How to shift the eigenvalues of a quantum Hermitian operator G to ±r?

Question

Consider a gate $\mathcal{G}(\mu)=e^{-i \mu G}$ generated by a Hermitian operator G. If G has just two distinct eigenvalues(which can be repeated) we can, without loss of generality, shift the eigenvalues to ±r, as the global phase is unobservable. Note that any single qubit gate is of this form.

I dont know How to shift the eigenvalues of a quantum Hermitian operator G to ±r ?

Schuld, Maria, et al. "Evaluating analytic gradients on quantum hardware." Physical Review A 99.3 (2019): 032331.

Answer 1

If G has eigenvalues a and b, $$ G-\frac{a+b}{2} {\mathbb I} $$ will have eigenvalues $\pm \frac{a-b}{2}$. But adding the extra identity term to G merely adds a global phase $-i\mu (a-b)/2$ to $\cal G$ ...