#import "qcs.typ": *
#set page(
  paper: "a4",
  header: align(center)[
    QCS - DM4 - Augustin LUCAS
  ],
)

#question("Assignment Q1",
  ([
    + Let $U=mat(u_(00),u_(01);u_(10),u_(11))$. Write a matrix representation of $U[1]$ and $U[2]$ for $n=2$.
    + For $n=3$, write a matrix representation of CNOT$[3,1]$.
  ])
)

+ $U[1] &= U times.circle I
  &= mat(u_(00),u_(01);u_(10),u_(11)) times.circle mat(1,0;0,1)
  &= mat(u_(00)I,u_(01)I;u_(10)I,u_(11)I)
  &= mat(u_(00),0,u_(01),0;0,u_(00),0,u_(01);u_(10),0,u_(11),0;0,u_(10),0,u_(11))$

  $U[2] &= I times.circle U
  &= mat(U,0;0,U) = mat(u_(00),u_(01),0,0;u_(10),u_(11),0,0;0,0,u_(00),u_(01);0,0,u_(01),u_(11))$



+ $"CNOT"[3,1]$ corresponds to the following circuit:

  #{
    import "@preview/quill:0.5.0": *

    quantum-circuit(
      lstick($v_0$), 1, targ(), 2, rstick($$), [\ ],
      lstick($v_1$), 1, rstick($$), 2, [\ ],
      lstick($v_2$), 1, ctrl(-2), 2, rstick($$)
    )
  }

  Which corresponds to the following matrix:
  
  $"CNOT"[3,1] = mat(
    1,0,0,0,0,0,0,0;
    0,0,0,0,0,1,0,0;
    0,0,1,0,0,0,0,0;
    0,0,0,0,0,0,0,1;
    0,0,0,0,1,0,0,0;
    0,1,0,0,0,0,0,0;
    0,0,0,0,0,0,1,0;
    0,0,0,1,0,0,0,0;
  )$


#question("Assignment Q2",
  ([
    Let $A=1/sqrt(2) mat(-i,-1;1,i)$ and $B=mat(0,1;-1,0)$.
    Which 2-qubit gate can you apply on the first qubits at the end of the circuit
    #{
      import "@preview/quill:0.5.0": *

      quantum-circuit(
        lstick($$), ctrl(2), 1, ctrl(2), 1, [\ ],
        lstick($$), 1, ctrl(1), 1, ctrl(1), [\ ],
        lstick($$), $B^(-1)$, $A^(-1)$, $B$, $A$, rstick($$)
      )
    }
    to get a Toffoli gate ?
  ]),
)

The circuit corresponds to applying the following gate:

$C &= mat(I_2,0,0,0;0,I_2,0,0;0,0,B^(-1),0;0,0,0,B^(-1))
      mat(I_2,0,0,0;0,A^(-1),0,0;0,0,I_2,0;0,0,0,A^(-1))
      mat(I_2,0,0,0;0,I_2,0,0;0,0,B,0;0,0,0,B)
      mat(I_2,0,0,0;0,A,0,0;0,0,I_2,0;0,0,0,A) 
&= mat(I_2,0,0,0;0,I_2,0,0;0,0,I_2,0;0,0,0,B^(-1)A^(-1) B A)$

And $B^(-1)A^(-1) B A = B A B A = mat(0,i;i,0)$.

Let $D = i I_2$. Then, $C mat(I_2,0,0,0;0,I_2,0,0;0,0,I_2,0;0,0,0,D) = "Toffoli"$

Which corresponds to the following circuit:
#{
  import "@preview/quill:0.5.0": *

  quantum-circuit(
    lstick($$), ctrl(2), 1, ctrl(2), 1, [\ ],
    lstick($$), 1, ctrl(1), 1, ctrl(1), [\ ],
    lstick($$), $B^(-1)$, $A^(-1)$, $B$, $A$, $D$, rstick($$)
  )
}