Fig. 2: Mean squared error and condition number vs number of outcomes.
From: Potential and limitations of quantum extreme learning machines
Mean squared error (MSE) associated to reconstruction of \({{{{{{{\rm{Tr}}}}}}}}({{{{{{{\mathcal{O}}}}}}}}\rho )\) for some single-qubit observable \({{{{{{{\mathcal{O}}}}}}}}\) in the first scenario configuration, with \({M}_{{{{{{{{\rm{tr}}}}}}}}}=100\) and Mtest = 1000 states used during training and testing phase, respectively. In all plots, different colours refer to different numbers of samples Ntrain, Ntest used to estimate the probabilities. The target observable is chosen at random, and kept fixed in all shown simulations. Choosing different observables does not significantly affect the behaviour of these plots. a Condition number of the probability matrix \(\langle \tilde{{{{{{{{\boldsymbol{\mu }}}}}}}}},{{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{{{{{{{\rm{tr}}}}}}}}}\rangle\) as a function of the number of measurement outcomes. b MSE as a function of the number of measurement outcomes, when both train and test probabilities are estimated with the same finite precision. c As above, but now the test probabilities are estimated with infinite precision. d As above, but now the training probabilities are estimated with infinite precision. In this last case, the large error corresponding to four outcomes is due to the amplification of the statistical error in the vector of probabilities p by the map W. The amount of amplification is described by the condition number in Eq. (19).
