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Next Generation Reservoir Computing (NG-RC) is a modern class of model-free machine learning that enables an accurate forecasting of time series data generated by dynamical systems. We demonstrate that NG-RC can accurately predict full many-body quantum dynamics, instead of merely concentrating on the dynamics of observables, which is the conventional application of reservoir computing. In addition, we apply a technique which we refer to as skipping ahead to predict far future states accurately without the need to extract information about the intermediate states. However, adopting a classical NG-RC for many-body quantum dynamics prediction is computationally prohibitive due to the large Hilbert space of sample input data. In this work, we propose an end-to-end quantum algorithm for many-body quantum dynamics forecasting with a quantum computational speedup via the block-encoding technique. This proposal presents an efficient model-free quantum scheme to forecast quantum dynamics coherently, bypassing inductive biases incurred in a model-based approach.

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Inspired by random walk on graphs, diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional dataset. In recent years, among its many applications, DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, classical DM algorithm is computationally prohibitive for a large dataset, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for DM, termed quantum diffusion map (qDM). Our qDM takes as an input N classical data vectors, performs an eigen-decomposition of the Markov transition matrix in time $O(\log^3 N)$, and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total runtime of $O(N^2 \text{polylog}\, N)$. Lastly, quantum subroutines in qDM for constructing a Markov transition operator, and for analyzing its spectral properties can also be useful for other random walk-based algorithms.

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