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研究生:李易展
研究生(外文):Lee, Yi-Chan
論文名稱:量子資訊科學衍生的限制與可能性
論文名稱(外文):The constraint and possibility derived from quantum information science
指導教授:李瑞光李瑞光引用關係
指導教授(外文):Lee, Ray Kuang
學位類別:博士
校院名稱:國立清華大學
系所名稱:物理系
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:英文
論文頁數:94
中文關鍵詞:量子資訊非侷域性宇稱-時間對稱錯誤更正碼
外文關鍵詞:quantum informationnonlocalityparity-time symmetryerror correction code
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  • 收藏至我的研究室書目清單書目收藏:1
量子資訊科學是一個跨領域的學科,其在物理、應用數學、電腦科學等相關領域發展了許多有趣的連結。量子資訊科學的概念亦在這些領域中產生了許多不同的影響。在量子資訊科學中,即使最大量子糾纏態具備了古典物理系統沒有的非侷域性,其應用仍舊被要求遵守狹義相對論的光速為資訊傳遞最高速原則。在這篇論文中,我們利用了此量子科學資訊中的基本原則來檢視宇稱-時間對稱量子理論($\mathcal{PT}$對稱量子理論)。我們的結果顯示對於$\mathcal{PT}$對稱量子理論的不適當解讀會造成資訊傳遞超越光速的情況,同時定下了發展這個新穎的量子理論必須遵守的限制。在這篇論文的第二部分,我們研究了在Kitaev蜂巢模型上實現量子錯誤更正碼的可能。我們首先證明了該模型基態的準簡併性(quasi-degeneracy)以及其近似侷域不可分辨性(approximate local-indistinguishability)。我們更進一步的提出了在古典電腦上模擬該錯誤更正碼在熱平衡過程中的表現的可能性。藉著結合量子資訊科學以及物理兩個領域的方法,我們的結果顯示出兩個領域的結合有更多的可能性。
Quantum information science, a interdisciplinary field, develops many interesting connections between the fields of physics, mathematics, computer science and other related subjects. The intuitive ideas in quantum information also has different influence in these areas. In this thesis, we use no-signalling condition in special relativity theory which is required to be satisfied in quantum information science, even under the condition that a maximally entangled state is given, to test parity-time symmetry quantum theory ($\mathcal{PT}$-symmetric theory). Our result shows that improper interpretation of $\mathcal{PT}$-symmetry quantum mechanics will cause the violation of no-signalling principle and set the constraints which should be obeyed for developing this novel theory. In the second part of thesis, we study the error correction in quantum information science on Kitaev honeycomb model, and establish the connection with the researches done by physicists. Our result formally proves the quasi-degeneracy and the approximate local-indistinguishability of the ground states of Kitaev honeycomb model. We further point out that the simulation of the information lifetime under the thermalization process is possible. By combining the methods from physics society and quantum information society, we shows that there are more possibility in the connection between these two fields.
I Impossible and possible alternatives of parity-time symmetric Hamiltonian 1
1 Introduction 2
2 Background knowledge of parity-time symmetric Hamiltonian 5
2.1 The first PT -symmetric Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Simplest two-level PT symmetric Hamiltonian . . . . . . . . . . . . . . . . . . . 8
2.3 Proposed applications of finite dimensional PT-symmetric Hamiltonian . . . . . 9
2.3.1 Ultra-fast spin flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.2 Distinguish non-orthogonal state with single-shot measurement . . . . . . 10
2.4 Equivalence of PT symmetric Hamiltonian and Hermitian Hamiltonian in finite
dimensional system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Violation of no-signalling principle 14
3.1 No-signalling principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Transmission of one bit classical information through PT box . . . . . . . . . . . 15
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 General parity-time symmetric two-level system 19
4.1 Naimark dilation of PT -symmetric Hamiltonian . . . . . . . . . . . . . . . . . . 19
4.2 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Future work and conclusion of PT -symmetric Hamiltonian 26

II Quantum error correction on Kitaev honeycomb model 28
6 Introduction 29
7 Background knowledge of QECC and KHM 32
