48 lines
1.4 KiB
Plaintext
48 lines
1.4 KiB
Plaintext
#import "qcs.typ": *
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#set page(
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paper: "a4",
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header: align(center)[
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QCS - DM7 : Decide satisfiability - Augustin LUCAS
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],
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)
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#v(10%)
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#question("Assignment Q1", [
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#align(center, {
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import "@preview/quill:0.5.0": *
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quantum-circuit(
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lstick($"input:" psi = alpha ket(0)+beta ket(1)$), 2, slice(label: $ket(phi_0)$), targ(),
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slice(label: $ket(phi_1)$), 2, rstick("output"), [\ ],
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lstick(ket(0)), $H$, 1, ctrl(-1), 2, rstick($X$)
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)
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})
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Compute Kraus representation of map input $->$ output.
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])
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#v(2%)
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$ket(phi_0) &= (alpha ket(0)+beta ket(1)) times.circle (ket(0)+ket(1))/sqrt(2) \
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&= alpha/sqrt(2) ket(00) + alpha/sqrt(2) ket(01) + beta/sqrt(2) ket(11) + beta/sqrt(2) ket(10)$
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$ket(phi_1) = (alpha ket(0) +beta ket(1))/sqrt(2) times.circle ket(0)
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+ (alpha ket(1) +beta ket(0))/sqrt(2) times.circle ket(1)$
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This corresponds on the first qubit to the circuit "do a bit flip with probability $1/2$",
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which Kraus representation is: $Phi(psi) = (1/sqrt(2) I) psi (1/sqrt(2) I) + (1/sqrt(2) X) psi (1/sqrt(2) X)$
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#v(10%)
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#question("Assignment Q2", [
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Let $M=mat(a,b;c,d)$, $a,b,c,d in bb(C)$.
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Compute $Phi(M)$ where $Phi$ phase flips with probability $1/2$.
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])
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#v(2%)
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#align(center, $Phi(M) &= (1-1/2)M + 1/2 Z M Z \
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&= 1/2mat(a,b;c,d)+1/2 mat(1,0;0,-1) mat(a,b;c,d) mat(1,0;0,-1) \
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&= 1/2 (mat(a,b;c,d)+mat(a,b;-c,-d) mat(1,0;0,-1)) \
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&= 1/2 (mat(a,b;c,d)+mat(a,-b;-c,d)) \
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&= mat(a,0;0,d)$
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)
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