In Problems 1-5, we will work through an example of QFTM for M=6
What is ω? You can use e, pi, and M in your response. Please do not subtitute 6 for M.
What is QFT6 of 12√(|0⟩+|3⟩)?
Specify the probability amplitudes. You should use w to denote ω, but please fully simplify your answer such that you only use ω and ω2 and no higher-degree terms.
What is QFT6 of 12√(|1⟩+|4⟩)?
Specify the probability amplitudes. You should use w to denote ω, but please fully simplify your answer such that you only use ω and ω2 and no higher-degree terms.
What is QFT6 of 13√(|0⟩+|2⟩+|4⟩)?
Specify the probability amplitudes. You should use w to denote ω, but please fully simplify your answer such that you only use ω and ω2 and no higher-degree terms.
What is QFT6 of 13√(|1⟩+|3⟩+|5⟩)?
Specify the probability amplitudes. You should use w to denote ω, but please fully simplify your answer such that you only use ω and ω2 and no higher-degree terms.
What is your question?
4) 1/sqrt2, 0, 0, 1/sqrt2, 0, 0
PLEASE ANSWER ALL PARTS
Hey guys please answer to 2,3,5
Problem 2 1/sqrt(3)(I0>+I2>+I4>)
Problem 5 1/sqrt(2) , 0 , 0, -(1/sqrt(2)) ,0 ,0
To calculate the Quantum Fourier Transform (QFT) for a given value of M, we first need to determine the value of ω. In this case, M is given as 6.
The value of ω can be calculated using the formula ω = e^(2πi/M), where e is the base of the natural logarithm (approximately 2.71828), and π is the mathematical constant pi (approximately 3.14159).
So, ω = e^(2πi/6).
Now, let's calculate the QFT for the given states.
Problem 1:
QFT6 of 12√(|0⟩+|3⟩)
To calculate the QFT6 of this state, we need to perform the QFT operation on the state |0⟩+|3⟩.
Let's go through the steps to calculate it.
Step 1: Calculate the value of ω
Using the formula mentioned above, ω = e^(2πi/6).
Simplifying further, ω = e^(πi/3).
Step 2: Apply the QFT operation
The QFT operation on a state |x⟩ is given by the formula:
QFT6(|x⟩) = 1/√(6) * Σ(y=0 to 5) ω^(xy) * |y⟩.
Let's substitute the values:
QFT6(12√(|0⟩+|3⟩)) = 1/√(6) * [ω^(0*0) * |0⟩ + ω^(0*1) * |1⟩ + ω^(0*2) * |2⟩ + ω^(0*3) * |3⟩ + ω^(0*4) * |4⟩ + ω^(0*5) * |5⟩].
Simplifying further:
QFT6(12√(|0⟩+|3⟩)) = 1/√(6) * [1 * |0⟩ + ω^0 * |1⟩ + ω^0 * |2⟩ + ω^0 * |3⟩ + ω^0 * |4⟩ + ω^0 * |5⟩].
Since ω^0 = 1, we can simplify this expression as:
QFT6(12√(|0⟩+|3⟩)) = 1/√(6) * [1 * |0⟩ + 1 * |1⟩ + 1 * |2⟩ + 1 * |3⟩ + 1 * |4⟩ + 1 * |5⟩].
Therefore, the probability amplitudes for QFT6(12√(|0⟩+|3⟩)) are:
1/√(6) for |0⟩,
1/√(6) for |1⟩,
1/√(6) for |2⟩,
1/√(6) for |3⟩,
1/√(6) for |4⟩,
1/√(6) for |5⟩.
To get the final answer, you need to fully simplify these expressions using ω and ω^2, without including higher-degree terms.