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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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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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 ...
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Why after transpiling a Qiskit circuit we obtain a different result?

I am trying to obtain the correct circuit transpiled for the ibmq_london device, as I want to know what the real gates applied in the quantum computer are. I am implementing the QFT circuit for 5 ...
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QFT on timeseries Data and compare results with classical FFT

Intention - To learn and apply QFT on time-series data and compare the result with classical FFT. Data Used - Small timeseries data of 16 points - ...
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Is there a quantum operation to change a phase $e^{(0.q_0 q_1 q_2 q_3)}$ into $e^{(0.q_1 q_2 q_3)}$?

Given a set of four qubits, say $q_{0},q_{1},q_{2},q_{3}$ which represent a $4$-bit binary number with $q_{0}$ as the MSB. After applying QFT on these qubits the phase of $q_{0}$using the concept of ...
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How to find a circuit for the roots of QFT?

After reading about using quantum gates instead of ancillas, it asserts that every quantum circuit has a square root. Theoretically, they do, but is there a practical method to generate the quantum ...
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Indexing an “unknown” quantum state

Assuming I have a state $$|x\rangle = \frac{1}{\sqrt{n}}\sum_n |x_n\rangle$$ where $|x_n\rangle$ are quantum state vectors $$|x_n\rangle = \frac{1}{\|x_n\|}\sum_i x_{in}|i\rangle$$ and that I have a ...
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Proof of QFT for a Periodic Function

For Mosca Keynes, ex 7.1.5: You are asked to prove: $\text{QFT}^{-1}_{mr}|\phi_{r,b}\rangle = \frac{1}{\sqrt{r}}\sum_{k=0}^{r-1}e^{-2\pi i \frac{b}{r}k}|mk\rangle$ where $|\phi_{r,b}\rangle = \...
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How to obtain the density matrix using tomography in the real device?

I am trying to run the QFT algorithm for n=5 (n number of qubits). The number of experiments that it generates is bigger than the one allowed by the IBM devices (i.e. 75). Therefore, I have tried to ...
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QFT of 3-bit system

Recently I was learning about QFT(Quantum Fourier Transform). I was learning how QFT is applied with H and cROT gates. I was playing with QFT here. I was testing with 3-Qubit set as you can see in the ...
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Question Regarding Quantum Period-Finding Fourier Transform Approximation

I am following the 5.4.1 Period-Finding Algorithm in Nielsen and Chuang as shown below: My confusion lies with the second expression of point 3 in the procedure. Why is the second expression an ...
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How to check if a quantum circuit is deterministic?

I'm trying to find a way to check if a given quantum circuit is essentially a classical one (up to changes in phase). Given a description of a quantum circuit by a list (of size $l$) of ordered ...
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How does Inverse QFT work in Quantum Phase Estimation?

I'm trying to implement Quantum Phase Estimation from qiskit textbook. Below is the implementation circuit taken from the above-mentioned site: The output at position 2 will be as follows: $$|\psi ...
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Quantum computers don't try all the possible solutions, so how does the QFT really work?

Scott Aaronson is fond of saying "Quantum computers do not solve hard search problems instantaneously by simply trying all the possible solutions at once." That is, they are not non-deterministic ...
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How to implement exponentiation of a gate without breaking complexity?

In the application of QFT for quantum phase estimation (QPE) of a unitary $\mathbf{U}$, one has to perform successive controlled operations using powers of $\mathbf{U}$. In order not to break the ...
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What is the probability to get all qubits equal zero after QFT

Question from exam: Bob built a quantum computer wiht 10 qubits. All qubits are set to zeroes. Bob performed a quantum Fourier transform on the system and then measured the system. what is the ...
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qiskit: IQFT acting on subsystem of reversed-ordered qubits state

I have a state psi as an ndarray of shape (2 ** 3,) s.t. psi[0]= amplitude of 000 ...
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Is it possible to demonstrate a quadratic speed-up of a quantum algorithm on a classical computer?

In article Quantum computational finance: Monte Carlo pricing of financial derivatives the authors said that: Firstly: While a practical quantum computer has yet to become a reality, we can ...
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What does “decoherence attenuates the density matrix” mean?

I'm reading the paper Implementation of the Quantum Fourier Transform. On page 4, they write To a first approximation, decoherence during the course of the QFT attenuates the entire density matrix. ...
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Does the Quantum Fourier Transform (QFT) preserve entanglement?

It is well known that entanglement in a quantum state is not affected when you perform a combination of 1-qubit unitary transformations. I have seen that the QFT can be decomposed into product of 1-...
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Does the quantum Fourier transform have many applications beyond period finding?

(This is a somewhat soft question.) The quantum Fourier transform is formally quite similar to the fast Fourier transform, but exponentially faster. The QFT is famously at the core of Shor's ...
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Why can the QFT be replaced by Hadamard gates?

I'm studying Shor's Algorithm. In the book, author explains QFT can be replaced by Hadamard gates? Why this process is possible?? Thank you everybody. This is QPE. I attach part of book!!
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Cannot replicate results in article on pricing financial derivatives on IBM Q

I am trying to implement a circuit for searching for the largest eigenvalue and respective eigenvector of an operator, i.e. phase estimation, introduced in article Towards Pricing Financial ...
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Implementing QFT for Shor's Algorithm

I’m trying to get a Quantum Fourier Transform working with the rest of a compiled version of Shor’s algorithm, attempting to factor $N=21$. In the following image, there’s an initialization phase (...
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Can a QFT be implemented as a physical change of the measurement basis?

Suppose you had an "analog" quantum computer, where a register would store a wavefunction $\psi(x)$ where $x$ is a continuous variable (with $0\leq x\leq 1$, say). Instead of gates, you would somehow ...