quantum-compiling/BGCDecompose.m at main · JorenB/quantum-compiling · GitHub

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function [v w] = BGCDecompose(u)

x = matToCart3(u);

n = norm(x);

a = [n;0;0];

xu = cart3ToMat(a);

s = similarityMatrix(u,xu);

as = conj(transpose((s)));

[vs ws] = xGroupFactor(xu);

v = s*vs*as;
w = s*ws*as;

end


function x = matToCart3(u)

sx1 = -1 * imag(u(1,2));
sx2 = real(u(2,1));
sx3 = imag(u(2,2));

costh = 1/2*real(u(1,1)+u(2,2));
sinth = sqrt(sx1^2 + sx2^2+sx3^2);

x = [0; 0; 0];
if sinth == 0
    x(1) = 2*acos(costh);
else
    th = atan2(sinth,costh);

    x(1) = 2*th*sx1/sinth;
    x(2) = 2*th*sx2/sinth;
    x(3) = 2*th*sx3/sinth;

end
end


function U = cart3ToMat(x)

th = sqrt(dot(x,x));

if th == 0
    U = eye(2);
else
    U = cos(th/2)*eye(2) - 1i*sin(th/2)/th*(x(1) * [0 1; 1 0] + x(2)*[0, -1i; 1i, 0] + x(3)*[1, 0; 0, -1]);
end

end


function S = similarityMatrix(u, xu)

a = matToCart3(u);
b = matToCart3(xu);

na = norm(a);
nb = norm(b);

ab = dot(a,b);

s = cross(b,a);
ns = norm(s);

% if the cross product is small, u is already a rotation about approximately the x-axis
if abs(ns) < constants.RE
    S = eye(2);

else
	s = acos(ab/(na*nb))/ns * s;

    S = cart3ToMat(s);
end

end


function [V W] = xGroupFactor(xu)

a = matToCart4(xu);

st = (0.5 - 0.5*a(1))^(1/4);

ct = sqrt(1-st^2);

theta = 2*asin(st);
alpha = atan(st);

b = [0;0;0]; c = [0;0;0];

b(1) = theta*st*cos(alpha);
c(1) = b(1);

b(2) = theta*st*sin(alpha);
c(2) = b(2);

b(3) = theta*ct;
c(3) = -b(3);


B = cart3ToMat(c);
W = cart3ToMat(b);

aB = conj(transpose(B));

V = B;

W = similarityMatrix(W, aB);

end