10.4 Phase estimation

Let U be an N by N square matrix with complex entries. From section 5.9

det(U − λ I_N) = 0

are the eigenvalues {λ₁, λ₂, …, λ_N} of U. Some of the λ_j may be equal. If a particular eigenvalue λ shows up k times among the N, we say λ has multiplicity k.

Each eigenvalue λ_j corresponds to an eigenvector v_j so that

U v_j = λ_j v_j

can take each v_j to be a unit vector.

When U is unitary, we can say even more: each λ_j has absolute value 1 and so can be represented as

λ_j = e^{2π ϕ_j i} where 0 ≤ ϕ_j < 1.

This is slightly different notation than we’ve used before, but it is common. It is the product 2π ϕ that is the full radian measure of the rotation.

Now we are ready to pose the question whose solution we outline in this section:

Let U be a quantum transformation...