Publication: Thermalization in Kitaev’s quantum double models via Tensor Network techniques
Loading...
Official URL
Full text at PDC
Publication Date
2021
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Citations
Exportar
Abstract
We show that the Davies generator associated to any 2D Kitaev’s quantum double model has a non-vanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] R: Alicki, M. Fannes, M. Horodecki, On thermalization in Kitaev’s 2D model. Journal of Physics A: Mathematical and Theoretical, 42(6), 065303 (2009).
[2] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, On thermal stability of topological qubit in Kitaev’s 4D model. Open Systems & Information Dynamics, 17(01), 1-20 (2010).
[3] A. Anshu, Improved local spectral gap thresholds for lattices of finite size. Physical Review B 101, 165104 (2020).
[4] F. G. Brandao, A. W. Harrow, M. Horodecki, Local random quantum circuits are approximate polynomial-designs. Communications in Mathematical Physics, 346(2), 397-434 (2016).
[5] S. Bravyi, J. Haah, Quantum self-correction in the 3d cubic code model. Physical review letters, 111(20), 200501 (2003).
[6] S. Bravyi, B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. New Journal of Physics, 11(4), 043029 (2009)
[7] B. J. Brown, D. Loss, J. Pachos, C.N. Self, J. Wootton, Quantum memories at finite temperature. Reviews of Modern Physics, 88(4), 045005 (2016)
[8] B. J. Brown, A. Al-Shimary, and J. K Pachos, Entropic barriers for twodimensional quantum memories, Phys. Rev. Lett. 112, 120503 (2014).
[9] F. Cesi. Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields, Probability Theory and Related Fields 120.4, 569–584 (2001).
[10] J.I. Cirac, D. Poilblanc, N. Schuch, F. Verstraete, Entanglement spectrum and boundary theories with projected entangled-pair states. Physical Review B, 83(24), 245134 (2011).
[11] J.I. Cirac, D. Perez-Garcia, N. Schuch, F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, and theorems, arXiv:2011.12127 (2020).
[12] E. B. Davies, Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974).
[13] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory. L. Math. Phys., 43(9), 4452-4505 (2002).
[14] D. Gosset, E. Mozgunov. Local gap threshold for frustration-free spin systems, Journal of Mathematical Physics 57, 091901 (2016).
[15] J. Haah, Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A, 83(4), 042330 (2011).
[16] M. Lemm, E. Mozgunov. Spectral gaps of frustration-free spin systems with boundary, Journal of Mathematical Physics 60, 051901 (2019).
[17] M. Lemm, Finite-size criteria for spectral gaps in D-dimensional quantum spin systems. Analytic Trends in Mathematical Physics 741, 121 (2020).
[18] M.J. Kastoryano, F. Brandao, Quantum Gibbs Samplers: the commuting case. Communications in Mathematical Physics, 344(3), 915-957 (2016).
[19] M. J. Kastoryano and A. Lucia, Divide and conquer method for proving gaps of frustration free Hamiltonians. J. Stat. Mech. 033105 (2018)
[20] M. J. Kastoryano, A. Lucia, and D. Perez-Garcia, Locality at the boundary implies gap in the bulk for 2D PEPS. Commun. Math. Phys. 366, 895 (2019).
[21] A. Y. Kitaev, A. Shen and M. N. Vyalyi. Classical and quantum computation. (No. 47). American Mathematical Soc. (2002)
[22] S. Knabe, Energy gaps and elementary excitations for certain VBS-quantum antiferromagnets. Journal of Statistical Physics, 52(3):627–638 (1988).
[23] A. Kómár, O. Landon-Cardinal, K. Temme, Necessity of an energy barrier for self-correction of Abelian quantum doubles. Physical Review A, 93(5), 052337 (2016).
[24] T. Kuwahara, A.M. Alhambra, A. Anshu, Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states, arXiv:2007.11174.
[25] O. Landon-Cardinal and D. Poulin, Local topological order inhibits thermal stability in 2D, Phys. Rev. Lett., 110, 090502 (2013).
[26] M.A. Levin, X.G. Wen, String-net condensation: A physical mechanism for topological phases. Physical Review B, 71(4), 045110 (2005).
[27] R. Orus, Tensor networks for complex quantum systems. Nature Reviews Physics, 1(9), 538-550 (2019).
[28] D. Pérez-García, A. Pérez-Hernández, Locality estimes for complex time evolution in 1D, arXiv:2004.10516 (2020).
[29] D. Pérez-García, F. Verstraete, J. Cirac, M. M. Wolf. PEPS as unique ground states of local Hamiltonians. Quantum Information and Computation. 8. (2007)
[30] M. Pretko, X. Chen, Y. You, Fracton phases of matter. International Journal of Modern Physics A, 35(06), 2030003 (2020).
[31] K. Temme, Thermalization Time Bounds for Pauli Stabilizer Hamiltonians. Commun. Math. Phys. 350, 603–637 (2017).
[32] F. Verstraete, V. Murg, J.I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Advances in Physics, 57(2), 143-224 (2008).
[33] F. Verstraete, M.M. Wolf, D. Pérez-García, J.I. Cirac. Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States. Phys. Rev. Lett. 96, 220601 (2006).