Surface codes: Towards practical large-scale quantum computation

This article provides an introduction to surface code quantum computing. We estimate the size and speed of a surface code quantum computer. We then the concept of the stabilizer, using two qubits, and extend this to stabilizers acting on a two-dimensional array of physical qubits, on we implement the surface code. We next describe how logical qubits are in the surface code array and give numerical estimates of their-tolerance. We outline how logical qubits are physically moved on the, how qubit braid transformations are constructed, and how a braid between logical qubits is equivalent to a controlled-NOT. We then describe the-qubit Hadamard, S and T operators, completing the set of required gates a universal quantum computer. We conclude by briefly discussing physical of the surface code. We include a number of appendices in which provide supplementary information to the main text.