A quantum Gram-Schmidt and QR decomposition algorithm is proposed with O(N^2) claimed complexity, but the algorithm as written cannot be implemented because it destroys the state it needs to keep.
Matrix QR Decomposition Let A ∈ CN ×M be an arbitrary matrix with full rank satisfying N ≥ M
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Quantum Algorithm for Vector Set Orthogonal Normalization and Matrix QR Decomposition with Polynomial Speedup
A quantum Gram-Schmidt and QR decomposition algorithm is proposed with O(N^2) claimed complexity, but the algorithm as written cannot be implemented because it destroys the state it needs to keep.