Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 58 / 2 / b

Write $a=⟨\psi |w⟩=2^{-N/2}$, which is real and satisfies $0<a<1$ for $N\ge 1$. The invariant two-dimensional span has orthonormal eigenvectorswith eigenvalues $\Delta (1+a)$ and $\Delta (1-a)$. In this basis,Transfer therefore requires the two evolved eigenphases to differ by an odd multiple of $\pi$:The first positive time isEquivalently, exponentiating on the invariant span gives the full evolutionAt $t_0$ this is $-i{e}^{-i\Delta t_0}|w⟩$, so the requested global phase can be chosen as $\alpha =-\Delta t_0-\pi /2$ modulo $2\pi$. Since the two vectors are linearly independent, their relative coefficient cannot produce exact transfer at an earlier positive time. This is continuous-time quantum search, with time proportional to the square root of the search-space dimension for fixed $\Delta$. Restoring units multiplies the stated time by $\hslash$. The degenerate zero-qubit case would already have $\psi =w$, so there would be no smallest positive transfer time; the formula concerns the nontrivial $N\ge 1$ search.