Majorana fermion codes. (English) Zbl 1448.82012
Summary: We initiate the study of Majorana fermion codes (MFCs). These codes can be viewed as extensions of Kitaev’s one-dimensional (1D) model of unpaired Majorana fermions in quantum wires to higher spatial dimensions and interacting fermions. The purpose of MFCs is to protect quantum information against low-weight fermionic errors, that is, operators acting on sufficiently small subsets of fermionic modes. We examine to what extent MFCs can surpass qubit stabilizer codes in terms of their stability properties. A general construction of 2D MFCs is proposed that combines topological protection based on a macroscopic code distance with protection based on fermionic parity conservation. Finally, we use MFCs to show how to transform any qubit stabilizer code to a weakly self-dual CSS code.
MSC:
| 82B23 | Exactly solvable models; Bethe ansatz |
| 81V70 | Many-body theory; quantum Hall effect |
| 82D77 | Quantum waveguides, quantum wires |
References:
| [1] | Kitaev A 2003 Fault-tolerant quantum computation by anyons Ann. Phys.303 2 · Zbl 1012.81006 · doi:10.1016/S0003-4916(02)00018-0 |
| [2] | Wen X G and Niu Q 1990 Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces Phys. Rev. B 41 9377 · doi:10.1103/PhysRevB.41.9377 |
| [3] | Bravyi S, Hastings M B and Michalakis S 2010 Topological quantum order: stability under local perturbations arXiv:1001.0344 |
| [4] | Bravyi S and Hastings M B 2010 A short proof of stability of topological order under local perturbations arXiv:1001.4363 |
| [5] | Dennis E, Kitaev A, Landahl A and Preskill J 2002 Topological quantum memory J. Math. Phys.43 4452-505 · Zbl 1060.94045 · doi:10.1063/1.1499754 |
| [6] | Bacon D 2006 Operator quantum error correcting subsystems for self-correcting quantum memories Phys. Rev. A 73 012340 · doi:10.1103/PhysRevA.73.012340 |
| [7] | Bravyi S and Terhal B M 2009 A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes New J. Phys.11 043029 · doi:10.1088/1367-2630/11/4/043029 |
| [8] | Aliferis P, Gottesman D and Preskill J 2006 Quantum accuracy threshold for concatenated distance-3 codes Quantum Inf. Comput.6 97-165 · Zbl 1152.81671 |
| [9] | Chesi S, Loss D, Bravyi S and Terhal B M 2010 Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes New J. Phys.12 025013 · Zbl 1360.82045 · doi:10.1088/1367-2630/12/2/025013 |
| [10] | Alicki R, Horodecki M, Horodecki P and Horodecki R 2008 On thermal stability of topological qubit in Kitaev’s 4D model arXiv:0811.0033 |
| [11] | Nussinov Z and Ortiz G 2008 Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems Phys. Rev. B 77 064302 · doi:10.1103/PhysRevB.77.064302 |
| [12] | Castelnovo C and Chamon C 2007 Entanglement and topological entropy of the toric code at finite temperature Phys. Rev. B 76 184442 · doi:10.1103/PhysRevB.76.184442 |
| [13] | Iblisdir S, Perez-Garcia D, Aguado M and Pachos J 2009 Scaling law for topologically ordered systems at finite temperature Phys. Rev. B 79 134303 · doi:10.1103/PhysRevB.79.134303 |
| [14] | Alicki R, Fannes M and Horodecki M 2009 On thermalization in Kitaev’s 2D model J. Phys. A: Math. Gen.42 065303 arXiv:0810.4584 · Zbl 1159.81011 · doi:10.1088/1751-8113/42/6/065303 |
| [15] | Kitaev A 2000 Unpaired Majorana fermions in quantum wires Proc. Mesoscopic and Strongly Correlated Electron Systems Conf. (July 2000, Chernogolovka, Russia) (arXiv:cond-mat/0010440) |
| [16] | Freedman M et al 2010 Projective ribbon permutation statistics: a remnant of non-Abelian braiding in higher dimensions arXiv:1005.0583 |
| [17] | Fu L and Kane C L 2008 Superconducting proximity effect and Majorana fermions at the surface of a topological insulator Phys. Rev. Lett.100 096407 · doi:10.1103/PhysRevLett.100.096407 |
| [18] | Lutchyn R M, Sau J D and Das Sarma S 2010 Majorana fermions and topological phase transition in semiconductor/superconductor heterostructures arXiv:1002.4033 |
| [19] | Terhal B M and DiVincenzo D P 2002 Classical simulation of noninteracting-fermion quantum circuits Phys. Rev. A 65 032325 · doi:10.1103/PhysRevA.65.032325 |
| [20] | Bonderson P, Das Sarma S, Freedman M and Nayak C 2010 A blueprint for a topologically fault-tolerant quantum computer arXiv:1003.2856 |
| [21] | Bravyi S and Kitaev A 2002 Fermionic quantum computation Ann. Phys.298 210-26 · Zbl 0995.81012 · doi:10.1006/aphy.2002.6254 |
| [22] | Fidkowski L and Kitaev A 2009 The effects of interactions on the topological classification of free fermion systems arXiv:0904.2197 |
| [23] | Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) · Zbl 1049.81015 |
| [24] | Verstraete F and Cirac J I 2005 Mapping local Hamiltonians of fermions to local Hamiltonians of spins J. Stat. Mech. P09012 · Zbl 1456.82200 |
| [25] | Bombin H and Martin-Delgado M A 2007 Quantum measurements and gates by code deformation arXiv:0704.2540 |
| [26] | Kitaev A 2006 Anyons in an exactly solved model and beyond Ann. Phys.321 2 · Zbl 1125.82009 · doi:10.1016/j.aop.2005.10.005 |
| [27] | Kogut J B 1979 An introduction to lattice gauge theory and spin systems Rev. Mod. Phys.51 659-713 · doi:10.1103/RevModPhys.51.659 |
| [28] | Wegner F 1971 Duality in generalized Ising models and phase transitions without local order parameter J. Math. Phys.12 2259-72 · doi:10.1063/1.1665530 |
| [29] | Castelnovo C and Chamon C 2008 Topological order in a three-dimensional toric code at finite temperature Phys. Rev. B 78 155120 · doi:10.1103/PhysRevB.78.155120 |
| [30] | Bravyi S, Poulin D and Terhal B M 2010 Tradeoffs for reliable quantum information storage in 2D systems Phys. Rev. Lett.104 050503 · doi:10.1103/PhysRevLett.104.050503 |
| [31] | Kay A 2008 The non-equilibrium reliability of quantum memories arXiv:0807.0287 |
| [32] | Bombin H and Martin-Delgado M A 2006 Topological quantum distillation Phys. Rev. Lett.97 180501 · doi:10.1103/PhysRevLett.97.180501 |
| [33] | Cimasoni D and Reshetikhin N 2007 Dimers on surface graphs and spin structures. I Commun. Math. Phys.275 187 · Zbl 1135.82006 · doi:10.1007/s00220-007-0302-7 |
| [34] | Cimasoni D and Reshetikhin N 2008 Dimers on surface graphs and spin structures. II Commun. Math. Phys.281 445 · Zbl 1168.82012 · doi:10.1007/s00220-008-0488-3 |
| [35] | Bravyi S 2009 Contraction of matchgate tensor networks on non-planar graphs Contemp. Math.482 179 · Zbl 1171.81318 · doi:10.1090/conm/482/09419 |
| [36] | Bombin H 2010 Topological order with a twist: Ising anyons from an Abelian model arXiv:1004.1838 |
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