# Tag Info

### Why does the "Phase Kickback" mechanism work in the Quantum phase estimation algorithm?

A first remark This same phenomenon of 'control' qubits changing states in some circumstances also occurs with controlled-NOT gates; in fact, this is the entire basis of eigenvalue estimation. So not ...
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### Why does the "Phase Kickback" mechanism work in the Quantum phase estimation algorithm?

Imagine you have an eigenvector $|u\rangle$ of $U$. If you have a state such as $|1\rangle|u\rangle$ and you apply controlled-$U$ to it, you get out $e^{i\phi}|1\rangle|u\rangle$. The phase isn't ...

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### Why do we use the quantum superposition for a period instead of factors in Shor's algorithm?

A QFT can't arbitrarily raise the probability of any state you want to any value you want. Once you create a superposition, you need to find some way to make destructive interference occur between the ...
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### How to describe, or encode, the input vector x of Quantum Fourier Transform?

Formula 5.2 refers to an encoding we call amplitude encoding. Imagine you have a vector $x$ with components $x_i$, the components are then encoded as amplitudes of a quantum state. This encoding is ...
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### What is the intuition of using Hadamard gate in quantum fourier transform?

The intuition, roughly speaking, is that the only way that you're going to get some difference between classical and quantum computing is if you are able to prepare qubits in a superposition. If you ...
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### Intuitively, what does the quantum Fourier transform do?

Let's see what QFT does on two qubit (and then on three qubit) computational basis states and try to gain some insights. The QFT action on $|j\rangle$ basis state: QFT |j\rangle = \frac{1}{2^{\frac{...
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### How is the fractional binary notation used in the QFT?

A positive integer $y$ has a binary representation $y_{n-1}\ldots y_{0}$ where $y_k \in \{0,1\}$. For example, for $n=3$, the number $5$ in binary is $\color{red}{101}$. If we do a binary expansion of ...

### How to describe, or encode, the input vector x of Quantum Fourier Transform?

You don't convert a classical input to the r.h.s. of Eq. (5.2). The r.h.s. of Eq. (5.2) is something you get as the output of a preceding quantum computation as a quantum state, such as in Shor's ...
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### Does the Quantum Fourier Transform (QFT) preserve entanglement?

TLDR: the Fourier transform is entangling. We can immediately agree on two things: if you input a computational basis state (separable) to the Fourier transform, it outputs a separable state the ...

### What is the probability to get all qubits equal zero after QFT

By looking to the circuit for the QFT presented in the M. Nielsen and I. Chuang textbook (Figure 5.1.) we can notice that all controlled rotations can be neglected because for each control rotation ...
From linear algebra we know that $(AB)^{-1} = B^{-1} A^{-1}$. This is because $(AB)*(AB)^{-1} = ABB^{-1}A^{-1} = AIA^{-1} = I$. Hence, if you have the circuit to generate the Bell state from the state ...