7.1 Quantum error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.1.1 Bit-flip code and Phase-flip code . . . . . . . . . . . . . . . . . . . . . . . 34
7.1.2 Shor code and discretization of errors . . . . . . . . . . . . . . . . . . . . 36
7.1.3 Stabilizer formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.1.4 Toric code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.1.5 Quantum error-correction condition . . . . . . . . . . . . . . . . . . . . . 46
7.2 Kitaev honeycomb model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8 Quasi-degeneracy and approximate local indistinguishability 56
8.1 Quasi-degeneracy of ground states . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.2 Local indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.3 Approximate local-indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . 59
8.3.1 Proof of approximate local-indistinguishability . . . . . . . . . . . . . . . 60
8.3.2 Argument for arbitrary local operators . . . . . . . . . . . . . . . . . . . . 65
9 Numerical simulation results 68
9.1 Methods of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.2 Results of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2.1 Logical error rate in medium- and low-temperature regions . . . . . . . . 73
9.2.2 Logical error rate in high-temperature region . . . . . . . . . . . . . . . . 77
10 Conclusion and future work of QECC on KHM 84
11 Conclusion of thesis 86
Bibliography 87
[1] Herman Feshbach. Unified theory of nuclear reactions. Ann. Phys., 5(4):357-390, 1958. ISSN 0003-4916. doi: http://dx.doi.org/10.1016/0003-4916(58)90007-1. URL http://www.sciencedirect.com/science/article/pii/0003491658900071.
[2] M. B. Plenio and P. L. Knight. The quantum-jump approach to dissipative dynamics in quantum optics. Rev. Mod. Phys., 70:101-144, Jan 1998. doi: 10.1103/RevModPhys.70.101.
URL http://link.aps.org/doi/10.1103/RevModPhys.70.101.
[3] Carl M. Bender and Stefan Boettcher. Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett., 80:5243-5246, Jun 1998. doi: 10.1103/PhysRevLett.80.5243. URL http://link.aps.org/doi/10.1103/PhysRevLett.80.5243.
[4] Carl M. Bender, Dorje C. Brody, and Hugh F. Jones. Complex extension of quantum mechanics. Phys. Rev. Lett., 89:270401, Dec 2002. doi: 10.1103/PhysRevLett.89.270401. URL http://link.aps.org/doi/10.1103/PhysRevLett.89.270401.
[5] A Ruschhaupt, F Delgado, and J G Muga. Physical realization of pt -symmetric potential
scattering in a planar slab waveguide. Journal of Physics A: Mathematical and General, 38(9):L171, 2005. URL http://stacks.iop.org/0305-4470/38/i=9/a=L03.
[6] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Ziad H. Musslimani. Theory of coupled optical pt-symmetric structures. Opt. Lett., 32(17):2632-2634, Sep 2007. doi: 10.1364/OL.32.002632. URL http://ol.osa.org/abstract.cfm?URI=ol-32-17-2632.
[7] Christian E. Ruter, Konstantinos G. Markris, Ramy El-Ganainy, Demetrios N. Cristodoulides, Mordechai Segev, and Detlef Kip. Observation of parity-time symmetry in optics. Nature Physics, 6:192, 2010. URL http://www.nature.com/nphys/journal/v6/n3/full/nphys1515.html.
[8] Hamidreza Ramezani, Tsampikos Kottos, Ramy El-Ganainy, and Demetrios N. Christodoulides. Unidirectional nonlinear PT-symmetric optical structures. Phys. Rev. A, 82:043803, Oct 2010. doi: 10.1103/PhysRevA.82.043803. URL http://link.aps.org/doi/10.1103/PhysRevA.82.043803.87
[9] Stefano Longhi. PT-symmetric laser absorber. Phys. Rev. A, 82:031801, Sep 2010. doi:10.1103/PhysRevA.82.031801. URL http://link.aps.org/doi/10.1103/PhysRevA.82.
031801.
[10] Fakhroddin Nazari, Mina Nazari, and Mohammad Kazem Moravvej-Farshi. A 2*2 spatial optical switch based on pt-symmetry. Opt. Lett., 36(22):4368-4370, Nov 2011. doi: 10.1364/OL.36.004368. URL http://ol.osa.org/abstract.cfm?URI=ol-36-22-4368.
