We consider a model of quantum computation in which the set of operations is limited to nearest-neighbor interactions on a 2D lattice. We model movement of qubits with noisy SWAP operations. For this architecture we design a fault-tolerant coding scheme using the concatenated [[7,1,3]] Steane code. Our scheme is potentially applicable to ion-trap and solid-state quantum technologies. We calculate a lower bound on the noise threshold for our local model using a detailed failure probability analysis. We obtain a threshold of 1.85 × 10-5 for the local setting, where memory error rates are one-tenth of the failure rates of gates, measurement, and preparation steps. For the analogous nonlocal setting, we obtain a noise threshold of 3.61 × 10-5. Our results thus show that the additional SWAP operations required to move qubits in the local model affect the noise threshold only moderately.@en

We analyze and study the effects of locality on the fault-tolerance threshold for quantum computation. We analytically estimate how the threshold will depend on a scale parameter which characterizes the scale-up in the size of the circuit due to encoding. We carry out a detailed seminumerical threshold analysis for concatenated coding using the seven-qubit CSS code in the local and the "nonlocal" setting. First, we find that the threshold in the local model for the code has a dependence, which is in correspondence with our analytical estimate. Second, the threshold, beyond the dependence, does not depend too strongly on the noise levels for transporting qubits. Beyond these results, we find that it is important to look at more than one level of concatenation in order to estimate the threshold and that it may be beneficial in certain places, like in the transportation of qubits, to do error correction only infrequently.@en