We would like to find the eigenvalue e 2 π i θ {\displaystyle e^{2\pi i\theta }} of | ψ ⟩ {\displaystyle |\psi \rangle } , which in this case is equivalent to estimating the phase θ {\displaystyle \theta } , to a finite level of precision. We can write the eigenvalue in the form e 2 π i θ {\displaystyle e^{2\pi i\theta }} because U is a unitary operator over a complex vector space, so its eigenvalues must be complex numbers with absolute value 1.