- #1
Peter_Newman
- 155
- 11
Good day everybody,
I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for.
In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's clear to me, that may look like this:
$$|\psi\rangle=\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}|x\rangle$$
In order to imagine the geometrically better one can divide then this vector and exactly that I have a question. So the superposition contains a solution (or solutions) and not solutions, that can be represented as a sum in the form:
$$|\psi\rangle=\frac{1}{\sqrt{N}}\left(\sum_{x \in L}^{'}|x\rangle + \sum_{x \notin L}^{''}|x\rangle\right)$$
Sum over ' means solutions and '' means no solutions
But now in the literature this form came here:
$$|\alpha\rangle \equiv \frac{1}{\sqrt{N-M}}\sum_{x}^{''}|x\rangle$$
$$|\beta\rangle \equiv \frac{1}{\sqrt{M}}\sum_{x}^{'}|x\rangle$$
Why are these states defined, how does one get to this equation?
Then it goes on:
In the book its called "simple algebra shows that the initial state ##|\psi\rangle## may be represented as:"
$$|\psi\rangle=\sqrt{\frac{N-M}{N}}|\alpha\rangle + \sqrt{\frac{M}{N}}|\beta\rangle$$
My question is, how do you get there? I can not quite understand that.
If anyone knows how to get this shape, it would be really nice if someone could help me a bit.
I hope that my question is right here.
I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for.
In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's clear to me, that may look like this:
$$|\psi\rangle=\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}|x\rangle$$
In order to imagine the geometrically better one can divide then this vector and exactly that I have a question. So the superposition contains a solution (or solutions) and not solutions, that can be represented as a sum in the form:
$$|\psi\rangle=\frac{1}{\sqrt{N}}\left(\sum_{x \in L}^{'}|x\rangle + \sum_{x \notin L}^{''}|x\rangle\right)$$
Sum over ' means solutions and '' means no solutions
But now in the literature this form came here:
$$|\alpha\rangle \equiv \frac{1}{\sqrt{N-M}}\sum_{x}^{''}|x\rangle$$
$$|\beta\rangle \equiv \frac{1}{\sqrt{M}}\sum_{x}^{'}|x\rangle$$
Why are these states defined, how does one get to this equation?
Then it goes on:
In the book its called "simple algebra shows that the initial state ##|\psi\rangle## may be represented as:"
$$|\psi\rangle=\sqrt{\frac{N-M}{N}}|\alpha\rangle + \sqrt{\frac{M}{N}}|\beta\rangle$$
My question is, how do you get there? I can not quite understand that.
If anyone knows how to get this shape, it would be really nice if someone could help me a bit.
I hope that my question is right here.
Last edited: