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Variational Bayes (VB) is a critical method in machine learning and statistics, underpinning the recent success of Bayesian deep learning.
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Bausch Bausch 2020 proposes a recurrent method with post-selection in the context of quantum Recurrent Neural Networks and argues that, for approximate post-selection it is possible to reduce this overhead. Another possible approach is to run the iterative algorithm without post-selection for T T steps and then amplify the desired end state using Quantum Amplitude Amplification (QAA) ( Brassard et al. 2002 ) . Even with these improvements, the exponential dependence on the number of iterations T / z T/z (where z = 2 z=2 for QAA method) remains
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Note that the conjugate gradient method Shewchuk et al. 1994 ; Saad 2003 , sometimes used as a classical benchmark for quantum linear systems algorithms Clader et al. 2013 , does not apply to the natural gradient problem. The conjugate gradient method requires that the inverted matrix is symmetric and positive definite. The design matrix T T is not symmetric and its symmetrized version ( OPEN ) 0 T T † 0 ) \begin{pmatrix})0&T\\ T^{\dagger}&0\end{pmatrix} is not positive definite
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The algorithm leverages the density matrix exponentiation method of Lloyd et al. Lloyd et al. 2014
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