[11] Carl M. Bender, Dorje C. Brody, Hugh F. Jones, and Bernhard K. Meister. Faster
than hermitian quantum mechanics. Phys. Rev. Lett., 98:040403, Jan 2007. doi: 10.1103/PhysRevLett.98.040403. URL ttp://link.aps.org/doi/10.1103/PhysRevLett.98.040403.
[12] Carl M. Bender, Dorje C. Brody, Joo Caldeira, Uwe G罕nther, Bernhard K. Meister,
and Boris F. Samsonov. Pt-symmetric quantum state discrimination. Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences, 371:20120160, 2013. doi: 10.1098/rsta.2012.0160. URL http://rsta.royalsocietypublishing.org/content/371/1989/20120160.
[13] S. Ibanez, S. Martinez-Garaot, Xi Chen, E. Torrontegui, and J. G. Muga. Shortcuts to adiabaticity for non-hermitian systems. Phys. Rev. A, 84:023415, Aug 2011. doi: 10.1103/PhysRevA.84.023415. URL http://link.aps.org/doi/10.1103/PhysRevA.84.023415.
[14] Boyan T. Torosov, Giuseppe Della Valle, and Stefano Longhi. Non-hermitian shortcut to adiabaticity. Phys. Rev. A, 87:052502, May 2013. doi: 10.1103/PhysRevA.87.052502. URL
http://link.aps.org/doi/10.1103/PhysRevA.87.052502.
[15] J. Anandan and Y. Aharonov. Geometry of quantum evolution. Phys. Rev. Lett., 65:1697-1700, Oct 1990. doi: 10.1103/PhysRevLett.65.1697. URL http://link.aps.org/doi/10.
1103/PhysRevLett.65.1697.
[16] Norman Margolus and Lev B. Levitin. The maximum speed of dynamical evolution. Physica D: Nonlinear Phenomena, 120(1-2):188-195, 1998. doi: http://dx.doi.org/10.1016/
S0167-2789(98)00054-2.
[17] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. ISBN 9780521635035. URL http://books.google.com.
tw/books?id=65FqEKQOfP8C.
[18] Uwe Gunther and Boris F. Samsonov. Naimark-dilated PT -symmetric brachistochrone. Phys. Rev. Lett., 101:230404, Dec 2008. doi: 10.1103/PhysRevLett.101.230404. URL http://link.aps.org/doi/10.1103/PhysRevLett.101.230404.
[19] Uwe Gunther and Boris F. Samsonov. Pt-symmetric brachistochrone problem, lorentz boosts, and nonunitary operator equivalence classes. Phys. Rev. A, 78:042115, Oct 2008. doi: 10.1103/PhysRevA.78.042115. URL http://link.aps.org/doi/10.1103/PhysRevA.78.042115.
[20] Liang Hao Chao Zheng and Gui Lu Long. Observation of a fast evolution in a paritytime-symmetric system. Phil. Trans. R. Soc. A, 371:20120053, 2013. URL http://rsta.
royalsocietypublishing.org/content/371/1989/20120053.abstract.
[21] Ali Mostafazadeh. Pseudo-hermiticity versus pt-symmetry. ii. a complete characterization of non-hermitian hamiltonians with a real spectrum. Journal of Mathematical Physics, 43:2814, 2002. URL http://link.aip.org/link/doi/10.1063/1.1461427.
[22] Ali Mostafazadeh. Pseudo-hermiticity versus pt-symmetry iii: Equivalence of pseudohermiticity and the presence of antilinear symmetries. Journal of Mathematical Physics, 43(8):3944-3951, 2002. doi: 10.1063/1.1489072. URL http://link.aip.org/link/?JMP/
43/3944/1.
[23] Ali Mostafazadeh. Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian. Journal of Mathematical Physics, 43:205, 2002. URL http://link.aip.org/link/doi/10.1063/1.1418246.
[24] Ali Mostafazadeh. Exact pt-symmetry is equivalent to hermiticity. J. Phys. A: Math. Gen., 36(25):7081, 2003. URL http://stacks.iop.org/0305-4470/36/i=25/a=312.
