# OK to post homework
# James Betti, 9 Feb 2025, Assignment 4

UM := proc(α,β,γ,δ) local Ry,Rz,A;
    Ry := t->[[cos(t/2),-sin(t/2)],[sin(t/2),cos(t/2)]];
    Rz := t->[[exp(-I*t/2),0],[0,exp(I*t/2)]];
    A := evalm(exp(I*α)*multiply(Rz(β),Ry(γ),Rz(δ)));
    [[A[1,1],A[1,2]],[A[2,1],A[2,2]]];  # multi-index -> list of lists
end:

# Form a random 2 by 2 unitary matrix using Box 1.1 of Nielsen–Chuang.
RUM := proc(K) local ra:
    ra := rand(-K..K);
    UM(ra(),ra(),ra(),ra());
end:

# Modified IsUM to call `simplify` and `evalc`.
IsUMSimp:=proc(A) local n,i,j,P;
    n := nops(A);
    P := multiply(A,CT(A));
    P := simplify(evalc([seq([seq(P[i,j],j=1..n)],i=1..n)]));
    evalb(P=[seq([0$(i-1),1,0$(n-i)],i=1..n)]);
end:

IsUMSimp(RUM(20));                      # should be true
IsUMSimp(UM(a,b,c,d));                  # should be true