Question

If a unitary operator U has an eigenvector |ψ⟩ with eigenvalue e^(2πiθ), what does the Phase Estimation algorithm output?

Accepted Answer

The Phase Estimation algorithm outputs an estimate of the phase value theta represented as a binary fraction. The algorithm uses two quantum registers: the first register contains t ancillary qubits initialized to the state |0⟩, and the second register contains the eigenvector |ψ⟩ of the unitary operator U. The process applies a series of controlled-U operations where U is raised to increasing powers of two, followed by an inverse Quantum Fourier Transform on the first register. If theta can be expressed exactly as a fraction with a denominator of 2 raised to the power of t, the algorithm outputs the binary string representing the integer 2 to the power of t multiplied by theta with a probability of 100 percent. If theta cannot be expressed exactly in this form, the algorithm outputs the best t-bit approximation of theta, with the probability of measuring the correct value peaking near the true value due to constructive interference. In this context, a unitary operator is a transformation that preserves the inner product of vectors, ensuring that its eigenvalues always have a magnitude of one and can be written in the form e to the power of 2 pi i theta. The phase theta is a real number between 0 and 1 that characterizes the rotation applied by the operator.

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If a unitary operator U has an eigenvector |ψ⟩ with eigenvalue e^(2πiθ), what does the Phase Estimation algorithm output?

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