Question

What is the quantum Fourier transform modulo M of the uniform superposition $\frac{\mathbf{1}}{\sqrt{\mathbf{M}}}\mathbf{\sum }_{\mathbf{j}\mathbf{=}\mathbf{0}}^{\mathbf{M}\mathbf{-}\mathbf{1}}\mathbf{ }\mathbf{|}\mathbf{j}\mathbf{⟩}$?

Short answer

The quantum Fourier Transform modulo M of the uniform superposition is

$|\beta ⟩=\frac{1}{\sqrt{k}}\sum _{j=0}^{k=1} \left|\frac{jM}{k}\right.>=|0⟩$

Step-by-step solution

Explain Quantum Fourier Transform

A quantum circuit takes some amount of qubits as input and outputs the same number of qubits. The quantum circuits take input qubits in $\mathbf{|}\mathit{\alpha }\mathbf{⟩}$state and output the state $\begin{array}{r}|\beta ⟩\end{array}$resulting in Quantum Fourier Transform.

Calculation of Quantum Fourier Transform Modulo M

Consider a function f with n input bits and m output bitts. The input to the quantum circuit is the uniform superposition over the n bit strings x. The uniform superposition poses, equal periods.

The Fourier sampling is represented by k, for the uniform superposition, the representation would be as follows,

$a⟩=\frac{1}{\sqrt{M}}\sum _{j=0}^{M-1} |j⟩$

Input uniform superposition to the quantum,

$\begin{array}{r}\text{a)}=\frac{1}{\sqrt{k}}\sum _{j=1}^{k=1} \frac{jM}{k}\\ \beta ⟩=\frac{1}{\sqrt{1}}\sum _{j=0}^{k=1} \frac{OM}{1}\\ |\beta ⟩=|0⟩\end{array}$

Therefore, The quantum Fourier Transform modulo M of the uniform superposition is $|\beta ⟩=\frac{1}{\sqrt{k}}\sum _{j=0}^{k=1} \left|\frac{jM}{k}\right.>=|0⟩$