An optimal dissipative encoder for the toric code

Abstract

We consider the problem of preparing specific encoded resource states for the toric code by local, time-independent interactions with a memoryless environment. We propose a construction of such a dissipative encoder which converts product states to topologically ordered ones while preserving logical information. The corresponding Liouvillian is made up of four-local Lindblad operators. For a qubit lattice of size $L\times L$, we show that this process prepares encoded states in time $O(L)$, which is optimal. This scaling compares favorably with known local unitary encoders for the toric code which take time of order $\Omega (L^2)$ and require active time-dependent control.

12 pages, 2 figures, v2: updated to match published version

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• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
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• Size-Driven Quantum Phase Transitions
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• Engineering autonomous error correction in stabilizer codes at finite temperature
• Bounds on Autonomous Quantum Error Correction
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• Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains