Branching MERA codes: a natural extension of polar codes

We introduce a new class of circuits for constructing efficiently decodable error-correction codes, based on a recently discovered contractible tensor network. We perform an in-depth study of a particular example that can be thought of as an extension to Arikan's polar code. Notably, our numerical simulation show that this code polarizes the logical channels more strongly while retaining the log-linear decoding complexity using the successive cancellation decoder. These codes also display improved error-correcting capability with only a minor impact on decoding complexity. Efficient decoding is realized using powerful graphical calculus tools developed in the field of quantum many-body physics. In a companion paper, we generalize our construction to the quantum setting and describe more in-depth the relation between classical and quantum error correction and the graphical calculus of tensor networks.

[1]  H. Pishro-Nik,et al.  On bit error rate performance of polar codes in finite regime , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

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

[3]  Alexander Vardy,et al.  How to Construct Polar Codes , 2011, IEEE Transactions on Information Theory.

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

[5]  Xiao-Gang Wen,et al.  Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions , 2008, 0806.3509.

[6]  Rüdiger L. Urbanke,et al.  Polar codes: Characterization of exponent, bounds, and constructions , 2009, 2009 IEEE International Symposium on Information Theory.

[7]  Alexander Vardy,et al.  Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes , 2010, IEEE Transactions on Information Theory.

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

[9]  Emre Telatar,et al.  Polarization for arbitrary discrete memoryless channels , 2009, 2009 IEEE Information Theory Workshop.

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

[11]  Alexander Vardy,et al.  List decoding of polar codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[12]  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.

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

[14]  Emre Telatar,et al.  On the rate of channel polarization , 2008, 2009 IEEE International Symposium on Information Theory.

[15]  Rolf Johannesson,et al.  Fundamentals of Convolutional Coding , 1999 .

[16]  Andrew J. Ferris,et al.  Tensor Networks and Quantum Error Correction , 2013, Physical review letters.

[17]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[18]  Rüdiger L. Urbanke,et al.  Polar Codes are Optimal for Lossy Source Coding , 2009, IEEE Transactions on Information Theory.

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