Simple magic state calculations using `Improved Simulation of Stabilizer Circuits'

We employed the techniques from [Phys. Rev. A $\mathbf{\text{70}}$, 052328 (2004)/arXiv:0406196] to analytically study two set of quantum circuits containing one $T$ gate/magic $\mathrm{\mid }T⟩=\frac{\mathrm{\mid }0⟩+\sqrt{i}\mathrm{\mid }1⟩}{\sqrt{2}}$ state. These include the $T$ state via gate teleportation and magic state injection on the rotated surface code.