Lecture 9. Computations on stabilizer circuits with magic

In this Lecture we briefly discussed how to perform universal quantum computations using stabilizer circuits with magic. An arbitrary unitary operation can be approximated with any accuracy by some composition of Clifford gates and $T$-gates. By the Solovey-Kitaev theorem, such approximation can be efficient as the precision grows. So, the circuits of the class $\mathrm{Clifford}+T$ are universal. The $T$ gate translates Pauli observables to the Clifford group. The class of unitary operations with this property is called 3-rd level of Clifford hierarchy. Such gates can be fault-tolerantly implemented by protocols of gate teleportation and state injection. Thus, the $T$-gate can be implemented using a stabilizer circuit and the magic state $|T\rangle = T |+\rangle$. Adding magic states to the stabilizer circuits allows us to increase its computational capabilities.