander S Holevo. One-mode bosonic Gaussian channels: a full weak-degradability classification. New J. Phys., 8 (12): 310, 2006. 10.1088/1367-2630/8/12/310.
https://doi.org/10.1088/1367-2630/8/12/310

[172] J. Solomon Ivan, Krishna Kumar Sabapathy, and R. Simon. Operator-sum representation for bosonic Gaussian channels. Phys. Rev. A, 84: 042311, Oct 2011. 10.1103/PhysRevA.84.042311.
https://doi.org/10.1103/PhysRevA.84.042311

[173] Krishna Kumar Sabapathy, J Solomon Ivan, and R Simon. Robustness of non-Gaussian entanglement against noisy amplifier and attenuator environments. Phys. Rev. Lett., 107 (13): 130501, 2011. 10.1103/PhysRevLett.107.130501.
https://doi.org/10.1103/PhysRevLett.107.130501

[174] Raul Garcia-Patron, Carlos Navarrete-Benlloch, Seth Lloyd, Jeffrey H Shapiro, and Nicolas J. Cerf. Majorization theory approach to the Gaussian channel minimum entropy conjecture. Phys. Rev. Lett., 108 (11): 110505, 2012. 10.1103/PhysRevLett.108.110505.
https://doi.org/10.1103/PhysRevLett.108.110505

[175] Tommaso F. Demarie, Trond Linjordet, Nicolas C. Menicucci, and Gavin K. Brennen. Detecting topological entanglement entropy in a lattice of quantum harmonic oscillators. New J. Phys., 16 (8): 085011, 2014. 10.1088/1367-2630/16/8/085011.
https://doi.org/10.1088/1367-2630/16/8/085011

[176] Matteo G.A. Paris. Displacement operator by beam splitter. Phys. Lett. A, 217 (2-3): 78–80, July 1996. 10.1016/0375-9601(96)00339-8.
https://doi.org/10.1016/0375-9601(96)00339-8

[177] Nathan Killoran, Josh Izaac, Nicolás Quesada, Ville Bergholm, Matthew Amy, and Christian Weedbrook. Strawberry fields: A software platform for photonic quantum computing. Quantum, 3: 129, March 2019. 10.22331/q-2019-03-11-129.
https://doi.org/10.22331/q-2019-03-11-129

Cited by

[1] Blayney W. Walshe, Ben Q. Baragiola, Rafael N. Alexander, and Nicolas C. Menicucci, “Continuous-variable gate teleportation and bosonic-code error correction”, Physical Review A 102 6, 062411 (2020).

[2] Shahnawaz Ahmed, Carlos Sánchez Muñoz, Franco Nori, and Anton Frisk Kockum, “Classification and reconstruction of optical quantum states with deep neural networks”, arXiv:2012.02185.

[3] Ivan H. Deutsch, “Harnessing the Power of the Second Quantum Revolution”, arXiv:2010.10283.

[4] Filip Rozpędek, Kyungjoo Noh, Qian Xu, Saikat Guha, and Liang Jiang, “Quantum repeaters based on concatenated bosonic and discrete-variable quantum codes”, arXiv:2011.15076.

[5] Mikkel V. Larsen, Christopher Chamberland, Kyungjoo Noh, Jonas S. Neergaard-Nielsen, and Ulrik L. Andersen, “A fault-tolerant continuous-variable measurement-based quantum computation architecture”, arXiv:2101.03014.

[6] Namrata Shukla, Stefan Nimmrichter, and Barry C. Sanders, “Squeezed comb states”, Physical Review A 103 1, 012408 (2021).

[7] Leonardo Assis Morais, Till Weinhold, Marcelo P. de Almeida, Adriana Lita, Thomas Gerrits, Sae Woo Nam, Andrew G. White, and Geoff Gillett, “Precisely determining photon-number in real-time”, arXiv:2012.10158.

[8] Kosuke Fukui and Nicolas C. Menicucci, “An efficient, concatenated, bosonic code for additive Gaussian noise”, arXiv:2102.01374.

[9] Lucas J. Mensen, Ben Q. Baragiola, and Nicolas C. Menicucci, “Phase-space methods for representing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits”, arXiv:2012.12488.

[10] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Terry Rudolph, and Chris Sparrow, “Fusion-based quantum computation”, arXiv:2101.09310.

[11] Ulysse Chabaud, Ganaël Roeland, Mattia Walschaers, Frédéric Grosshans, Valentina Parigi, Damian Markham, and Nicolas Treps, “Certification of non-Gaussian states with operational measurements”, arXiv:2011.04320.

The above citations are from SAO/NASA ADS (last updated successfully 2021-02-06 02:31:13). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2021-02-06 02:31:11).

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.

Generative Data Intelligence

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

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