$($a$)$ Draw a classical circuit, with $3$ input bits and $2$ output bits, that consists of XOR

gates and FANOUT, and maps

$0007->000017->011007->100107->11$

$1117->001107->010117->101017->11$

$($Aside: Note that the above map may be used to identify where a single bit flip

has occurred $($or that no bit flip has occurred$),$ assuming the input state was

either $000$ or $111).$

$2$

$($b$)$ Give a reversible $5-$bit circuit for computing the above function while keeping the

input intact. E$.$g$.$ the circuit should map $100007->10010.$ Use only CNOT

gates.