Quantum Mechanics and Quantum Computation

[superscientist]

Homework Statement

q1]] 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.)

a) I and X
b) X and Z
c) H and X
d) CNOT and X applied to the target
e) qubit Real rotation by 30∘ and real rotation by 45∘

q2]] 1] What is ZX applied to |0>?
2] What is ZX applied to H|0>?
3] Suppose we have a qubit in the state |ψ⟩. We know that if we measure it in the standard basis, the probability of getting a 0 is 29. Now, if we instead first apply a Hadamard gate and then measure the resulting qubit H|ψ⟩ in the sign basis, what is the probability of getting a +?

Homework Equations

The Attempt at a Solution

 

[member 553083]

q1]] a) I and X - Yes b) X and Z - No c) H and X - No d) CNOT and X applied to the target - No e) qubit Real rotation by 30∘ and real rotation by 45∘ - Yes q2]] 1] ZX|0> = -i|1> 2] ZXH|0> = i|1> 3] Probability of getting a + = (1/2)*(1+cos(2θ))