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June  2019, 1(2): 157-176. doi: 10.3934/fods.2019007

## Estimation and uncertainty quantification for the output from quantum simulators

 1 Computational Science and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA 2 Department of Statistics and Applied Probability, National University of Singapore, Singapore 3 School of Mathematics, University of Manchester, Manchester, UK, M13 9PL

* Corresponding author

Published  May 2019

The problem of estimating certain distributions over {0, 1}d is considered here. The distribution represents a quantum system of d qubits, where there are non-trivial dependencies between the qubits. A maximum entropy approach is adopted to reconstruct the distribution from exact moments or observed empirical moments. The Robbins Monro algorithm is used to solve the intractable maximum entropy problem, by constructing an unbiased estimator of the un-normalized target with a sequential Monte Carlo sampler at each iteration. In the case of empirical moments, this coincides with a maximum likelihood estimator. A Bayesian formulation is also considered in order to quantify uncertainty a posteriori. Several approaches are proposed in order to tackle this challenging problem, based on recently developed methodologies. In particular, unbiased estimators of the gradient of the log posterior are constructed and used within a provably convergent Langevin-based Markov chain Monte Carlo method. The methods are illustrated on classically simulated output from quantum simulators.

Citation: Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1 (2) : 157-176. doi: 10.3934/fods.2019007
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Truth (left) and reconstruction after $K = 10^4$ steps (middle) for known truth. The error as a function of iteration is given in the right plot
The reconstruction with $1000$ (left) and $50$ (right) observations are given in the top row, along with the corresponding convergence plots in the second row. The bottom row shows the actual second moments $\widehat m$ (left), with $M = 50$ observations, which are used to train the model, and the moments under the reconstruction (right)
Convergence of SGD with the debiased estimator described in Section 6.3 (left), the original unbiased estimator $\widehat F$ (middle), and the simple consistent but biased estimator (right)
Illustration of pairwise marginal UQ for the posterior on the diagonal of $\Lambda$ with $M = 1000$ observations and $d = 4$ qubits. The true value of the parameters is indicated in red
Illustration of pairwise marginal UQ for the posterior on the diagonal of $\Lambda$ with $M = 10^6$ observations and $d = 4$ qubits. The true value of the parameters is indicated in red
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