Cellular-automaton decoders for topological quantum memories

A new error correction method for quantum computing memories is based on local computing elements. Michael Herold from the Freie Universitat Berlin in Germany, with colleagues in Germany, Denmark and the UK, sought to address the challenge of maintaining information stored in topological quantum memories. Without a stable memory, delicate quantum states can decay quickly, introducing errors in stored information. Error correction is an important process in stabilizing topological memories, but was previously conceived as a system-wide process. The proposed practical error correction mechanism relies on parallel cellular operations within the topological quantum memory, so that the local operations replace the need for a complex system-wide scheme. The concept has the further benefit of being compatible with classical hardware, and it is easily scalable.

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

[2]  Keisuke Fujii,et al.  Measurement-Free Topological Protection Using Dissipative Feedback , 2014, 1401.6350.

[3]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[4]  James R. Wootton,et al.  Self-correcting quantum memory with a boundary , 2012, 1206.0991.

[5]  Austin G. Fowler,et al.  Quantifying the effects of local many-qubit errors and nonlocal two-qubit errors on the surface code , 2014 .

[6]  F. Barahona.  On the computational complexity of Ising spin glass models , 1982 .

[7]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[8]  M. J. Oliveira,et al.  Isotropic majority-vote model on a square lattice , 1992 .

[9]  Eric Dennis.  Purifying Quantum States: Quantum and Classical Algorithms , 2005 .

[10]  Amin Shokrollahi,et al.  LDPC Codes: An Introduction , 2004 .

[11]  Lise Ponselet,et al.  Phase transitions in probabilistic cellular automata , 2013, 1312.3612.

[12]  D. D. Awschalom,et al.  Decoherence-protected quantum gates for a hybrid solid-state spin register , 2012, Nature.

[13]  Fern H. E. Watson,et al.  Estimating Overheads for Topological Quantum Codes , 2013 .

[14]  Maury Bramson,et al.  A Useful Renormalization Argument , 1991 .

[15]  Peter GBcsl.  Reliable Cellular Automata with Self-organization , 1997 .

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

[17]  James R. Wootton,et al.  Efficient Markov chain Monte Carlo algorithm for the surface code , 2013, 1302.2669.

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

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

[20]  Austin G. Fowler,et al.  Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time , 2013, Quantum Inf. Comput..

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

[22]  Schumacher,et al.  Noncommuting mixed states cannot be broadcast. , 1995, Physical review letters.

[23]  Lawrence Gray,et al.  The positive rates problem for attractive nearest neighbor spin systems on ℤ , 1982 .

[24]  James R. Wootton.  A Simple Decoder for Topological Codes , 2013, Entropy.

[25]  Nicholas Pippenger,et al.  Developments in "The synthesis of reliable organ-isms from unreliable components , 1990 .

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

[27]  James R. Wootton,et al.  High threshold error correction for the surface code. , 2012, Physical review letters.

[28]  Daniel Gottesman,et al.  What is the Overhead Required for Fault-Tolerant Quantum Computation? , 2013 .

[29]  H. Rieger,et al.  Finite-size scaling analysis of exact ground states for ±J spin glass models in two dimensions , 1997 .

[30]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[31]  Lawrence Gray.  Toom’s Stability Theorem in Continuous Time , 1999 .

[32]  Martin J. Wainwright,et al.  An Efficient 10GBASE-T Ethernet LDPC Decoder Design With Low Error Floors , 2010, IEEE Journal of Solid-State Circuits.

[33]  R. Bhatia.  Matrix Analysis , 1996 .

[34]  Lawrence Gray,et al.  A Reader's Guide to Gacs's “Positive Rates” Paper , 2001 .

[35]  Austin G. Fowler,et al.  Experimental demonstration of topological error correction , 2009, Nature.

[36]  J. Myrheim,et al.  On the theory of identical particles , 1977 .

[37]  A. Kitaev,et al.  Quantum codes on a lattice with boundary , 1998, quant-ph/9811052.

[38]  Daniel Loss,et al.  Quantum memory coupled to cavity modes , 2010, 1011.3762.

