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 unitary $U:|x_n\rangle \mapsto e^{2\pi i\theta_n}|x_n\rangle$ such that I can use the phase estimation procedure to get the state $$|x\rangle = \frac{1}{\sqrt{n}}\sum_n |x_n\rangle|\theta_n\rangle$$

Question: I am wondering whether there is a way to compute the state $$|x\rangle = \frac{1}{\sqrt{n}}\sum_n |x_n\rangle|n\rangle$$ I was thinking to modify the Phase Estimation Algorithm, but I still find it difficult to understand if I can prepare a unitary $U = \sum_n e^{2\pi in}|x_n\rangle\langle x_n|$ for instance.
I am not insterested in ordering the vectors $|x_n\rangle$ in any way, I just wonder if there is a way to index them easily. I don't know if this problem has been raised before in literature and I don't know where to look at. I'd be glad if someone had some insights.

  • $\begingroup$ Do you want to index the vectors $\{|x_n\rangle\}$ classically? If yes, then an orthonormal basis for the $n$-qubit space can be efficiently indexed using only $n$ classical bits. Consider labelling states of the form $|10111 \cdots 1\rangle$ with the classical bit string $10111 \cdots 1$? If not, can you clarify what you're looking for? $\endgroup$ Jun 20 '20 at 0:45
  • $\begingroup$ what are the $|\theta_n\rangle$ here? $\endgroup$
    – glS
    Jun 21 '20 at 9:48

The root of the issue here is how do you map between the values $\theta_n$ and $n$. A priori there is no way of doing this because the values $n$ are a completely abstract labelling. It wouldn't make any difference if I rearranged all the labels $n$.

So, you have defined the $n$s to be a particular order that you want. Presumably as part of that, you know how to identify, given a $\theta_n$, what the value of $n$ is. Whatever mental process that you go through to identify it, you need to translate that into a circuit which you would apply on the ancilla register.

Incidentally, are the coefficients $x_{in}$ known? If so, you should be able to construct a transformation directly rather than having to use phase estimation.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.