feat(fundamentals): add fundamentals

layout.typ

@@ -47,21 +47,12 @@
 
   // Formatting for regular text
   set par(justify: true, leading: 1em, first-line-indent: indent, spacing: 1em)
-  show heading: set block(spacing: 1.5em)
+  show heading: set block(spacing: 0.7em)
 
   set terms(separator: [ -- ])
 
   // Headings
   set heading(numbering: "1.1.")
-  show heading: it => {
-    if it.level == 1 {
-      text(12pt, align(center, upper(it)))
-    } else {
-      text(12pt, it)
-    }
-    ""
-    v(-1cm)
-  }
 
   // Start page numbering
   set page(numbering: "1", number-align: center)
@@ -190,7 +181,7 @@
   ) // TODO: make the same style as LaTeX: 1. | (a) | i. | A.
 
 
-  outline(depth: 3, indent: 1cm, title: text(size: 14pt, "SATURS"))
+  outline(depth: 3, indent: 1cm, title: text(size: 14pt, "Saturs"))
 
   body
 }

main.typ

@@ -5,3 +5,33 @@
 #import "layout.typ": indent-par, project
 
 #show: project.with(title: [Kvantu skaitļošana], authors: ("Kristofer Solo",))
+
+= Fundamentals
+== Qubit (Kvantu bits)
+=== Basis states
+$ ket(0)=vec(1, 0), ket(1)=vec(0, 1) $
+
+=== Superposition
+A qubit can be in a linear combination of basis states:
+$ket(psi)=alpha ket(0)+ beta ket(1)$, where $alpha, beta in CC$ are probability amplitudes.
+
+=== Normalization
+$ abs(alpha)^2 + abs(beta)^2 = 1 $
+$abs(alpha)^2$ is the probability of measuring $ket(0)$, $abs(beta)^2$ is the
+probability of measuring $ket(1)$.
+
+=== Bloch Sphere
+Geometric representation of a single qubit state:
+$ ket(psi)=cos theta/2 ket(0)+ e^(i phi) sin theta/2 ket(1) $
+
+== Measurement (Mērījumi)
+- Projective measurement in the computational basis ${ket(0), ket(1)}$.
+
+- If state is $ket(psi)=alpha ket(0) + beta ket(1)$:
+  - Outcome $0$: probability $P(0)=abs(braket(0, psi))^2=abs(alpha)^2$.
+    Post-measurement state: $ket(0)$.
+  - Outcome $1$: probability $P(1)=abs(braket(1, psi))^2=abs(beta)^2$.
+    Post-measurement state: $ket(1)$.
+- Measurement collapses the superposition.
+- Measurement operators: $M_0=ket(0)bra(0)$, $M_1 = ket(1)bra(1)$.
+  $sum_m M_m^dagger M_m=I$.