Quantum Codes From Classical Graphical Models

We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or graphical models in classical information theory and machine learning. It enables us to formulate the description of the recently-introduced ‘coherent parity check’ quantum error correction codes entirely within the language of classical information theory. This makes our construction accessible without requiring background in quantum error correction or even quantum mechanics in general. More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes.

[1]  Nir Friedman,et al.  Probabilistic Graphical Models - Principles and Techniques , 2009 .

[2]  R. Barends,et al.  Superconducting quantum circuits at the surface code threshold for fault tolerance , 2014, Nature.

[3]  Manuela Herman,et al.  Quantum Computing: A Gentle Introduction , 2011 .

[4]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[5]  Gilles Zémor,et al.  Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength , 2009, IEEE Transactions on Information Theory.

[6]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[7]  S. Brierley,et al.  Accelerated Variational Quantum Eigensolver. , 2018, Physical review letters.

[8]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[9]  T. Beth,et al.  Codes for the quantum erasure channel , 1996, quant-ph/9610042.

[10]  Andrew W. Cross,et al.  Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits. , 2017, Physical review letters.

[11]  Vaidman,et al.  Error prevention scheme with four particles. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[12]  Brendan J. Frey,et al.  Factor graphs and the sum-product algorithm , 2001, IEEE Trans. Inf. Theory.

[13]  David J. C. MacKay,et al.  Information Theory, Inference, and Learning Algorithms , 2004, IEEE Transactions on Information Theory.

[14]  Ying Li,et al.  Topological quantum computing with a very noisy network and local error rates approaching one percent , 2012, Nature Communications.

[15]  Stephen Brierley,et al.  A Generalised Variational Quantum Eigensolver , 2018 .

[16]  James L. Park.  The concept of transition in quantum mechanics , 1970 .

[17]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[18]  V. Kendon,et al.  Protecting quantum memories using coherent parity check codes , 2017, Quantum Science and Technology.

[19]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem , 2014, 1412.6062.

[20]  Stefan Zohren,et al.  Graphical structures for design and verification of quantum error correction , 2016, Quantum Science and Technology.

[21]  M. Mariantoni,et al.  Surface codes: Towards practical large-scale quantum computation , 2012, 1208.0928.

[22]  Hartmut Neven,et al.  Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle , 2016, ArXiv.

[23]  Alain Glavieux,et al.  Reflections on the Prize Paper : "Near optimum error-correcting coding and decoding: turbo codes" , 1998 .

[24]  J. Gambetta,et al.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets , 2017, Nature.

[25]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm , 2014, 1411.4028.

[26]  D. Gottesman.  The Heisenberg Representation of Quantum Computers , 1998, quant-ph/9807006.

[27]  Igor Devetak,et al.  Correcting Quantum Errors with Entanglement , 2006, Science.

[28]  Austin G. Fowler,et al.  Surface code quantum computing by lattice surgery , 2011, 1111.4022.

[29]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[30]  Pascal O. Vontobel.  Stabilizer quantum codes: A unified view based on Forney-style factor graphs , 2008, 2008 5th International Symposium on Turbo Codes and Related Topics.

[31]  L. Brown.  Dirac ’ s The Principles of Quantum Mechanics * , 2006 .

[32]  D. Poulin,et al.  Quantum Graphical Models and Belief Propagation , 2007, 0708.1337.

[33]  A. Glavieux,et al.  Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1 , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[34]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[35]  Robert B. Griffiths,et al.  Quantum Error Correction , 2011 .

[36]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[37]  Radford M. Neal,et al.  Near Shannon limit performance of low density parity check codes , 1996 .

[38]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[39]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[40]  H. Neven,et al.  Quantum Algorithms for Fixed Qubit Architectures , 2017, 1703.06199.

[41]  David J. C. MacKay,et al.  Sparse-graph codes for quantum error correction , 2004, IEEE Transactions on Information Theory.

[42]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[43]  Joshua Roffe,et al.  The coherent parity check framework for quantum error correction , 2019 .

[44]  Andrew W. Cross,et al.  Quantum optimization using variational algorithms on near-term quantum devices , 2017, Quantum Science and Technology.

[45]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .