I have a problem interpreting reshape with a matrix product

Here is my code:

if true
  %{
Pi = zeros(n^2,n^2);
Psi0 = zeros(n^2,1);
for k=1:n
  aux = zeros(n,1);
  aux(k) = 1;
  psi{k} = kron(aux,sqrt(G(:,k)));
  P{k} = psi{k} * psi{k}';
  Pi = Pi + P{k};
  Psi0 = Psi0 + psi{k};

 end
Psi0 = 1/sqrt(n)*Psi0;

 %construct the swap operator S

 S = zeros(n^2,n^2);
 for k=1:n
     for j=1:n
         S((k-1)*n+j,(j-1)*n+k) = 1;
    end
 end

 % construct time evolution operator U:

 U = S * (2*Pi - eye(n^2));
 U2 = U^2;

 %%Perform the actual iterative dynamics:

 % # of double-steps of the quantum walk:

 steps = 1000;

 state = reshape(Psi0,n,n);
 for k=1:steps
     state = reshape(U2*state(:),n,n);
     for j=1:n
         p(k,j) = norm(state(j,:))^2;
     end

end

} end

In the end where there's "state = reshape(U2*state(:),n,n);" I can't understand how this product can be possible.. U2 is an (n^2,n^2) matrix and state is now (n,n);

Can anyone explain me this?

Moreover I'm trying to translate this in Python in order to apply it to network analysis, but with Python the command reshape from numpy (which I suppose is similar to MATLAB reshape) doesn't work at all: it tells me "ValueError: shapes (n^2,n^2) and (n,n) not aligned: n^2 (dim 1) != n (dim 0)" ; in this sense it approves my suggestions!

I would be thankful if anyone can help me! Thanks a lot!!

Simone

Answer 1

[David Goodmanson]

Hello Simone,

   state = reshape(Psi0,n,n);                % Psi0 is n^2 x 1 column vector
                                             % and state is n x n                                                   
   for k=1:steps
       state = reshape(U2*state(:),n,n);     % state(:) is n^2 x 1 column vector.
                                             % U2 is n^2 x n^2 matrix.
                                             % their product is n^2 x 1 column vector.
                                             % so reshape to n x n works.

[Simone Azeglio]

Thank you so much ! So, there is no difference between Psi0 and state(:), isn't it ?

[David Goodmanson]

That's true. State(:) just reads things out columnwise into a single column vector.

[Simone Azeglio]

Thank you David !