[25] Ali Mostafazadeh. Pseudounitary operators and pseudounitary quantum dynamics. Journal of Mathematical Physics, 45(3):932-946, 2004. doi: 10.1063/1.1646448. URL http://link.aip.org/link/?JMP/45/932/1.
[26] Ali Mostafazadeh. Quantum brachistochrone problem and the geometry of the state space in pseudo-hermitian quantum mechanics. Phys. Rev. Lett., 99:130502, Sep 2007. doi: 10.1103/PhysRevLett.99.130502. URL http://link.aps.org/doi/10.1103/PhysRevLett.99.130502.
[27] Anthony Chefles. Quantum state discrimination. Contemporary Physics, 41(6):401-424, 2000. doi: 10.1080/00107510010002599. URL http://dx.doi.org/10.1080/
00107510010002599.
[28] Carl M Bender and Sergii Kuzhel. Unbounded C-symmetries and their nonuniqueness. Journal of Physics A: Mathematical and Theoretical, 45(44):444005, 2012. URL http://stacks.iop.org/1751-8121/45/i=44/a=444005.
[29] T. Kato. Perturbation Theory for Linear Operators. Classics in Mathematics. U.S. Government Printing Office, 1995. ISBN 9783540586616. URL https://books.google.com.
tw/books?id=8ji2kN D3BwC.
[30] Eric G. Cavalcanti, Nicholas C. Menicucci, and Jacques L. Pienaar. The preparation problem in nonlinear extensions of quantum theory. arxiv: 1206.2725, 2012.
[31] Daniel S. Abrams and Seth Lloyd. Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems. Phys. Rev. Lett., 81:3992?995, Nov 1998. doi: 10.1103/PhysRevLett.81.3992.
[32] Steven Weinberg. Testing quantum mechanics. Annals of Physics, 194(2):336-386, 1989.
ISSN 0003-4916. doi: http://dx.doi.org/10.1016/0003-4916(89)90276-5. URL http://www.sciencedirect.com/science/article/pii/0003491689902765.
[33] Steven Weinberg. Precision tests of quantum mechanics. Phys. Rev. Lett., 62:485-488, Jan 1989. doi: 10.1103/PhysRevLett.62.485. URL http://link.aps.org/doi/10.1103/PhysRevLett.62.485.
[34] N. Gisin. Weinberg non-linear quantum mechanics and supraluminal communications.
Physics Letters A, 143(1-2):1-2, 1990. ISSN 0375-9601. doi: 10.1016/0375-9601(90)90786-N. URL http://www.sciencedirect.com/science/article/pii/037596019090786N.
[35] N. Gisin. Stochastic quantum dynamics and relativity. Helvetica Physica Acta, 62:363, 1989. URL http://retro.seals.ch/digbib/view?rid=hpa-001:1989:62::379&id=&id2=&id3=.
[36] Adrian Kent. Nonlinearity without superluminality. Phys. Rev. A, 72:012108, Jul 2005. doi: 10.1103/PhysRevA.72.012108.
[37] M. Znojil. Comment on letter "local PT symmetry violates the no-signaling principle" by
Yi-Chan Lee et al, Phys. Rev. Lett. 112, 130404 (2014). ArXiv e-prints, April 2014.
[38] H. F. Jones. Scattering from localized non-hermitian potentials. Phys. Rev. D, 76:125003, Dec 2007. doi: 10.1103/PhysRevD.76.125003. URL http://link.aps.org/doi/10.1103/PhysRevD.76.125003.
[39] H. F. Jones. Interface between hermitian and non-hermitian hamiltonians in a model calculation. Phys. Rev. D, 78:065032, Sep 2008. doi: 10.1103/PhysRevD.78.065032. URL http://link.aps.org/doi/10.1103/PhysRevD.78.065032.
[40] Miloslav Znojil. Scattering theory with localized non-hermiticities. Phys. Rev. D, 78: 025026, Jul 2008. doi: 10.1103/PhysRevD.78.025026. URL http://link.aps.org/doi/10.1103/PhysRevD.78.025026.
[41] Uwe Gunther, Ingrid Rotter, and Boris F Samsonov. Projective hilbert space structures at exceptional points. J. Phys. A: Math. Theor., 40(30):8815, 2007. URL http://stacks.iop.org/1751-8121/40/i=30/a=014.
