Quantum Computation: extracting information from entangled state

[AVMB]

Hello,

I have a state of two registers (A and B) which is in the following form respect to the computational basis:

$$|v> = \frac{1}{N}\sum_{k=0}^{N-1} |k>_A |f(k)>_B$$

I don't know how to prepare this state, it is externally given. I would like to extract the value of $$f(\hat{x})$$ for some known $$\hat{x}$$ .

An obvious solution is to measure both registers respect the computational basis and check whether result obtained from A is $$\hat{x}$$ . This would succed with probability 1/N .

I thought of using some form of Grover-like amplitude amplification to increase the success probability, but since I don't have access to an operator which produces |v> , I can't costruct the appropriate reflection operators.

Any thoughts?

 

[eljose79]

Hello,

Thank you for sharing your question with the forum. It seems like you are trying to extract the value of f(\hat{x}) from a state that is externally given to you. In this situation, there are a few different approaches you could take. One option is to use quantum state tomography to reconstruct the state |v> and then measure it in the computational basis to obtain the value of f(\hat{x}). This method may be more time consuming and resource intensive, but it would give you the exact value of f(\hat{x}).

Another approach is to use a quantum algorithm called quantum phase estimation (QPE). This algorithm can be used to estimate the eigenvalues of a unitary operator, which in your case would be the operator that produces |v>. By estimating the eigenvalues, you can then infer the value of f(\hat{x}). However, QPE requires knowledge of the operator, so it may not be applicable in your situation.

If you are limited in terms of available operators, another option is to use a quantum algorithm called quantum amplitude estimation (QAE). This algorithm can estimate the amplitudes of a superposition state, which in your case would be the amplitudes of |v>. By estimating the amplitudes, you can then infer the value of f(\hat{x}). QAE does not require knowledge of the operator, but it does require a quantum circuit that prepares the state |v>.

I hope these suggestions are helpful to you. Please let me know if you have any further questions or if I can assist you in any other way. Good luck with your research!