(Solved): A controlled swap gate CSWAP is a gate acting on three systems: a control qubit (two dimensional), ...

A controlled swap gate CSWAP is a gate acting on three systems: a control qubit (two dimensional), and two target systems (which, for this problem, may be of arbitrary, but equal, dimension). If the control qubit is the left-most system, then the CSWAP acts as follows: $$\begin{array}{l}\mathrm{CSWAP}?0,a,b?=?0,a,b?\\ \mathrm{CSWAP}?1,a,b?=?1,b,a?\end{array}$$ (a) Consider the algorithm shown in Figure 1. The CSWAP gate is depicted as a vertical line passing through three wires with a solid circle on the control wire (top most) and an " $$x$$ " on each of the two target wires. This algorithm is meant to allow an experimenter to estimate $$??a?b?{?}^2$$ for arbitrary (normalized) $$?a?$$ and $$?b?$$. Prove that the probability of observing a $$?0?$$ after the final measurement on the top (qubit) wire can give you this quantity. Namely, show that: $$P(0)=\frac{1}{2}+\frac{1}{2}??a?b?{?}^2$$ (Hint: Recall that $$?x,y?z,w?=?x?z???y?w?$$. Note that, if given a sufficient number of states prepared in the form $$?a???b?$$, one could repeat this experiment multiple times to obtain an estimate of the desired inner-product.