Quantum Computing and Communications: An Engineering Approach

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 − qI} = 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} = q₁q₂ 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^jε_i, 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 ...