Measurement-based quantum computation with the toric code states

We study measurement-based quantum computation (MQC) using as a quantum resource the planar code state on a two-dimensional square lattice (planar analog of the toric code). It is shown that MQC with the planar code state can be efficiently simulated on a classical computer if at each step of MQC the sets of measured and unmeasured qubits correspond to connected subsets of the lattice. The simulation scheme is built upon Barahona's algorithm for computing the partition function of the Ising model on a planar graph. Our results provide a simulation method for MQC centered around planarity of graphs.