Hint for Quantum Computing Question Regarding QFT's and Eigenvalue Estimation

Expert SummarizerIn summary, the student is seeking help with a quantum computing problem involving a given circuit and an observable M. They have attempted to use the inverse Quantum Fourier Transform and spectral decomposition to solve the problem, but have not found a satisfactory solution. The expert suggests analyzing the circuit and understanding its relation to the operator U, which in turn can help determine the eigenvalues of M. They also advise breaking down complex problems into smaller, more manageable parts.
  • #1
ArjSiv
6
0

Homework Statement


I've pasted the actual question below:
http://www.zeta-psi.com/aj/qip5b.png [Broken]

I don't think there are many quantum computing specific things here other than the circuit (which I can derive easily if I can figure out the algorithm)

Homework Equations



The Quantum Fourier Transform: http://en.wikipedia.org/wiki/Quantum_fourier_transform" [Broken]
Helpful Identities:
http://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysis" [Broken]
[tex]M=\sum_i m_i P_i[/tex], I'm assuming [tex]P_i^n = P_i[/tex] where [tex]n >= 1[/tex] is.
Also, I'm assuming that [tex]M^2 \neq I[/tex], in that case I could have used the identity [tex]e^{iAx} = cos(x)I + isin(x)A[/tex] but I can't.

The Attempt at a Solution


I've tried letting [tex]|x> = \sum_{i=0}^{N-1} |i>[/tex], then from there, I can perform an inverse QFT on [tex]|x>U^x|\psi_j>[/tex] where [tex]|\psi_j>[/tex] is some eigenvalue of [tex]U[/tex] (and thus also [tex]U^x[/tex]) to get me [tex]|\omega>[/tex] which I could use in replacement of M in the definition of U.

Assuming I'm not making a trivial mistake, I'm assuming the observable itself I want to find is [tex]M|\psi_j>[/tex], which I can then use to find [tex]M|\psi>[/tex] (since spectral decomposition let's me write this as the sum of eigenvectors).

I think the key somehow revolves around writing the eigenvalues of [tex]M|\psi_j>[/tex] in terms of the eigenvalues of U for each [tex]|\psi>[/tex], assuming that they even have the same eigenvector bases.

I've also tried expanding [tex]e^{2\pi i M/N}[/tex] and I was able to (partially) factor out the [tex]M[/tex], but I wasn't sure where to go from there.
 
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  • #2


Dear student,

Thank you for sharing your thoughts and attempts at solving this problem. I can see that you have a good understanding of the Quantum Fourier Transform and its properties. However, I believe that your approach may not be the most efficient way to solve this problem.

Instead of starting with the inverse QFT, I suggest that you first try to understand the circuit given in the question. It may seem complex at first, but if you break it down into smaller parts, you will see that it is actually composed of simpler gates such as the Hadamard gate and the controlled-phase gate.

Once you have a better understanding of the circuit, you can then try to figure out how it relates to the observable M. Remember that the circuit is essentially a representation of the operator U, which is related to M through the identity M = U^x. By analyzing the circuit, you should be able to determine the eigenvalues of M and how they are related to the eigenvalues of U.

I hope this helps guide you in the right direction. Remember to always break down complex problems into smaller, more manageable parts. Good luck with your studies!
 

1. What is a QFT in quantum computing?

A QFT, or quantum Fourier transform, is a mathematical operation that is used in quantum computing to efficiently extract information from a superposition of quantum states. It is similar to the classical Fourier transform, but it operates on quantum states rather than classical signals.

2. What is the significance of eigenvalue estimation in quantum computing?

Eigenvalue estimation is a crucial part of quantum computing because it allows us to determine the energy levels of a quantum system. This information is important for understanding the behavior of quantum systems and designing efficient quantum algorithms.

3. How does quantum computing use QFT and eigenvalue estimation together?

Quantum computing uses QFT and eigenvalue estimation together in algorithms such as the quantum phase estimation algorithm. This algorithm uses QFT to extract the eigenvalues of a quantum system and then uses eigenvalue estimation to estimate the energy levels of the system.

4. Can classical computers perform QFT and eigenvalue estimation?

No, classical computers are not capable of performing QFT or eigenvalue estimation. These operations require the use of quantum states and quantum operations, which cannot be simulated by classical computers.

5. Are there any challenges or limitations to using QFT and eigenvalue estimation in quantum computing?

Yes, there are several challenges and limitations to using QFT and eigenvalue estimation in quantum computing. These include the need for precise control over quantum states, the susceptibility to errors and decoherence, and the difficulty of scaling these operations to larger systems.

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