Alice has a very important piece of information, which is encoded as an integer k. She is storing this number in a quantum state |ψ⟩=∑xαx|x⟩ of n qubits in such a way that if you apply QFT2n to |ψ⟩ and then measure in the standard basis, you get k with probability 99%. Her archenemy Eve sneaked into Alice's lab and, not knowing how to perform QFT, decided to destroy the information instead. She started pushing random buttons and it resulted in adding a number y to the qubits i.e.~the resulting state is |ψ′⟩=∑xαx|x+y(mod2n)⟩. Which of the following statements are true? Check all that apply.

Goodness.

please answer

YOU HAVE TO CHECK

If Alice now applies QFT2n and measure, she will still get k with probability 99%.

AND

If she does as in the previous option, she will get k with probability 99%.

thanx

To answer this question, we need to understand the given scenario and the concepts involved.

1. Alice initially stores the information encoded as an integer k in a quantum state |ψ⟩ of n qubits.
2. Alice performs the Quantum Fourier Transform (QFT) on |ψ⟩.
3. After applying QFT, Alice measures the qubits in the standard basis, and the outcome is k with probability 99%.

Now, Eve interferes by adding a number y to the qubits. So, the resulting state becomes |ψ′⟩ = ∑x αx |x+y(mod 2^n)⟩.

We are asked to determine which of the following statements are true. Let's analyze each statement:

1. The probability of measuring k in state |ψ′⟩ is 99%: This statement is not necessarily true. By adding y, Eve disrupts the original encoding, which might affect the probability of measuring k. Without knowing the specific values of k, αx, and y, we cannot determine if the probability remains 99%.

2. The QFT operation applied to |ψ⟩ can be undone to recover the original information: This statement is true. The QFT operation is a reversible transformation, so applying the inverse QFT to |ψ′⟩ will allow the recovery of the original information.

3. The integer y added by Eve can be determined by applying an inverse QFT and measuring the qubits: This statement is false. The information about y is lost when Eve adds it to the qubits. Applying an inverse QFT and measuring the qubits will not reveal the value of y.

Therefore, the correct answer is:
- Statement 2: The QFT operation applied to |ψ⟩ can be undone to recover the original information.

Please note that without further information about the specific values of k, αx, and y, we cannot determine the validity of statement 1. Statement 3 is false in general.