Exercise $6\mathrm{.}1\mathrm{.}2$

$($a$)$ Describe complex numbers ${}_i,i=0,1,$dots,$N-1$ satisfying

${\displaystyle \underset{i}{\overset{?}{}}}|{}_i{|}^2=1$ and $|{\displaystyle \underset{i}{\overset{?}{}}}{}_i{|}^2=0.$

$($b$)$ Describe complex numbers ${}_i,i=0,1,$dots,$N-1$ satisfying

${\displaystyle \underset{i}{\overset{?}{}}}|{}_i{|}^2=\frac{1}{N}$ and $|{\displaystyle \underset{i}{\overset{?}{}}}{}_i{|}^2=1.$

Quantum interference has already been seen in Section $1\mathrm{.}6$ where we examined

the photon and beam$-$splitter apparatus. We can revisit this example in the lan$-$

guage of quantum circuits to provide a concrete example of interference in a

quantum computation. Consider the quantum circuit in Figure $6\mathrm{.}4.$ This circuit

does not perform a purely quantum computation, because we make a measure$-$

ment immediately after the first Hadamard gate $($recall the definition of the

Hadamard gate $H,$ from Section $4\mathrm{.}3).$