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q6:Now, consider the case where N4 elements are marked instead of just one. If we run one iteration of Grover's algorithm and measure, what is the probability that we see a marked element?
q7: Which of the following observables correspond to a standard basis measurement?
q8: Which of the following observables correspond to a sign basis measurement?
q5: uppose we ran m steps of Grover's algorithm on some function f (which has one marked element y) and the resulting superposition was exactly |y⟩.
(a) What was the state after the (m−1)-th step? Note that you can describe the superposition by specifying two numbers, αy and αx for x≠y. Use K to denote the total number of elements. (Sorry for not using the conventional letter N, but EdX grader doesn't seem to allow the use of that letter.) Please fully simply your answer.
(b) Now, if we run one more step (total of m+1 steps), what is the resulting superposition?
(c) What if you now apply another phase inversion?
q6: To calculate the probability of seeing a marked element after running one iteration of Grover's algorithm with N/4 elements marked, we need to take into account the amplitude of the marked elements in the superposition.
1. First, we need to calculate the number of iterations, known as R, required for Grover's algorithm. In this case, with N/4 marked elements, R is given by:
R = π/(4θ) , where θ = arcsin(sqrt(N/4))
2. Once R is known, we can calculate the amplitude of the marked elements after one iteration using the formula:
α_m = sin((2R+1)θ)
3. Finally, the probability of measuring a marked element is given by the square of the amplitude:
P_m = |α_m|^2
You can substitute the values of N and calculate the probability using the above formulas.
q7: Observables corresponding to a standard basis measurement are those that measure the state in terms of the computational basis states, i.e., the states |0⟩ and |1⟩. To determine if an observable corresponds to a standard basis measurement, check if its eigenbasis consists of the computational basis states.
q8: Observables corresponding to a sign basis measurement are those that measure the state in terms of the sign basis states, i.e., the states |+⟩ and |-⟩. To determine if an observable corresponds to a sign basis measurement, check if its eigenbasis consists of the sign basis states.
q5:
(a) To determine the state after the (m-1)-th step of Grover's algorithm, we can use the following formulas:
- α_y = -α_x = sin((2m-1)θ)
- α'_y = cos(θ)α_y - sin(θ)α_x
- α'_x = sin(θ)α_y + cos(θ)α_x
Here, α_y and α_x represent the amplitudes of the marked and unmarked elements respectively, and θ is the angle parameter.
(b) To determine the resulting superposition after one more step (m+1 steps total), we can use the same formulas as above but substitute m with (m+1) in the equations.
(c) If we apply another phase inversion after the m+1 steps, it will undo the effect of the previous phase inversion and revert the superposition back to the state after m steps. So, the resulting superposition will be the same as the one obtained after the (m-1)-th step.