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# Spatial quantum error correction threshold

Mar 2014

We consider a spatial analogue of the quantum error correction threshold.Given individual time-independent subsystems in which quantum information iscoherent over sufficiently long lengths, we show how the information can bekept coherent for arbitrarily long lengths by forming time-independentcomposite systems. The subsystem coherence length exhibits threshold behavior.When it exceeds a length ${\xi}_{th}$, meaningful information can be extractedfrom the ground state of the composite system. Otherwise, the information isgarbled. The threshold transition implies that the parent Hamiltonian of theground state has gone from gapped to gapless. Ramifications of the constructionfor PEPS and for adiabatic quantum computation are noted.

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