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MAX-k-SAT in one dimension is also in P, and can in fact be reduced to MAX-2-SAT with large enough variables. Despite the intuition that one-dimensional systems should not be too difficult, we prove: Theorem 2. 1-DIM 12-STATE HAMILTONIAN is QMA-complete. The theorem implies a striking qualitative difference between the quantum and the classical one-dimensional versions of the same problem — one is easy, whereas the other is complete for a class which seems to be strictly larger than NP. This might seem 46 D.

Aspenberg, M. Yampolsky Now the proof goes precisely as in [Mi5]. Set Ui = PˆNi −1 , numbered so that U0 = PˆN −1 (−a). Equip every Ui with the Poincaré distance ρi (x, y). Note that for each i > 0 there are exactly two univalent branches g1i and g2i of Ra−1 defined on Ui , each of which carries Ui into a proper subset of some U j . It follows that for each puzzle piece PNi −1 , i > 0, the branch gki shrinks the Poincaré distance by some definite factor λ < 1. Since the orbit z 0 , z 1 , z 2 , .

Define the parameter annuli An = n \ n+1 . Fix the number N = m j from now on and let N = . Define a map defined on , by H = H N : a → h˜ a−1 (−a). it is injective, which follows We see that H N : A N → AaN0 . 9. The next issue is to show that the map H N is quasiconformal with a definite bound on the dilatation independent of N . Here the proof is again the same as in [Ro1]; let us differentiate the relation h˜ aN (H N (a)) = −a. Then we get ∂h aN (H N (a))∂ H N (a) + ∂h aN ∂ H N (a) = 0. This implies that the Beltrami coefficient µ(a) = ∂ H N/∂ H N satisfies |µ(a)| = |∂h aN (H N (a))| KN − 1 = < 1, a |∂h N (H N (a))| KN + 1 where K N is the dilatation of h aN .