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 ma
trices. 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.
Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these results, we step
back into the classical domain, and explore its usefulness in designing classical algorithms. We achieve an algorithm for solving the major linear-algebraic problems in time $O(n^{omega+ u})$ for any $ u>0$, where $omega$ is the optimal matrix-product constant. Thus our algorithm is optimal w.r.t. matrix multiplication, and comparable to the state-of-the-art algorithm for these problems due to Demmel et. al. Being derived from quantum intuition, our proposed algorithm is completely disjoint from all previous classical algorithms, and builds on a combination of low-discrepancy sequences and perturbation analysis. As such, we hope it motivates further exploration of quantum techniques in this respect, hopefully leading to improvements in our understanding of space complexity and numerical stability of these problems.
