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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Find the conditions under which the state $|\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle$ is unentangled

Show that the state $ |\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle $ is unentangled if $a \in \{ 0,1,...,2^n - 1\} $ and $|\phi\rangle$ can be expressed in the form $ \...
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Why don't we use exact QFT in Shor's algorithm?

In DLP ($g \equiv x^r$ (mod $p$) with known order of $x$ as $p$), Shor algorithm applies QFT to the state $$\frac{1}{p}\sum_{a, b}^{p-1}|a, b, g^ax^b⟩$$ Here QFT is of size $q$ that satisfies $(p-1)\...
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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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Rephase gate in implementaion of QFT of arbitray size

Talking about how exact size of QFT is achieved, both paper 1 and paper 2 skipped the implementation of gate $U$ that can do: $$U|\alpha, \beta⟩ \mapsto exp\left(\frac{i2\pi}{N} \alpha\beta\right)|\...
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Implementation of the Phase Estimation algorithm

I've been working on implementing quantum phase estimation in Qiskit for a $2^n \times 2^n$ Hamiltonian as part of my bachelor project, I'm using Trotterization as my Hamiltonian simulation of choice ...
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What is the relation between Hadamard transformation and QFT?

I am new to the field and I can't help having a feeling that Hadamard and Fourier Transform are somehow related, but it is not clear to me how. Any explanation on how these two are related would be ...
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In Shor's algorithm, how is ${\rm QFT}_n|x\rangle$ split into its even and odd components?

I am auditing a course on quantum computing. Since this is not paid, I dont have any staff support to ask questions. Therefore I am asking the stackoverflow community to help me with it. This is ...
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What are the input and output of QFT and IQFT, respectively?

I have read two opposite explanations about QFT and IQFT from 2 books for beginners of Quantum Computing. Which one is correct? The first book said, if we input an n-qubit non-superposition state into ...
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What is the quantum query complexity of the period finding routine of Shor's algorithm?

It seems like it should be a function of N - O(log N), to minimise probability of getting a multiple of the period. However, Prof Preskill's lec notes mention: Thus we solve Period Finding if the ...
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What is the quantum Fourier transform of $\alpha|0\rangle+\beta|1\rangle$?

Given $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ and $|\alpha|^2 + |\beta|^2 = 1$, what would the quantum Fourier transform of $|\psi\rangle$ be? I know it is of the form $\frac{1}{\sqrt{2}}(...
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Qiskit QFT matrix does not match with DFT matrix

The unitary matrix associated with the QFT circuit in Qiskit does not match the actual DFT matrix. In fact, all the imaginary components have their sign flipped (...
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Canonical construction of Logical Fourier Gate

For physical $d$-dimensional qudits we can define $$X= \sum_{i=0}^{d-1} |i+1\rangle \langle i |$$ and $$Z = \sum_{i=0}^{d-1} \omega^i |i\rangle \langle i |,$$ with $\omega=e^{2\pi i/d}$. The Fourier ...
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In the hidden subgroup problem for finite Abelian groups, where does the state $\frac{1}{\sqrt{|G|}}\sum_{g\in G} |g,0\rangle$ come from?

I am new to the concept of HSP. Previously, I saw how to solve hidden subgroup problem over $\mathbb{Z}_2^n$, which was Simon's algorithm. Over there the first step was to apply $H^{\otimes n}$, which ...
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How to use QFT operation in Q#?

I see the QFT operation in the document given by Microsoft, but I don't know how to call it. ...
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Quantum Fourier Transform in the Period Finding Problem

I am trying to prove that when applying the inverse QFT to the following state: we get the following result: However, I get a wrong prefactor. Can anyone tell me where I went wrong? Here my ...
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Calculating the QFFT according to Coppersmith

I have come through an interesting paper by Don Coppersmith (https://arxiv.org/pdf/quant-ph/0201067.pdf), and I was wondering what was your view on the $Q_{JK}$ ("twiddle") transformation, ...
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Intuition behind the use of inverse FFT in Quantum Circuit for Hamming weight

I have found this question from MIT problemset. I could only design the circuit for 3 qubits. But they have a general solution that shows this circuit. I am trying to understand this circuit (from ...
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Efficient QFT-based QPEA complexity

The HHL algorithm lies on an implementation of the Quantum Phase Estimation algorithm. One popular implementation is based on the Quantum Fourier Transform which can be divided in three steps. Let $U$ ...
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How to generalize the relationship HXH = Z for higher dimensions

Concerning the Hadamard gate and the Pauli $X$ and $Z$ gates for qubits, it is straightforward to show the following relationship via direct substitution: $$ HXH = Z.\tag{1}$$ And I would like to ...
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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 ...
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Quantum Fourier Transform for general cyclic groups

The QFT on the group $\mathbb{Z}_N$ is given by \begin{equation} QFT\,|k\rangle =\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi i\,jk/N}|j\rangle\,. \end{equation} The usual circuit implements the QFT ...
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Abelian Hidden Subgroup Problem for arbitrary cyclic p-Groups

I had asked a question similar to this one here regarding how to handle the HSP for groups whose cyclic decomposition contains factors whose order is not a power of two. I also had some prior ...
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How to write a classical version of Shor's algorithm

For learning purposes, I would like to write a classical version of Shor's algorithm. From what I have read, what makes this algorithm fast is the quantum FFT, which is used to find the period of the ...
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Quantum circuit, Fourier Transform/Decomposition?

Broadly speaking, can we say that quantum circuits are like Fourier Transform/Decomposition? We use qbit like waves, tune it with quantum gates, to find answer. https://ars.els-cdn.com/content/image/3-...
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Constructing arbitrary functions for the Abelian HSP

My question might be similar to Hidden subgroup problem. However, I'm not exactly sure though. In addition, that question doesn't have an answer. I'm trying to create some simple instances of the ...
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How to apply QFT to a quantum state in superposition?

Given the following quantum state: $\frac{1}{2}(|0000\rangle + |0100\rangle + |1000\rangle + |1100\rangle)$ How do I apply a QFT (given by the formula below) to that state in superposition? $QFT_n|j\...
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Intuitively, what does the quantum Fourier transform do?

I somewhat understand its practical use in phase estimation and algorithms like Shor's algorithm but is there some more intuitive way of understanding what it does? More concretely, I'd like to know ...
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Prove that QFT and Walsh-Hadamard gates give the same output when acting on $\lvert x\rangle\lvert 0\rangle$ [duplicate]

I know that $QFT_n|0\rangle$ is equivalent to $H_n|0\rangle$ (mathematical proof). And it is also easy to prove that $QFT_1$ is equivalent to $H_1$ (applied to one QuBit). From looking at the circuit ...
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Are there any common applications where one can replace FFT with Quantum Fourier Transform?

I want to apply QFT to some common applications like on wave equations. I haven't found any applications of QFT except Shor's algorithm and I am yet to build an intuition for its use cases. I am a ...
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N-Qubit Hadamard vs Quantum Fourier Transform

Both Simon's algorithm and the algorithm for period finding begin by placing qubits in the equal superposition state, but Simon's algorithm uses the n-qubit Hadamard $H^{\otimes n}$ while the period ...