Suppose we initially had n qubits in the state |ψ⟩=∑x∈{0,1}nαx|x⟩, on which we were going to perform Fourier sampling (i.e. perform Hadamard transform H⊗n and sample). Unfortunately, a clumsy lab assistant mistakenly applied X to one of the qubits and forgot which one it was.

Question

3a. Suppose we initially had n qubits in the state |ψ⟩=∑x∈{0,1}nαx|x⟩, on which we were going to perform Fourier sampling (i.e. perform Hadamard transform H⊗n and sample). Unfortunately, a clumsy lab assistant mistakenly applied X to one of the qubits and forgot which one it was.
.βx′=βx
.βx′=−βx
.βx′=(−1)xiβx
.βx′=(−1)x⋅xβx
βx′=βx⊕ei

Answer 1

Hi Elviria, do you have any answer for this exam?

Answer 2

ƒÀx�Œ=(−1)xiƒÀx

Answer 3

what is that?

Answer 4

Another way to view it is to say that |ψ′⟩=Xi|ψ⟩ where Xi means X applied on the i-th qubit. We would like to claim that the application of Xi to the original state is effectively the same as applying some A to the i-th qubit after the Hadamard transform, i.e, H⊗n|ψ′⟩=H⊗nXi|ψ⟩=AiH⊗n|ψ⟩. What is A?

I
X
Z
H
XH

Answer 5

3A .βx′=(−1)xiβx