Quantum Data Fitting Algorithm for Non-sparse Matrices
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We propose a quantum data fitting algorithm for non-sparse matrices, which is based on the Quantum Singular Value Estimation (QSVE) subroutine and a novel efficient method for recovering the signs of eigenvalues. Our algorithm generalizes the quantum data fitting algorithm of Wiebe, Braun, and Lloyd for sparse and well-conditioned matrices by adding a regularization term to avoid the over-fitting problem, which is a very important problem in machine learning. As a result, the algorithm achieves a sparsity-independent runtime of O(κ^2√(N)polylog(N)/(ϵκ)) for an N× N dimensional Hermitian matrix F, where κ denotes the condition number of F and ϵ is the precision parameter. This amounts to a polynomial speedup on the dimension of matrices when compared with the classical data fitting algorithms, and a strictly less than quadratic dependence on κ.
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