# Questions tagged [quantum-fourier-transform]

Quantum Fourier Transform (QFT) is a linear transformation on quantum bits and is the quantum analogue of the discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem. (Wikipedia)

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### What are quantum algorithms with only one possible outcome with probability equal to one?

I would like to study circuits with only one possible outcome. Quantum phase estimation, Bernstein-Vazirani, and in part Deutsch-Jozsa (for constant functions) come to mind - do you know any other ...
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### Qiskit's QFT not returning the expected state

I'm currently going through the lab problems for the Qiskit course here. I'm trying to finish lab set number 3 on QPE but I can't seem to get the desired output, even from the solution notebook. I'm ...
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### Represent the $n$-qubit $2^n\times2^n$ size Hadamard/quantum Fourier transform unitary square matrix as product of $k$ two-level unitary matrices

I wish to know if it is possible to express the n-qubit Hadamard unitary square matrix of size $2^n * 2^n$ as a product of 'k' two-level unitary square matrices where 'k' is of the order of polynomial ...
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### Where is the extra qubit after QFT$^{-1}$ coming from in Shor’s algorithm?

Hello sorry if this is a stupid question arising from my ignorance but I have been looking at the modular adder for Shor's algorithm according to this website. Here is what the gate looks like: The ...
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### Quantum fourier transform with classical vibrations

Is there any difference in effect between a quantum circuit and a carefully constructed analogue one relying on interference? For example, why couldn't I take a series of $N$ carefully shaped pipes, ...
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### Two possible ways how to implement Shor's Algorithm

Among many paper describing circuit solving period finding problem and discrete logarithm problem (DLP) (for simplity, let's say $g \equiv x^r$ (mod n) and try to find $r$), there are two variants ...
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### Why are these two QFT circuits equivalent?

I am new to quantum computing and have been trying to understand the Quantum Fourier Transform (QFT). Through my research using both the Qiskit textbook and other sources, I see differences in how the ...
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### How do I find the state of each qubit at the end of the circuit?

I have this Quantum Fourier Transform (QFT) and I want to know how to find the final state of each qubit if q0, q1, ...
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### Lieb-Robinson Bound in 2nd quantized description?

Background Let us restrict our discussion to bosons and adopt the convention First Quantised $\leftrightarrow$ Second Quantised Theory (we are following these Ashok Sen's Quantum Field Theory I of ...
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### Why can I not apply a control gate/function to a gate like T, S, S dagger, ... (using IBM Quantum Experience)? Is there another option?

I am trying to use the circuit composer of the IBM QE. I am doing the inverse QFT on 3 qubits and therefore need a control on T and S dagger gates, but it won't let me. Does anyone know why or know a ...
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### Show that Quantum Fourier Transform maps Bell states to Bell states

Given Bell states $\mathcal{B} = \{\left\vert \phi^{\pm} \right\rangle, \left\vert \psi^{\pm} \right\rangle\}$, show that Quantum Fourier Transform(QFT) maps $\mathcal{B}\rightarrow \mathcal{B}$ by ...
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### Application of QFT to Order-finding

In the Nielsen & Chuang book, section 5.3.1 page 226, there is a statement which goes like this:- (statement-1) The quantum algorithm for order-finding is just the phase estimation algorithm ...
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### Qubit ordering in qiskit

I am confused about the qubit ordering in circuit diagrams and endianness used in qiskit. As far as I understand, qiskit uses little endian (least significant qubit is rightmost) and while drawing ...
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### How does the QFT represent the frequency domain?

QFT is often explained through the classical analogue which converts a certain function from the time domain to the frequency domain. When looking at the discrete Fourier transform, it makes sense to ...
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### A simple question about QFT and CNOT

Given two $d$-dimensional states $QFT|i\rangle(i\in\{0,1,2,...,d-1\})$ and $|\varphi\rangle=|0\rangle$. If I perform $CNOT(QFT|i\rangle,|\varphi\rangle)$, and then perform $QFT^{-1}|\varphi\rangle$, ...
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### Trying to perform Quantum Phase Estimation on T-gate

I'm trying to perform QPE on the T-gate in Quirk but I'm not getting the correct result. For the T-gate, I should be measuring (001) with 100% probability, but instead, I'm getting the following: I'...
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### What is the matrix for a SWAP operation on two qubits?

Say we want to swap qubits $a$, $b$ in the same register, where $a,b \in \left \{ 0, 1,\cdots, n-1 \right \}$. What would be the corresponding matrix. For those interested, I'm curious about this ...
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### Measuring in the computational basis in the single qubit gate QFT implementation

I've come across this paper about a single-qubit-gate-only QFT implementation. In the paper it is claimed that measuring a qubit after applying the Hadamard gate (it isn't called Hadamard gate in the ...
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### Shor's algorithm: what to do after reading the QFT's result twice?

I asked about how to identify the period looking at a Fourier transform plot. The answer seems to be to run the fourier transform multiple times getting multiple values associated to high ...
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### What is this equation for coin operator is trying to do in this quantum walk for Non-regular graph? This coin operator is called Fourier coin

I am reading the following paper: Discrete-time quantum walk on complex networks for community detection by Kanae Mukai We define the Coin operator $C$ by: $C=C_1\otimes C_2....C_n$ , We define coin ...
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### Shor's algorithm: initialization of second register

I am trying to understand Shor's algorithm. I am not quite sure why the initialization, indicated as $|1\rangle$ in the below image at the bottom left is chosen as it is? I understand the modular ...
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### Qiskit Inverse of a quantum fourier transformation

In the photo provided the Quantum Fourier Transform is depicted in Qiskit before the barrier. I don't understand the result of inverse. Conceptually, should the inverse of the QFT be the same ...
The QFT on the group $\mathbb{Z}_N$ is given by $$QFT\,|k\rangle =\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi i\,jk/N}|j\rangle\,.$$ The usual circuit implements the QFT ...