Add DM4.typ

DM4.typ

@@ -0,0 +1,120 @@
+#set page(
+  paper: "a4",
+  header: align(center)[
+    QCS - DM4 - Augustin LUCAS
+  ],
+)
+#import "@preview/showybox:2.0.1": showybox
+#import "@preview/physica:0.9.3": bra, ket
+
+#showybox(
+  frame: (
+    border-color: blue.darken(50%),
+    title-color: blue.lighten(60%),
+    body-color: blue.lighten(80%)
+  ),
+  title-style: (
+    color: black,
+    weight: "regular",
+    align: center
+  ),
+  shadow: (
+    offset: 3pt,
+  ),
+  title: "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:0.4.0": *
+
+    quantum-circuit(
+      lstick($v_0$), 1, targ(), 2, rstick($$), [\ ],
+      lstick($v_1$), 1, rstick($$), 2, [\ ],
+      lstick($v_2$), 1, ctrl(-2), 2, rstick($$)
+    )
+  }
+
+  Which corresponds to the following matrix:
+  
+  $"CNOT"[3,1] = mat(
+    1,0,0,0,0,0,0,0;
+    0,0,0,0,0,1,0,0;
+    0,0,1,0,0,0,0,0;
+    0,0,0,0,0,0,0,1;
+    0,0,0,0,1,0,0,0;
+    0,1,0,0,0,0,0,0;
+    0,0,0,0,0,0,1,0;
+    0,0,0,1,0,0,0,0;
+  )$
+
+
+#showybox(
+  frame: (
+    border-color: blue.darken(50%),
+    title-color: blue.lighten(60%),
+    body-color: blue.lighten(80%)
+  ),
+  title-style: (
+    color: black,
+    weight: "regular",
+    align: center
+  ),
+  shadow: (
+    offset: 3pt,
+  ),
+  title: "Assignment Q2",
+  ([
+    Let $A=1/sqrt(2) mat(-i,-1;1,i)$ and $B=mat(0,1;-1,0)$.
+    Which 2-qubit gate can you apply on the first qubits at the end of the circuit
+    #{
+      import "@preview/quill:0.4.0": *
+
+      quantum-circuit(
+        lstick($$), ctrl(2), 1, ctrl(2), 1, [\ ],
+        lstick($$), 1, ctrl(1), 1, ctrl(1), [\ ],
+        lstick($$), $B^(-1)$, $A^(-1)$, $B$, $A$, rstick($$)
+      )
+    }
+    to get a Toffoli gate ?
+  ]),
+)
+
+The circuit corresponds to applying the following gate:
+
+$C &= mat(I_2,0,0,0;0,I_2,0,0;0,0,B^(-1),0;0,0,0,B^(-1))
+      mat(I_2,0,0,0;0,A^(-1),0,0;0,0,I_2,0;0,0,0,A^(-1))
+      mat(I_2,0,0,0;0,I_2,0,0;0,0,B,0;0,0,0,B)
+      mat(I_2,0,0,0;0,A,0,0;0,0,I_2,0;0,0,0,A) 
+&= mat(I_2,0,0,0;0,I_2,0,0;0,0,I_2,0;0,0,0,B^(-1)A^(-1) B A)$
+
+And $B^(-1)A^(-1) B A = B A B A = mat(0,i;i,0)$.
+
+Let $D = i I_2$. Then, $C mat(I_2,0,0,0;0,I_2,0,0;0,0,I_2,0;0,0,0,D) = "Toffoli"$
+
+Which corresponds to the following circuit:
+#{
+  import "@preview/quill:0.4.0": *
+
+  quantum-circuit(
+    lstick($$), ctrl(2), 1, ctrl(2), 1, [\ ],
+    lstick($$), 1, ctrl(1), 1, ctrl(1), [\ ],
+    lstick($$), $B^(-1)$, $A^(-1)$, $B$, $A$, $D$, rstick($$)
+  )
+}
+