[42] D. Marcuse. Theory of Dielectric Optical Waveguides. Elsevier Science, 2013. ISBN 9780323162364. URL https://books.google.com.tw/books?id=5KiVYLj0f1YC.
[43] D. Marcuse. Directional-coupler filter using dissimilar optical fibres. Electronics Letters, 21:726-727(1), August 1985. ISSN 0013-5194. URL http://digital-library.theiet.org/content/journals/10.1049/el 19850512.
[44] A.W. Snyder and J. Love. Optical Waveguide Theory. Springer US, 2012. ISBN 9781461328131. URL https://books.google.com.tw/books?id=DCXVBwAAQBAJ.
[45] R. B. B. Santos. Non-hermitian model for resonant cavities coupled by a chiral mirror. EPL (Europhysics Letters), 100(2):24005, 2012. URL http://stacks.iop.org/0295-5075/100/i=2/a=24005.
[46] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. doi: 10.1137/S0097539795293172. URL http://dx.doi.org/10.1137/S0097539795293172.
[47] Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 212-219, New York, NY, USA, 1996. ACM. ISBN 0-89791-785-5. doi: 10.1145/237814.237866. URL http://doi.acm.org/10.1145/237814.237866.
[48] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. ISBN 9781139495486. URL
http://books.google.com.tw/books?id=-s4DEy7o-a0C.
[49] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Phys. Rev. A, 68:022312, Aug 2003. doi:10.1103/PhysRevA.68.022312. URL http://link.aps.org/doi/10.1103/PhysRevA.68.022312.
[50] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52:R24932496, Oct 1995. doi: 10.1103/PhysRevA.52.R2493. URL http://link.aps.org/doi/10.1103/PhysRevA.52.R2493.
[51] Julio T. Monz-Thomas Nebendahl Volckmar Nigg Daniel Chwalla Michael Hennrich Markus Blatt Rainer Schindler, Philipp Barreiro. Experimental repetitive quantum error correction. Science, 332(6033):1059?061, 2011. doi: 10.1126/science.1203329. URL http://www.sciencemag.org/content/332/6033/1059.abstract.
[52] D. Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997.
[53] Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum hamming bound. Phys. Rev. A, 54:1862?868, Sep 1996. doi: 10.1103/PhysRevA.54.1862. URL
http://link.aps.org/doi/10.1103/PhysRevA.54.1862.
[54] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2-30, 2003. ISSN 0003-4916. doi: http://dx.doi.org/10.1016/S0003-4916(02)00018-0. URL http://www.sciencedirect.com/science/article/pii/S0003491602000180.
[55] S. B. Bravyi and A. Y. Kitaev. Quantum codes on a lattice with boundary. eprint arXiv:quant-ph/9811052, November 1998.
[56] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, Sep 2012. doi: 10.1103/PhysRevA.86.032324. URL http://link.aps.org/doi/10.1103/PhysRevA.86.032324.
[57] S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara. Subsystem surface codes with three-qubit check operators. ArXiv e-prints, July 2012.
[58] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2-111, 2006. ISSN 0003-4916. doi: http://dx.doi.org/10.1016/j.aop.2005.10.005. URL http://www.sciencedirect.com/science/article/pii/S0003491605002381.
[59] Courtney G Brell, Steven T Flammia, Stephen D Bartlett, and Andrew C Doherty. Toric codes and quantum doubles from two-body hamiltonians. New Journal of Physics, 13(5):053039, 2011. URL http://stacks.iop.org/1367-2630/13/i=5/a=053039.
[60] Jiannis K. Pachos. Quantum computation with abelian anyons on the honeycomb lattice. International Journal of Quantum Information, 04(06):947-954, 2006. doi: 10.1142/S0219749906002328. URL http://www.worldscientific.com/doi/abs/10.1142/S0219749906002328.
[61] M. Aguado, G. K. Brennen, F. Verstraete, and J. I. Cirac. Creation, manipulation, and detection of abelian and non-abelian anyons in optical lattices. Phys. Rev. Lett., 101:260501, Dec 2008. doi: 10.1103/PhysRevLett.101.260501. URL http://link.aps.org/
doi/10.1103/PhysRevLett.101.260501.
