feat: add qubit transformations

layout.typ

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main.typ

@@ -9,6 +9,8 @@
   "Kristofers Solo",
 ))
 
+#let distance = $space.quad space.quad$
+
 = Bre-Ket Notation
 == Ket $ket(psi)$
 Represents a column vector for a quantum state.
@@ -96,3 +98,132 @@ $
   P(1) & =abs((1-i)/sqrt(7))^2=(1^2+(-1)^2)/7=2/7
 $
 #tip[ Sum must be $1$. ]
+
+= Single Qubit Unitary Transformations
+Quantum gates are unitary matrices $U$.
+$ U U^dagger=U^dagger U=I $
+=== Properties
+
+Linearity
+$(U(alpha ket(psi_1)+beta ket(psi_2))=alpha U ket(psi_1)+beta U ket(psi_2))$
+and preserves vector length.
+=== Matix form
+If $U ket(0)=a ket(0)+b ket(1))$ and $U ket(1)=c ket(0) + d ket(1)$, then
+$ U=mat(a, c; b, d) $
+
+Columns (and rows) must be orthonormal vectors: \
+$arrow(v_1^*) dot arrow(v_2)=0$ and $abs(arrow(v_1))^2=1$.
+
+== Pauli Gates
+=== I (Identity)
+$
+  I=mat(1, 0; 0, 1) distance
+  cases(
+    I ket(0)=ket(0),
+    I ket(1)=ket(1)
+  )
+$
+
+=== X (NOT)
+Bit flip
+$
+  X=mat(0, 1; 1, 0) distance
+  cases(
+    X ket(0)=ket(1),
+    X ket(1)=ket(0)
+  )
+$
+
+=== Y Gate
+$
+  Y=mat(0, -i; i, 0) distance
+  cases(
+    Y ket(0)=-i ket(1),
+    Y ket(1)=i ket(0)
+  )
+$
+
+=== Z Gate
+Phase flip
+$
+  Z=mat(1, 0; 0, -1) distance
+  cases(
+    Z ket(0) = ket(0),
+    Z ket(1) = -ket(1),
+  )
+$
+
+== Hadamard Gate ($H$)
+Creates superpositions
+$
+  H=1/sqrt(2) mat(1, 1; 1, -1) distance
+  cases(
+    H ket(0)=1/sqrt(2) ket(0) + 1/sqrt(2) ket(1),
+    H ket(-1)=1/sqrt(2) ket(0) - 1/sqrt(2) ket(1)
+  ) \
+  cases(
+    H ket(0)=ket(+),
+    H ket(1)=ket(-)
+  ) distance
+  H H=H^2=I
+$
+
+== Phase Gates
+=== $S$ Gate $(sqrt(Z))$
+$ S= mat(1, 0; 0, i) distance S^2=Z $
+
+=== $T$ Gate $(pi/8))$
+$ T= mat(1, 0; 0, e^(i pi/4)) distance T^2=S $
+
+== Rotation Gates ($R_n (theta)$)
+$
+  R_x (theta)=
+  e^((-i theta X)/2)=
+  mat(
+    cos theta/2, -i sin theta/2;
+    -i sin theta/2, cos theta/2
+  )
+$
+
+$
+  R_y (theta)=
+  e^((-i theta Y)/2)=
+  mat(
+    cos theta/2, -sin theta/2;
+    sin theta/2, cos theta/2
+  )
+$
+
+$
+  R_z (theta)=
+  e^((-i theta Z)/2)=
+  mat(
+    e^((-i theta)/2), 0
+    0, e^((i theta)/2)
+  )
+$
+
+#tip[
+  $R_alpha: cases(
+    R_alpha ket(0):cos alpha ket(0)+sin alpha ket(1),
+    R_alpha ket(1):-sin alpha ket(0)+cos alpha ket(1),
+  )$. This is $R_y(-2 alpha)$.]
+
+== Game Compositions
+Applied right to left. $U V ket(psi)=U(V ket(psi))$.
+- $H Z H=X$
+- $H X H=Z$
+
+== Inverse Tranformation
+$ U^(-1)=U^dagger $
+
+== Non-Unitary Operations
+(Not physically realizable as closed system evolution)
+
+=== Qubit Deletion
+$
+  cases(
+    U ket(0) = ket(0),
+    U ket(1) = ket(0)
+  )
+$