$1.$ Consider function f$3$ where f$3(0)=1$ and fx$(1)=0.$

a$.$ What would be oracle associated with this function?

b$.$ Build the internal circuit of the oracle. $($There is no single way to do this.$)$

c$.$ What is the matrix associated with this quantum operation. $($Textbook mentions it$,$ but it is incorrect!$)$

$2.$ In lecture, we implemented the oracles associated with f$1($constant$)$ and

f$2($balanced$)$ and compared them when input to each was $|00\backslash$rangle

$.$ We saw the results were the same, i$.$e$.,|00\backslash$rangle $,$ and thus indistinguishable on a single run. Now, having defined the matrix for f$3$ in the previous question, implement the oracles associated with f$3($balanced$)$ and f$4($constant$),$ pass each input $|00\backslash$rangle and output the result. $($They both should output $|10\backslash$rangle and thus indistinguishable in a single run.$)$ Also, render the circuits. Both should look like the following:

q$[0]10>||--------------------$off$--$

$||$

q$[1]10>||--------------------0$n$---$

$3.$ Let's now consider a function that receives two bits as input and outputs a single bit. It works as follows:

f$(00)=$ f$(01)=1$ and f$(10)=$ f$(11)=0.$

a$.$ Is this function balanced, constant, or none?

b$.$ What would be oracle associated with this function?

c$.$ What is the matrix associated with this quantum operation. Hint: It should be a $8*8$ matrix.

$4.$ Now that we have defined the matrix for the function in the previous question, let's define its oracle in Strange. Pass the circuit input $|000\backslash$rangle and $|110\backslash$rangle$,$ output the result for each $($respectively $|100\backslash$rangle and $|110\backslash$rangle $,$ and render the circuit for each. The circuit snapshots are given below

q$[0]10>||--------------------$off$--$

$||$

q$[1]10>||--------------------0$ff$---$

q$[2]10>||--------------------$on$--$

q$[0]10>||--------------------0$ff$---$

q$[1]10>||--------------------$on$--$

q$[2]10>||--------------------0$n$---$