Computing the eigenvectors and eigenvalues of a given Hermitian matrix is arguably one of the most well-studied computational problems. Yet despite its immense importance, and a vast array of heuristic techniques, there is no algorithm that can provably approximate the spectral decomposition of any Hermitian matrix in asymptotic bit-complexity o(n 3 ). Inspired by the quantum computing paradigm, we introduce a new perspective on this problem that draws from the theory of quasi-random, or low-discrepancy sequences -a theory which has been unconnected to linear-algebra problems thus far. We analyze the discrepancy of an n-dimensional sequence formed by taking the fractional part of integer multiples of the vector of eigenvalues of the input matrix. This analysis gives rise to a conceptually new algorithm to compute an approximate spectral decomposition of any n × n Hermitian matrix. This algorithm can be implemented by (randomized) circuits of bit-complexity at most O(n ω+ν ) for any ν > 0. Because it is disjoint from previous algorithms, we hope it sheds new light on the complexity of approximate spectral decomposition, in terms of run-time, space complexity, and the number of random bits required.
Independently and concurrently, Ben-Or and Eldar REF gave an algorithm having boolean complexityÕ (n ω+ν ) for any ν > 0, to compute all the eigenvalues of a matrix.
The Quasi-Random Perspective on Matrix Spectral Analysis with Applications.
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Mathematics Computer Science