92 lines
2.3 KiB
Plaintext
92 lines
2.3 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 - DM4 - Augustin LUCAS
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],
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)
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#question("Assignment Q1",
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([
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+ Let $U=mat(u_(00),u_(01);u_(10),u_(11))$. Write a matrix representation of $U[1]$ and $U[2]$ for $n=2$.
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+ For $n=3$, write a matrix representation of CNOT$[3,1]$.
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])
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)
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+ $U[1] &= U times.circle I
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&= mat(u_(00),u_(01);u_(10),u_(11)) times.circle mat(1,0;0,1)
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&= mat(u_(00)I,u_(01)I;u_(10)I,u_(11)I)
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&= mat(u_(00),0,u_(01),0;0,u_(00),0,u_(01);u_(10),0,u_(11),0;0,u_(10),0,u_(11))$
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$U[2] &= I times.circle U
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&= mat(U,0;0,U) = mat(u_(00),u_(01),0,0;u_(10),u_(11),0,0;0,0,u_(00),u_(01);0,0,u_(01),u_(11))$
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+ $"CNOT"[3,1]$ corresponds to the following circuit:
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#{
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import "@preview/quill:0.5.0": *
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quantum-circuit(
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lstick($v_0$), 1, targ(), 2, rstick($$), [\ ],
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lstick($v_1$), 1, rstick($$), 2, [\ ],
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lstick($v_2$), 1, ctrl(-2), 2, rstick($$)
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)
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}
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Which corresponds to the following matrix:
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$"CNOT"[3,1] = mat(
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1,0,0,0,0,0,0,0;
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0,0,0,0,0,1,0,0;
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0,0,1,0,0,0,0,0;
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0,0,0,0,0,0,0,1;
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0,0,0,0,1,0,0,0;
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0,1,0,0,0,0,0,0;
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0,0,0,0,0,0,1,0;
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0,0,0,1,0,0,0,0;
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)$
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#question("Assignment Q2",
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([
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Let $A=1/sqrt(2) mat(-i,-1;1,i)$ and $B=mat(0,1;-1,0)$.
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Which 2-qubit gate can you apply on the first qubits at the end of the circuit
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#{
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import "@preview/quill:0.5.0": *
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quantum-circuit(
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lstick($$), ctrl(2), 1, ctrl(2), 1, [\ ],
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lstick($$), 1, ctrl(1), 1, ctrl(1), [\ ],
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lstick($$), $B^(-1)$, $A^(-1)$, $B$, $A$, rstick($$)
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)
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}
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to get a Toffoli gate ?
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]),
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)
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The circuit corresponds to applying the following gate:
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$C &= mat(I_2,0,0,0;0,I_2,0,0;0,0,B^(-1),0;0,0,0,B^(-1))
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mat(I_2,0,0,0;0,A^(-1),0,0;0,0,I_2,0;0,0,0,A^(-1))
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mat(I_2,0,0,0;0,I_2,0,0;0,0,B,0;0,0,0,B)
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mat(I_2,0,0,0;0,A,0,0;0,0,I_2,0;0,0,0,A)
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&= mat(I_2,0,0,0;0,I_2,0,0;0,0,I_2,0;0,0,0,B^(-1)A^(-1) B A)$
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And $B^(-1)A^(-1) B A = B A B A = mat(0,i;i,0)$.
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Let $D = i I_2$. Then, $C mat(I_2,0,0,0;0,I_2,0,0;0,0,I_2,0;0,0,0,D) = "Toffoli"$
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Which corresponds to the following circuit:
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#{
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import "@preview/quill:0.5.0": *
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quantum-circuit(
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lstick($$), ctrl(2), 1, ctrl(2), 1, [\ ],
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lstick($$), 1, ctrl(1), 1, ctrl(1), [\ ],
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lstick($$), $B^(-1)$, $A^(-1)$, $B$, $A$, $D$, rstick($$)
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)
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}
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