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Quantum Computing and Communications: An Engineering Approach by Ferenc Balazs, Sandor Imre

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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 {QqI} = 0, which seems to be a fairly hard task

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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,

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

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since qi = ei, hence the eigenvalues of the generalized Grover operator become

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

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resulting in

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

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