[62] Stefano Chesi, Daniel Loss, Sergey Bravyi, and Barbara M Terhal. Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes. New Journal of Physics, 12(2):025013, 2010. URL http://stacks.iop.org/1367-2630/12/i=2/a=025013.
Bibliography 93
[63] Keiji Liu Yu-xi Nori Franco Tanamoto, Tetsufumi Ono. Dynamic creation of a topologicallyordered hamiltonian using spin-pulse control in the heisenberg model. Scientic Reports,5:10076, 2015.
[64] G. Kells, J. K. Slingerland, and J. Vala. Description of kitaev's honeycomb model with toriccode
stabilizers. Phys. Rev. B, 80:125415, Sep 2009. doi: 10.1103/PhysRevB.80.125415. URL http://link.aps.org/doi/10.1103/PhysRevB.80.125415.
[65] Julien Vidal, Kai Phillip Schmidt, and S悶bastien Dusuel. Perturbative approach to an exactly solved problem: Kitaev honeycomb model. Phys. Rev. B, 78:245121, Dec 2008. doi:10.1103/PhysRevB.78.245121. URL http://link.aps.org/doi/10.1103/PhysRevB.78.245121.
[66] W. H. WOOTTERS, W. K. ZUREK. A single quantum cannot be cloned. Nature, 299:802-803, 1982. doi: 10.1038/299802a0. URL http://www.nature.com/nature/journal/v299/n5886/abs/299802a0.html.
[67] R. Kress. Numerical Analysis. Graduate Texts in Mathematics. Springer New York, 2012. ISBN 9781461205999. URL https://books.google.com.tw/books?id=Jv ZBwAAQBAJ.
[68] Spyridon Michalakis and JustynaP. Zwolak. Stability of frustration-free hamiltonians. Communications in Mathematical Physics, 322(2):277-302, 2013. ISSN 0010-3616. doi: 10.1007/s00220-013-1762-6. URL http://dx.doi.org/10.1007/s00220-013-1762-6.
[69] C. Daniel Freeman, C. M. Herdman, D. J. Gorman, and K. B. Whaley. Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime. Phys. Rev. B, 90:134302, Oct 2014. doi: 10.1103/PhysRevB.90.134302. URL http://link.aps.org/doi/10.1103/PhysRevB.90.134302.
[70] Stefano Chesi, Beat Rothlisberger, and Daniel Loss. Self-correcting quantum memory in a thermal environment. Phys. Rev. A, 82:022305, Aug 2010. doi: 10.1103/PhysRevA.82.
022305. URL http://link.aps.org/doi/10.1103/PhysRevA.82.022305.
[71] Benjamin J. Brown, Abbas Al-Shimary, and Jiannis K. Pachos. Entropic barriers for twodimensional quantum memories. Phys. Rev. Lett., 112:120503, Mar 2014. doi: 10.1103/PhysRevLett.112.120503. URL http://link.aps.org/doi/10.1103/PhysRevLett.112.120503.
[72] Vladimir Kolmogorov. Blossom v: a new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation, 1(1):43-67, 2009.
ISSN 1867-2949. doi: 10.1007/s12532-009-0002-8. URL http://dx.doi.org/10.1007/s12532-009-0002-8.
[73] Chenyang Wang, Jim Harrington, and John Preskill. Confinement-higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory. Annals
of Physics, 303(1):31-58, 2003. ISSN 0003-4916. doi: http://dx.doi.org/10.1016/S0003-4916(02)00019-2. URL http://www.sciencedirect.com/science/article/pii/S0003491602000192.
[74] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43(9):4452?505, 2002. doi: http://dx.doi.org/10.1063/1.1499754. URL http://scitation.aip.org/content/aip/journal/jmp/43/9/10.1063/1.1499754.
[75] M B Hastings. Quasi-adiabatic continuation in gapped spin and fermion systems: Goldstone's
theorem and flux periodicity. Journal of Statistical Mechanics: Theory and Experiment, 2007(05):P05010, 2007. URL http://stacks.iop.org/1742-5468/2007/i=05/a=P05010.
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