Linked-cluster technique for finding the distance of a quantum LDPC code

We present a linked-cluster technique for calculating the distance of a quantum LDPC code. It offers an advantage over existing deterministic techniques for codes with small relative distances (which includes all known families of quantum LDPC codes), and over the probabilistic technique for codes with sufficiently high rates.

[1]  David Poulin,et al.  Tradeoffs for reliable quantum information storage in 2D systems , 2010, Quantum Cryptography and Computing.

[2]  Austin G. Fowler,et al.  A primer on surface codes: Developing a machine language for a quantum computer , 2012 .

[3]  T. Schaetz,et al.  Simulating a quantum magnet with trapped ions , 2008 .

[4]  Jacques Stern,et al.  A method for finding codewords of small weight , 1989, Coding Theory and Applications.

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

[6]  Matthias Steffen,et al.  Simultaneous State Measurement of Coupled Josephson Phase Qubits , 2005, Science.

[7]  Markus Grassl,et al.  Searching for linear codes with large minimum distance , 2006 .

[8]  Leonid P. Pryadko,et al.  Fault-Tolerance of"Bad"Quantum Low-Density Parity Check Codes , 2012 .

[9]  Rodney M. Goodman,et al.  The complexity of information set decoding , 1990, IEEE Trans. Inf. Theory.

[10]  Wieb Bosma,et al.  Discovering Mathematics with Magma , 2006 .

[11]  Ilya Dumer,et al.  Suboptimal decoding of linear codes: partition technique , 1996, IEEE Trans. Inf. Theory.

[12]  O. Astafiev,et al.  Demonstration of conditional gate operation using superconducting charge qubits , 2003, Nature.

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

[14]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[15]  L. Landau Fault-tolerant quantum computation by anyons , 2003 .

[16]  Ilya Dumer Soft-decision decoding using punctured codes , 2001, IEEE Trans. Inf. Theory.

[17]  L. Pryadko,et al.  Fault tolerance of quantum low-density parity check codes with sublinear distance scaling , 2013 .

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

[19]  Helmut G. Katzgraber,et al.  Strong resilience of topological codes to depolarization , 2012, 1202.1852.

[20]  Zhi Ma,et al.  A finite Gilbert-Varshamov bound for pure stabilizer quantum codes , 2004, IEEE Transactions on Information Theory.

[21]  Leonid P. Pryadko,et al.  Improved quantum hypergraph-product LDPC codes , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[22]  Robert Raussendorf,et al.  Fault-tolerant quantum computation with high threshold in two dimensions. , 2007, Physical review letters.

[23]  Chin-Kun Hu,et al.  Exact cluster size distributions and mean cluster sizes for the q-state bond-correlated percolation model , 1987 .

[24]  Leonid P Pryadko,et al.  Universal set of scalable dynamically corrected gates for quantum error correction with always-on qubit couplings. , 2012, Physical review letters.

[25]  Amir Yacoby,et al.  Long-Distance Spin-Spin Coupling via Floating Gates , 2011, Physical Review X.

[26]  Raymond Laflamme,et al.  A Theory of Quantum Error-Correcting Codes , 1996 .

[27]  Michael S. Postol A Proposed Quantum Low Density Parity Check Code , 2001, quant-ph/0108131.

[28]  Markus Grassl,et al.  A New Minimum Weight Algorithm for Additive Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[29]  R. Blatt,et al.  Towards fault-tolerant quantum computing with trapped ions , 2008, 0803.2798.

[30]  I. Chuang,et al.  Quantum Teleportation is a Universal Computational Primitive , 1999, quant-ph/9908010.

[31]  L. Pryadko,et al.  Quantum Kronecker sum-product low-density parity-check codes with finite rate , 2012, 1212.6703.

[32]  Jeffrey S. Leon,et al.  A probabilistic algorithm for computing minimum weights of large error-correcting codes , 1988, IEEE Trans. Inf. Theory.

[33]  Po-Shen Loh,et al.  Probabilistic Methods in Combinatorics , 2009 .

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

[35]  J. Spencer Probabilistic Methods in Combinatorics , 1974 .

[36]  Iryna Andriyanova,et al.  New constructions of CSS codes obtained by moving to higher alphabets , 2012, ArXiv.

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

[38]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[39]  Isaac L. Chuang,et al.  Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations , 1999, Nature.

[40]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

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

[42]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[43]  Gilles Zémor,et al.  Quantum LDPC codes with positive rate and minimum distance proportional to n½ , 2009, ISIT.

[44]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[45]  Hideki Imai,et al.  Quantum Error Correction Beyond the Bounded Distance Decoding Limit , 2010, IEEE Transactions on Information Theory.