Lecture 10. Alternative ways of computing on quantum circuits

In this Lecture, we discussed some alternative methods to perform computations using quantum circuits. Considering such methods is useful both from a purely explorative point of view and for realising computations on various physical platforms. In measurement-based quantum computation (MBQC), a large entangled state, called a cluster state, is prepared, then sequentially locally measured, with subsequent measurements depending on the results of the previous ones. The correctness of this method is based on the one-bit teleportation protocol, and this method requires classical information processing during computation. On the other hand, any quantum circuit can be reduced to successive entangling measurements over an initially separable state. As a special case, it is sometimes interesting to study a one clean qubit model, in which one it is allowed to do a single entangled measurement over a maximally mixed state. Using the Hadamard test, it turns out to be possible to solve some range of interesting problems even in such a restricted model. In the theory of fault-tolerance, there are methods to correct errors on Clifford gates of the form $\langle H,S, C\mathrm{NOT}\rangle$. In order to achieve universality, one can implement the $T$-gate using magic state injection protocol. It turns out that arbitrary quantum computations can be performed by means of successive entangling Pauli measurements over the magic state.