Reliable computation with cellular automata

Gács, Péter

doi:10.1145/800061.808730

Cited by 39 publications

(67 citation statements)

References 7 publications

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“…Our proof borrows some techniques from [21,23,28], specifically Section 5.1 of [23]. For reader's convenience, we briefly summarize the RG decoding algorithm below (see Section VI for details).…”

Section: Appendix B: Threshold Theorem For Topological Stabilizer Codesmentioning

confidence: 99%

Quantum Self-Correction in the 3D Cubic Code Model

2013

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A big open question in the quantum information theory concerns feasibility of a selfcorrecting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction if the memory is put in contact with a cold enough thermal bath. In this paper we derive a rigorous lower bound on the memory time T mem of the 3D Cubic Code model which was recently conjectured to have a self-correcting behavior. Assuming that dynamics of the memory system can be described by a Markovian master equation of Davies form, we prove that T mem ≥ L cβ for some constant c > 0, where L is the lattice size and β is the inverse temperature of the bath. However, this bound applies only if the lattice size does not exceed certain critical value L * ∼ e β/3 . We also report a numerical Monte Carlo simulation of the studied memory indicating that our analytic bounds on T mem are tight up to constant coefficients. In order to model the readout step we introduce a new decoding algorithm which might be of independent interest. Our decoder can be implemented efficiently for any topological stabilizer code and has a constant error threshold under random uncorrelated errors.

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Section: Appendix B: Threshold Theorem For Topological Stabilizer Codesmentioning

confidence: 99%

Quantum Self-Correction in the 3D Cubic Code Model

2013

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“…Therefore it came as a surprise that P. Gacs constructed a model on the infinite lattice which violates the positive rates conjecture [157,158].…”

Section: Critical Phenomenamentioning

confidence: 99%

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

Schütz

2003

J. Phys. A: Math. Gen.

137

136

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Recent work on stochastic interacting particle systems with two particle species (or single-species systems with kinematic constraints) has demonstrated the existence of spontaneous symmetry breaking, long-range order and phase coexistence in nonequilibrium steady states, even if translational invariance is not broken by defects or open boundaries. If both particle species are conserved, the temporal behaviour is largely unexplored, but first results of current work on the transition from the microscopic to the macroscopic scale yield exact coupled nonlinear hydrodynamic equations and indicate the emergence of novel types of shock waves which are collective excitations stabilized by the flow of microscopic fluctuations. We review the basic stationary and dynamic properties of these systems, highlighting the role of conservation laws and kinetic constraints for the hydrodynamic behaviour, the microscopic origin of domain wall (shock) stability and the coarsening dynamics of domains during phase separation.

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“…Among the important topics that were left out are discussions of hardware implementations of CAs (e.g., Toffoli and Margolus, 1987); reliable computation in CAs (e.g., Gacs, 1986); a larger discussion of analysis of CAs in terms of computation theory (e.g., Wolfram, 1984b, Nordahl, 1989, Moore, 1996, computation in more complex CA architectures, including three-dimensional CAs (e.g., Tsalides, Hicks, and York, 1989), CAs with real-valued states (e.g., Garzon and Botelho, 1993), and systolic arrays (e.g., Kung, 1982), and the many applications CAs and CA-like architectures have found in parallel computation.…”

Section: Resultsmentioning

confidence: 99%