mirror of
https://github.com/kristoferssolo/Quantum-Computation-Cheatsheet.git
synced 2025-10-21 19:50:34 +00:00
38 lines
1.4 KiB
Plaintext
38 lines
1.4 KiB
Plaintext
#import "@preview/fletcher:0.5.7" as fletcher: diagram, edge, node
|
|
#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",))
|
|
|
|
= 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: $M_0=ket(0)bra(0)$, $M_1 = ket(1)bra(1)$.
|
|
$sum_m M_m^dagger M_m=I$.
|