Entanglement renormalization can be viewed as an encoding circuit for a family of approximate quantum error correcting codes. The logical information becomes progres-sively more well-protected against erasure errors at larger length scales. In particular, an approximate variant of holographic quantum error correcting code emerges at low energy for critical systems. This implies that two operators that are largely separated in scales behave as if they are spatially separated operators, in the sense that they obey a Lieb-Robinson type locality bound under a time evolution generated by a local Hamiltonian.
E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].
A.J. Ferris and D. Poulin, Tensor networks and quantum error correction, Phys. Rev. Lett. 113 (2014) 030501 [arXiv:1312.4578].
V. Giovannetti, S. Montangero and R. Fazio, Quantum multiscale entanglement renormalization ansatz channels, Phys. Rev. Lett. 101 (2008) 180503 [arXiv:0804.0520].
D. Kretschmann, D. Schlingemann and R.F. Werner, The information-disturbance tradeoff and the continuity of Stinespring’s representation, IEEE Trans. Informat. Theor. 54 (2008) 1708.
C. Bény and O. Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Phys. Rev. Lett. 104 (2010) 120501 [arXiv:0907.5391].
G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].
S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11 (2009) 043029 [arXiv:0810.1983].
S. Bravyi, D. Poulin and B. Terhal, Tradeoffs for reliable quantum information storage in 2d systems, Phys. Rev. Lett. 104 (2010) 050503 [arXiv:0909.5200].
S. Bravyi, M.B. Hastings and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett. 97 (2006) 050401 [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, 10th anniversary edition, Cambridge University Press, New York NY U.S.A., (2011).
M.M. Wolf, Quantum channels & operations: guided tour, http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf, (2012).
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ArXiv ePrint: 1701.00050
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Kim, I.H., Kastoryano, M.J. Entanglement renormalization, quantum error correction, and bulk causality. J. High Energ. Phys. 2017, 40 (2017). https://doi.org/10.1007/JHEP04(2017)040
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- Renormalization Group