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