[39]  B. Terhal,et al.  A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes , 2008, 0810.1983.

[40]  Jose A. Cuesta,et al.  General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions , 2003, cond-mat/0306354.

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

[42]  Jiannis K. Pachos,et al.  Engineering complex topological memories from simple Abelian models , 2009, 0908.0708.

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

[44]  Fernando Pastawski,et al.  Quantum memories based on engineered dissipation , 2010, 1010.2901.

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

[46]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[47]  A. G. Fowler,et al.  Threshold error rates for the toric and surface codes , 2009, 0905.0531.

[48]  James R. Wootton,et al.  Enhanced thermal stability of the toric code through coupling to a bosonic bath , 2013, 1309.0621.

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

[50]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[51]  David Avis,et al.  A survey of heuristics for the weighted matching problem , 1983, Networks.

[52]  Claudio Chamon,et al.  Toric-boson model: Toward a topological quantum memory at finite temperature , 2008, 0812.4622.

[53]  L. Van Hove,et al.  WITHDRAWN: SUR L'INTÉGRALE DE CONFIGURATION POUR LES SYSTÉMES DE PARTICULES Á UNE DIMENSION , 1966 .

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

[55]  D. Gottesman.  Theory of fault-tolerant quantum computation , 1997, quant-ph/9702029.

[56]  David Poulin,et al.  Fault-tolerant renormalization group decoder for abelian topological codes , 2013, Quantum Inf. Comput..

[57]  Maury Bramson,et al.  Perplexing Problems in Probability , 1999 .

[58]  J. T. Chalker,et al.  Two-dimensional random-bond Ising model, free fermions, and the network model , 2002 .

[59]  I. Chuang,et al.  Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance , 2001, Nature.

[60]  Dorit Aharonov,et al.  Fault-tolerant Quantum Computation with Constant Error Rate * , 1999 .

[61]  Adam C. Whiteside,et al.  Towards practical classical processing for the surface code: Timing analysis , 2012, 1202.5602.

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

[63]  Beni Yoshida,et al.  Feasibility of self-correcting quantum memory and thermal stability of topological order , 2011, 1103.1885.

[64]  J. Harrington,et al.  Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes , 2004 .

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

[66]  S. Bravyi,et al.  Quantum self-correction in the 3D cubic code model. , 2013, Physical review letters.

[67]  Péter Gács,et al.  Reliable computation with cellular automata , 1983, J. Comput. Syst. Sci..

[68]  Sean D. Barrett,et al.  Logical error rate scaling of the toric code , 2013, 1312.5213.

[69]  Frank Wilczek,et al.  Quantum Mechanics of Fractional-Spin Particles , 1982 .

[70]  James R. Wootton,et al.  Effective quantum-memory Hamiltonian from local two-body interactions , 2014 .

[71]  Tetsufumi Tanamoto,et al.  Dynamic generation of topologically protected self-correcting quantum memory , 2013, 1302.3998.

[72]  Austin G. Fowler,et al.  Threshold error rates for the toric and planar codes , 2010, Quantum Inf. Comput..

[73]  D. Loss,et al.  Self-correcting quantum memory in a thermal environment , 2009, 0908.4264.

[74]  Daniel Gottesman,et al.  Fault-tolerant quantum computation with constant overhead , 2013, Quantum Inf. Comput..

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

[76]  Leo P. Kadanoff,et al.  Determination of an Operator Algebra for the Two-Dimensional Ising Model , 1971 .

[77]  John M. Martinis,et al.  Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing , 2014 .

[78]  J Eisert,et al.  Precisely timing dissipative quantum information processing. , 2012, Physical review letters.

[79]  David Poulin,et al.  A renormalization group decoding algorithm for topological quantum codes , 2010, 2010 IEEE Information Theory Workshop.

[80]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[81]  Vladimir Kolmogorov,et al.  Blossom V: a new implementation of a minimum cost perfect matching algorithm , 2009, Math. Program. Comput..

[82]  Daniel Loss,et al.  Breakdown of surface-code error correction due to coupling to a bosonic bath , 2014, 1402.3108.

[83]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[84]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.