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I know that the wave like nature of the electrons allows the qubits to interfere with each other amplifying correct answers and canceling out wrong answers. But what kind of problems that use this kind of phenomena? I am still a beginner in the field so please point out any mistake I made along the way. And if there are any research papers or articles regarding my question please let me know.

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    $\begingroup$ answering the question in the title: "quantum interference" is a (the?) fundamental feature of the way we describe quantum phenomena. In this sense, any quantum phenomenon "uses" quantum interference, quantum computation being one such example. $\endgroup$ – glS May 30 at 17:45
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Welcome, Yousef.

A quick answer to your question: Grover's algorithm, among others.

Your first statement is a bit confused: "the wave like nature of the electrons allows the qubits to interfere with each other amplifying correct answers and canceling out wrong answers."

Indeed, interference is between "paths of quantum computation": some are amplified, others cancel out. You may see a quantum computation as the evolution over time of the state of a quantum system (a single qubit or a set of qubits). I suggest you to read the following interview to Scott Aaronson:

https://blogs.scientificamerican.com/cross-check/scott-aaronson-answers-every-ridiculously-big-question-i-throw-at-him/

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  • $\begingroup$ okayy so it isn't correct to say that the wave like nature is what allows this phenomena in quantum computation rather the superpositions created interfere with each other canceling each other or adding up am i correct? $\endgroup$ – yousef elbrolosy May 30 at 15:33
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An example of the interference in the quantum circuit model of computation:

Let's say we have some arbitrary one qubit state $|\psi \rangle = \alpha|0\rangle + \beta|1\rangle$. After applying the Hadamard gate we will have:

$$H |\psi \rangle = \frac{1}{\sqrt{2}}(\alpha|0\rangle + \alpha|1\rangle + \beta|0\rangle - \beta|1\rangle) = \frac{1}{\sqrt{2}}\big((\alpha +\beta) |0\rangle + (\alpha - \beta)|1\rangle\big)$$

because:

\begin{equation} H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \qquad H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) \\ H = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1 \end{pmatrix} \end{equation}

If $\alpha = \beta = \frac{1}{\sqrt{2}}$, we will have constructive interference for $|0\rangle$ state and destructive interference for $|1\rangle$ state:

\begin{equation} H \frac{1}{\sqrt{2}}( |0\rangle + |1\rangle) = \frac{1}{2}( |0\rangle + |1\rangle + |0\rangle - |1\rangle) = |0\rangle \end{equation}

Here is a video lecture (starting from 4:19) about a similar example (with another gate) for the interference.

One of the applications of the interference in QC can be found in the Deutch algorithm. The main concept is that in the algorithm depending on the property of the Oracle (a function that is in the consideration of the algorithm) we will have constructive interference for some outcome and destructive interference for another outcome. For more info: video lecture about the Deutch algorithm (from 50:20 to 1:10:00).

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