Tensor Networks and Quantum Error Correction

We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.

[1]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[2]  P. Morse.  Annals of Physics , 1957, Nature.

[3]  L. Goddard.  Information Theory , 1962, Nature.

[4]  Physical Review , 1965, Nature.

[5]  L. Kadanoff.  Scaling laws for Ising models near T(c) , 1966 .

[6]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[7]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[8]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[9]  A. Kitaev.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.

[10]  Jean-Pierre Tillich,et al.  Description of a quantum convolutional code. , 2003, Physical review letters.

[11]  J. Preskill,et al.  Confinement Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory , 2002, quant-ph/0207088.

[12]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[13]  Xiao-Gang Wen,et al.  String-net condensation: A physical mechanism for topological phases , 2004, cond-mat/0404617.

[14]  David Poulin.  Optimal and efficient decoding of concatenated quantum block codes , 2006 .

[15]  G. Vidal,et al.  Classical simulation of quantum many-body systems with a tree tensor network , 2005, quant-ph/0511070.

[16]  G. Vidal.  Entanglement renormalization. , 2005, Physical review letters.

[17]  David Poulin,et al.  On the iterative decoding of sparse quantum codes , 2008, Quantum Inf. Comput..

[18]  Xiao-Gang Wen,et al.  Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions , 2008 .

[19]  Igor L. Markov,et al.  Simulating Quantum Computation by Contracting Tensor Networks , 2008, SIAM J. Comput..

[20]  Erdal Arikan,et al.  Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels , 2008, IEEE Transactions on Information Theory.

[21]  G. Evenbly,et al.  Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law , 2009, 0903.5017.

[22]  David Poulin,et al.  Quantum serial turbo codes , 2009, IEEE Trans. Inf. Theory.

[23]  G. Vidal,et al.  Exact entanglement renormalization for string-net models , 2008, 0806.4583.

[24]  David Poulin,et al.  Fast decoders for topological quantum codes. , 2009, Physical review letters.

[25]  J. Ignacio Cirac,et al.  Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently , 2010 .

[26]  D. Pérez-García,et al.  PEPS as ground states: Degeneracy and topology , 2010, 1001.3807.

[27]  F. Gall,et al.  NP-hardness of decoding quantum error-correction codes , 2010, 1009.1319.

[28]  Joseph M Renes,et al.  Efficient polar coding of quantum information. , 2011, Physical review letters.

[29]  Saikat Guha,et al.  Polar Codes for Classical-Quantum Channels , 2011, IEEE Transactions on Information Theory.

[30]  F. Verstraete,et al.  Tree tensor networks and entanglement spectra , 2013, 1309.2255.

[31]  S Severini,et al.  8th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2013, May 21-23, 2013, Guelph, Canada , 2013, TQC.

[32]  A. Ferris.  Area law and real-space renormalization , 2013, 1301.2608.

[33]  Saikat Guha,et al.  Polar Codes for Degradable Quantum Channels , 2011, IEEE Transactions on Information Theory.

[34]  David Poulin,et al.  Degenerate Viterbi Decoding , 2012, IEEE Transactions on Information Theory.

[35]  Joseph M. Renes,et al.  Polar Codes for Private and Quantum Communication Over Arbitrary Channels , 2012, IEEE Transactions on Information Theory.

[36]  A. Ferris,et al.  Fourier transform for fermionic systems and the spectral tensor network. , 2013, Physical review letters.

[37]  G. Evenbly,et al.  Class of highly entangled many-body states that can be efficiently simulated. , 2012, Physical review letters.

[38]  Joseph M. Renes,et al.  Efficient Quantum Polar Codes Requiring No Preshared Entanglement , 2013, IEEE Transactions on Information Theory.