Which of the following pairs of quantum gates commute? Select all that apply. (Gates A and B commute if and only if for any input applying A and then B gives the same results as applying B and then A. This is the same as saying that the unitary transformations commute.)

3 of them commute

I and X
CNOT and X applied to the target qubit
Real rotation by 30∘ and real rotation by 45∘

To determine which pairs of quantum gates commute, we need to check if applying one gate before the other gives the same result as applying the gates in the opposite order. Let's consider each pair of gates:

- Gate A: Pauli-X (X), Gate B: Pauli-Y (Y)
To check if X and Y commute, we can evaluate the products XY and YX. Since XY = YX, X and Y commute.

- Gate A: Hadamard (H), Gate B: Pauli-Z (Z)
To check if H and Z commute, we can evaluate the products HZ and ZH. Since HZ = -ZH, H and Z do not commute.

- Gate A: Pauli-X (X), Gate B: Pauli-Z (Z)
To check if X and Z commute, we can evaluate the products XZ and ZX. Since XZ = -ZX, X and Z do not commute.

Therefore, the pairs of gates that commute are: X and Y.

Note: The commutation rules for quantum gates are crucial in designing quantum circuits and understanding the order in which gates should be applied.