On a Quantum Realization of the Bayesian Filtering Using the Log-Homotopy Flow

The representation of a discrete state space in a quantum machine can be very effective due to the superposition effect, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> qubits represent 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> states. To the authors' knowledge, the Bayesian update step using a quantum computing regime has been unsolved up to date. In this paper, a quantum realization of the Daum-Huang log-homotopy flow filter is presented as a solution of the filtering step in a Bayesian recursion for state estimation. The numerical example for a low-dimensional problem demonstrates the feasibility of the presented approach.