Quantum-Mechanical Methods for Processing the Results of Classical Experiments as an Alternative to Kalman Filter

Abstract

A problem of timeserie data analysis is considered. Whereas standard approaches such as Kalman filter are of linear quadratic estimation (LQE) type, the technique developed in this paper views system dynamics as a sequence of unitary transformations. The approach consists of two steps: 1. Convert the sequence of vector $\mathrm{x}^{(l)}$ observations $l=1\ldots M$ to a sequence of localized at $\mathrm{x}^{(l)}$ states $\vert \psi_{\mathrm{x}(l)}\rangle$. 2. Find unitary operator $\mathcal{U}$ ‘optimally converting $\vert \psi_{\mathrm{x}(l+1)}\}=\vert \mathcal{U}\vert \psi_{\mathrm{x}^{(l)}}\rangle;$: the problem is reduced to finding the maximum of a quadratic form on $U$ matrix elements subject to constraints that are quadratic forms on $U$ matrix elements as well. The approach is outlier-stable and can be applied to the processes with spikes and non-Gaussian noise. The approach is gauge-invariant, e.g. the result is the same when arbitrary non-degenerate linear transform is applied to input vector $\mathrm{x}^{(l)}$ components.