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Unformatted text preview: CSE 599d - Quantum Computing Quantum Phase Estimation and Arbitrary Size Quantum Fourier Transforms Dave Bacon Department of Computer Science & Engineering, University of Washington What use is the quantum Fourier transform? Well we’ve already seen Jordan’s algorithm which was like the Bernstein-Vazirani problem. But the most important use for the quantum Fourier transform was discovered by Peter Shor when he showed how to us it to efficiently factor numbers. In these notes we’ll develop the quantum machinery necessary to get to the point where we can understand Shor’s algorithm. I. QUANTUM PHASE ESTIMATION ALGORITHM Let’s look at the quantum Fourier transform. It performs the transform | x i → 1 √ N N- 1 X y =0 ω- xy N | y i (1) Or, looking at it in a different manner, transforms the states 1 √ N N- 1 X y =0 ω xy N | y i → | x i (2) Now what we learned from the Bernstein-Vazirani problem was that if we could get the information about the function into the ± 1 phase of each state, then we could use this, for the Hadamard basis states, to find a hidden string s for the function f ( x ) = s · x . What might the analogy be for the quantum Fourier transform? Well the first thing to notice is that there are roots of unity which are not just ± 1 here. Intuitively this means that a phase kickback trick will have to do phase kickback with roots of unity. Recall that the eigenvalues of unitary matrices are roots of unity. This leads us naturally to the quantum phase estimation algorithm. Suppose that you are in the following situation. You can prepare an eigenstate | ψ i of a unitary operator U . Further, you have the ability to apply a controlled- U 2 j operator. How can you use these tools to estimate the eigenvalue of this unitary. First recall that unitary matrices have eigenvalues which are roots of unity. Let’s call the eigenvalue corresponding to the eigenvector | ψ i , e i 2 πφ , where 0 ≤ φ < 1. Our goal is to find an estimate of φ . The quantum phase estimation algorithm performs exactly this task. How does this algorithm work? It begins by preparing doing a phase kickback trick. Consider the following quantum circuit on n qubits, | i H • ··· | i H • ··· ....
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- Fall '08
- Computer Science, Nuclear magnetic resonance, Shor's algorithm, Quantum information science, QFT, Unitary operator