Local Haar-random circuit qubit code[1] 

An \(n\)-qubit code whose codewords are a pair of approximately locally indistinguishable states produced by starting with any two orthogonal \(n\)-qubit states and acting with a random unitary circuit of depth polynomial in \(n\). Two states are locally indistinguishable if they cannot be distinguished by local measurements. A single layer of the encoding circuit is composed of about \(n/2\) two-qubit nearest-neighbor gates run in parallel, with each gate drawn randomly from the Haar distribution on two-qubit unitaries.

The above circuit elements act on nearest-neighbor qubits arranged in a line, i.e., a one-dimensional geometry (\(D=1\), while codes for higher-dimensional geometries require \(O(n^{1/D})\)-depth circuits [1]. Follow-up work [2] revealed that optimal code properties require only \(O(\sqrt{n})\)-depth circuits for that case, and \(O(\sqrt{n})\)-depth circuits for a two-dimensional square-lattice geometry. This result has in turn to other types of Pauli noise [3], with the previous result holding for erasure noise only.