feat(fundamentals): update

layout.typ

@@ -46,7 +46,7 @@
   show math.equation: set text(weight: 400)
 
   // Formatting for regular text
-  set par(justify: true, leading: 1em, first-line-indent: indent, spacing: 1em)
+  set par(justify: true, leading: 1em)
   show heading: set block(spacing: 0.7em)
 
   set terms(separator: [ -- ])
@@ -181,7 +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: indent, title: text(size: 14pt, "Saturs"))
 
   body
 }

main.typ

@@ -1,10 +1,45 @@
 #import "@preview/fletcher:0.5.7" as fletcher: diagram, edge, node
+#import "@preview/gentle-clues:1.2.0": *
 #import "@preview/physica:0.9.5": bra, braket, ket, ketbra
 #import "@preview/quill:0.6.1": *
 #import "@preview/quill:0.6.1" as quill: tequila as tq
 #import "layout.typ": indent-par, project
 
-#show: project.with(title: [Kvantu skaitļošana], authors: ("Kristofer Solo",))
+#show: project.with(title: [Quantum Computation Cheatsheet], authors: (
+  "Kristofers Solo",
+))
+
+= Bre-Ket Notation
+== Ket $ket(psi)$
+Represents a column vector for a quantum state.
+$ ket(psi)=alpha ket(0)+beta ket(1) <==> vec(alpha, beta) $
+
+=== Basis states
+$ ket(0)=vec(1, 0), ket(1)=vec(0, 1) $
+
+== Bra $bra(psi)$
+Represents a *conjugate transpose vector (kompleksi saistīts)* (row vector) of
+$ket(psi)$.
+$ "If " ket(psi) = vec(alpha, beta) ", then" ket(psi)=(a^* space.quad b^*) $
+
+== Scalar Product $braket(phi, psi)$
+Inner product of two states.
+$
+  "If " ket(phi) = gamma ket(0)+ delta ket(1) ", then"
+  braket(phi, psi)= gamma^* alpha + delta^* beta
+$
+
+=== Orthogonal states
+$ braket(phi, psi)=0 $
+
+== Projection $braket(i, psi)$
+Amplitude of the basis state $ket(i)$ in $ket(psi)$.
+
+For $ket(psi)=alpha ket(0) + beta ket(1) :
+braket(0, psi)=alpha,
+psi braket(1, psi)=beta$.
+
+Probability of measuring state $ket(i): P(i)=abs(braket(i, psi))^2$
 
 = Fundamentals
 == Qubit (Kvantu bits)
@@ -13,7 +48,8 @@ $ 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.
+$ ket(psi)=alpha ket(0)+ beta ket(1) ", where "alpha, beta in CC $
+are probability amplitudes.
 
 === Normalization
 $ abs(alpha)^2 + abs(beta)^2 = 1 $
@@ -25,13 +61,38 @@ 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 ${ket(0), ket(1)}$.
+- Projective measurement in the basis (e.g. computational ${ket(0), ket(1)}$ or
+  Hadamard ${ket(+), ket(-)}$).
 
 - 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: $M_0=ket(0)bra(0)$, $M_1 = ket(1)bra(1)$.
-  $sum_m M_m^dagger M_m=I$.
+- Measurement collapses the superposition (mērījums maina kvantu bitu (observer effect)).
+
+=== Measurement operators
+$
+  M_0=ket(0)bra(0) ,
+  M_1 = ket(1)bra(1) \
+  sum_m M_m^dagger M_m=I
+$
+
+=== Measuring in $ket(+), ket(-)$ basis
+$
+  ket(+)=1/sqrt(2)(ket(0)+ket(1)),
+  ket(-)=1/sqrt(2)(ket(0)-ket(1))
+$
+
+To measure $ket(0)$ in this basis: $ket(0)=1/sqrt(+)+1/sqrt(2)ket(-)$.
+$
+  P(+)=abs(braket(+, 0))^2=1/2,
+  P(-)=abs(braket(-, 1))^2=1/2
+$
+
+=== Example: $ket(psi)=(1+2i)/sqrt(7)ket(0)+(1-i)/sqrt(7)ket(1)$
+$
+  P(0) & =abs((1+2i)/sqrt(7))^2=(1^2+2^2)/7=5/7   \
+  P(1) & =abs((1-i)/sqrt(7))^2=(1^2+(-1)^2)/7=2/7
+$
+#tip[ Sum must be $1$. ]

requirements.typ

@@ -1,3 +1,4 @@
 #import "@preview/fletcher:0.5.7"
 #import "@preview/physica:0.9.5"
 #import "@preview/quill:0.6.1"
+#import "@preview/gentle-clues:1.2.0"