How to show numerically that a subgroup of SU(2) and CNOT gate can give us a finite group?

Imagine that I have a two-qubit system, and one of the qubits performs single-qubit rotations from a subgroup of SU(2), specifically 2O. In this subgroup, the single-qubit rotations are represented by H, S,X, Y, Z, and I, where I denotes no rotation. Additionally, I have a CNOT gate for this system. I am wondering how to show numerically that this subgroup and the CNOT gate together form a finite group.

P.S. I understand that this may seem like a trivial question, but I am also trying to grasp the methodology for solving these types of problems, particularly in studying groups and determining whether they are infinite or finite. If someone can provide a comprehensive answer to this question, it will help me understand the approach and process for handling more complex groups as well. I kindly request that any response refrain from phrases such as "with H and T gate, they perform a universal gate set, so if you include them this will perform infinite group" or "it is obvious this will not give us a universal gate set without the T gate and H, so it is finite.

• look into gap-system.org ...as long your groups are "algebraic" (no floating point definitions) it's a pretty powerful software. Jun 30, 2023 at 15:33
• Thank you! Will look at that. Can you give some tips about how to solve that kind of problem? Jul 1, 2023 at 1:07
• you define the set of matrices that generate the group; then you ask GAP to calculate the size of the group. It will give back the size of the group or infinity... Jul 1, 2023 at 2:50
• Excellen! Thank you! And for the interaction of CNOT, then I should include the CNOT gate matrices in the set Jul 1, 2023 at 3:59
• @quest if you solve the problem, kindly post the solution here. This is a good problem. Jul 8, 2023 at 3:38