#import "qcs.typ": *
#set page(
  paper: "a4",
  header: align(center)[
    QCS - DM7 : Decide satisfiability - Augustin LUCAS
  ],
)

#v(10%)
#question("Assignment Q1", [
  #align(center, {
    import "@preview/quill:0.5.0": *

    quantum-circuit(
      lstick($"input:" psi = alpha ket(0)+beta ket(1)$), 2, slice(label: $ket(phi_0)$), targ(),
      slice(label: $ket(phi_1)$), 2, rstick("output"), [\ ],
      lstick(ket(0)), $H$, 1, ctrl(-1), 2, rstick($X$)
    )
  })

  Compute Kraus representation of map input $->$ output.
])
#v(2%)

$ket(phi_0) &= (alpha ket(0)+beta ket(1)) times.circle (ket(0)+ket(1))/sqrt(2) \
&= alpha/sqrt(2) ket(00) + alpha/sqrt(2) ket(01) + beta/sqrt(2) ket(11) + beta/sqrt(2) ket(10)$

$ket(phi_1) = (alpha ket(0) +beta ket(1))/sqrt(2) times.circle ket(0)
+ (alpha ket(1) +beta ket(0))/sqrt(2) times.circle ket(1)$

This corresponds on the first qubit to the circuit "do a bit flip with probability $1/2$",
which Kraus representation is: $Phi(psi) = (1/sqrt(2) I) psi (1/sqrt(2) I) + (1/sqrt(2) X) psi (1/sqrt(2) X)$

#v(10%)

#question("Assignment Q2", [
  Let $M=mat(a,b;c,d)$, $a,b,c,d in bb(C)$.
  Compute $Phi(M)$ where $Phi$ phase flips with probability $1/2$.
])
#v(2%)

#align(center, $Phi(M) &= (1-1/2)M + 1/2 Z M Z \
  &= 1/2mat(a,b;c,d)+1/2 mat(1,0;0,-1) mat(a,b;c,d) mat(1,0;0,-1) \
  &= 1/2 (mat(a,b;c,d)+mat(a,b;-c,-d) mat(1,0;0,-1)) \
  &= 1/2 (mat(a,b;c,d)+mat(a,-b;-c,d)) \
  &= mat(a,0;0,d)$
)