Given a set of Pauli matrices ${X, Y, Z, I}$ and $n$ is the number representing the length of tensor products I want to produce from Paulie matrices. I construct a set of $2n-2$ such matrices, for example, the set $S=\{Z_1Z_2Y_1,Z_3Y_2,Y_3,Y_2\}$ with n=3 and $Y_2 = I\otimes Y \otimes I$ (the same notation with other elements). Then I also have a matrix group equipped with matrix product operation denoted $(M,.)$. This group has the following elements $\{Z_2Y_1,Z_3Y_2,Y_2,Y_3\}$. My question is: is there a way to check that the group $M$ generates S or another way is that given an element of group S, can I show that there is no combination of group $M$ that I can produce that element? I have a feeling that there are some theorems or algorithms about this. I truly appreciate any feedback. Thank you in advance.
2026-03-26 07:59:09.1774511949
How to check if a group of matrices generates a set of matrices?
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