Algebraic Proof of Correctness
To complete the algebraic analysis we need to find out what happens when we repeatedly apply . A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm is a linear combination of and . We can write the action of and in the space spanned by as:
So in the basis (which is neither orthogonal nor a basis of the whole space) the action of applying followed by is given by the matrix
This matrix happens to have a very convenient Jordan form. If we define, it is
- where
It follows that rth power of the matrix (corresponding to r iterations) is
Using this form we can use trigonometric identities to compute the probability of observing ω after r iterations mentioned in the previous section,
- .
Alternatively, one might reasonably imagine that a near-optimal time to distinguish would be when the angles 2rt and -2rt are as far apart as possible, which corresponds to or . Then the system is in state
A short calculation now shows that the observation yields the correct answer ω with error O(1/N).
Read more about this topic: Grover's Algorithm
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