Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of Phase Estimation algorithms performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in $O(log^2 n)$ parallel time with a total cost of $O(n^{omega+1})$ Boolean operations. This Boolean complexity matches the best known rigorous $O(log^2 n)$ parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases $(mod (2pi))$ is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in $O(log^2 n)$ parallel time and $O(n^omega)$ Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.
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