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#Steinspring quantum error correction code#
Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded.
In a regime where the recovery is successful to accuracy $\delta$ that is exponentially small in $\ell$, which is the case for perturbations of local commuting projector codes, our bound reads $kd^\bigr)$ in operator norm. Our tradeoff bounds relate the number of physical qubits $n$, the number of encoded qubits $k$, the code distance $d$, the accuracy parameter $\delta$ that quantifies how well the erasure channel can be reversed, and the locality parameter $\ell$ that specifies the length scale at which the recovery operation can be done. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes.
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