Surface code for low-density qubit array

Surface code is a promising candidate for the quantum error corrections needed for fault-tolerant quantum computations because it can operate on a two-dimensional grid of qubits. However, the gates and control lines become dense as more and more qubits are integrated, making their design and control difficult. This problem can be alleviated if the surface code can operate on sparse qubit arrays. Here, we give an solution for an array in which qubits are placed on edges as well as on nodes of a two-dimensional grid. The qubits on the edges are divided into two groups: those in one group act as the deputies of data qubits; the others act as deputies of the syndrome qubits. Syndrome outputs are obtained by multiplying the measured values of the syndrome and edge qubits. The procedure for the quantum part is the same as that of the ordinary surface code, making the surface code applicable to sparse qubit arrays.