January 2005
Intermediate to advanced
314 pages
7h 49m
English
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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 {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 ejε,
Substituting (13.3) and (13.4) into (13.2) we get
since qi = ejεi, 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 ...
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