# 13

# Derivations Related to the Generalized Grover Algorithm

## 13.1 EIGENVALUES OF THE GENERALIZED GROVER OPERATOR

To find the eigenvalues of *Q* one should solve the characteristic equation det {**Q** − *q***I**} = 0, which seems to be a fairly hard task

Therefore we follow a more pragmatic way. Applying the basis-independent product of eigenvalues in the form of det {**Q**} = *q*_{1}*q*_{2} as well as exploiting the form of eigenvalues of unitary operators *e*^{jε},

Substituting (13.3) and (13.4) into (13.2) we get

since *q*_{i} = *e*^{jεi}, hence the eigenvalues of the generalized Grover operator become

Furthermore, it is known that the trace of **Q** can be expressed as

resulting in

where the equality stands if both the real and ...