It was already possible to correct a positive fraction of qubits as long as you accept a constant distance; consider the Hamiltonian: H = <a_i1|X|a_i2> + <a_i2

It was already possible to correct a positive fraction of qubits as long as you accept a constant distance; consider the Hamiltonian: H = <a_i1|X|a_i2> + <a_i2|Z|a_i3> + <a_i1|Z|a_(i+1)1> + <a_i3|X|a_(i+1)3> [i = 1..N] This generates a [N, N/3, 2] code, with N/3 logical qubits per physical qubit. The trouble is that a no-go theorem states that for codes defined by operators which are local in any finite dimension, the code distance d = o(N); in particular, in two dimensions, d = O(√N). (IIRC; haven't done this in a while...)