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Do quantum computers only offer some computational benefit or advantage for complicated algorithms or would simple ordinary least squares solving or computing the maximum likelihood estimates or Newton Raphson also benefit?

Would Monte Carlo methods or Bayesian simulation based methods or Gibbs sampling be improved?

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We can expect that quantum computers will provide a significant advantage in various machine learning classic algorithms, and theoretical results for some of them are already here. For example, the quantum EM algorithm for mixture of gaussian has been studied and the complexity of the computation is given In the article Quantum Expectation-Maximization Algorithm written by Hideyuki Miyahara (first author). Their main result is that the algorithm is exponentially faster in the number of sample than it's classical counterpart.

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  • $\begingroup$ Is there some kind of quantum machine learning or quantum statistical journal ? $\endgroup$ – user10151 Feb 24 at 16:34
  • $\begingroup$ not that I know of but it might be. If you look at the references of the articles I gave, you can see that most references comes from physics and statistical journals, it's a clue that maybe there's no dedicated journal for it yet. $\endgroup$ – nathan raynal Feb 24 at 16:38
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Quantum computers bring quadratic speed-up for Monte Carlo method. See for example Quantum Risk Analysis where the authors discussed application of Monte Carlo for risk management in finance.

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