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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How to show that the QFT satisfies $\frac1{\sqrt N}\sum_j\prod_le^{2\pi i j_l k/2^l}|j_1...j_n⟩=\bigotimes_l \frac1{\sqrt2}(|0⟩+e^{2\pi i k/2^l}|1⟩)$?
I'm reading Ronald de Wolf's lecture notes, and in chapter 4.5 he writes that
$$
\frac{1}{\sqrt N}\sum\limits_{j=0}^{N-1}\prod\limits_{l=1}^{n}e^{2\pi i j_l k / 2^l}|j_1...j_n\rangle =
\bigotimes\...
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Difference between semiclassical QFT and QFT
In papers, one of them being An Experimental Study of Shor's Factoring Algorithm on IBM Q is stated that replacing QFT with the semiclassical QFT (Kitaev's approach) reduces the needed number of ...
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Does QFT exploit entanglement?
I was studying the quantum circuit for the Quantum Fourier Transform (QFT) on the Mike & Ike, and they write the result of the transformation as a product state.
More precisely they wrote the ...
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Hamiltonian of Qiskit QFT is not hermitian
I am trying to generate the Hamiltonian of a quantum Fourier transform by taking the log of the corresponding unitary using qiskit and scipy.
I don't find a hermitian matrix. Why?
...
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Equating the state of the Phase Estimation algorithm to $\frac{1}{2^{t/2}}\sum_{k=0}^{2^t-1} e^{2\pi i\phi k}|k\rangle$
It is stated in the Phase Estimation algorithm in Page 222, Quantum Computation and Quantum Information by Nielsen and Chuang that
It seems to say that taking the inverse Quantum Fourier transform of ...
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Making sense of the terms Polynomial and Exponential Precision in a Quantum Circuit
The quantum circuit construction of the quantum Fourier transform apparently requires gates of exponential precision in the number of qubits used. However, such precision is never required in any ...
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Formulation of quantum phase estimation in Nielsen and Chuang is different then from other sources?
In a chapter of Quantum Computation and Quantum Information by Nielsen and Chuang (10th edition) about quantum phase estimation I get a little confused. Namely:
Before applying inverse QFT our quantum ...
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Question regarding the notation of QFT
I have a question about the notation of QFT. I would like to present briefly what my problem is. So given is the QFT as a mapping with:
$$|j_1,...,j_n\rangle \rightarrow \frac{(|0\rangle + e^{2\pi i 0....
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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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386 views
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 ...
