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

I've probably read the chapter The quantum Fourier transform and its applications from Nielsen and Chuang (10 th anniversary edition) a couple of times before and this took this thing for granted, but ...
1k views

Why can the Discrete Fourier Transform be implemented efficiently as a quantum circuit?

It is a well known result that the Discrete Fourier Transform (DFT) of $N=2^n$ numbers has complexity $\mathcal O(n2^n)$ with the best known algorithm, while performing the Fourier transform of the ...
170 views

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 ...
790 views

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!!
389 views

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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Why should we use inverse QFT instead of QFT in Shor's algorithm?

Why should we use inverse QFT instead of QFT in Shor's algorithm? When I tried to simulate Shor's algorithm for small numbers, I got an answer even when I used just QFT instead of inverse QFT.
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Why is quantum Fourier transform required in Shor's algorithm?

I’m currently studying the Shor’s algorithm and am confused about the matter of complexity. From what I have read, the Shor’s algorithm reduces the factorization problem to the order-finding problem ...
2k views

Implementation of inverse QFT?

When implementing the inverse quantum Fourier transform, in addition to reversing the circuit, does one need to take the conjugate transpose of the phase shift gates in the circuit as well?
143 views

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 ...
159 views

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= amplitude of 000 ...
115 views

Hidden subgroup problem

Let $H$ be a hidden subgroup of $G_1$ that is indistinguishable from subgroup $H^{\prime}$ by quantum Fourier sampling. Now take a larger group $G_2$ such that it contains $G_1$. Now if I do quantum ...
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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 ...
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 ...