A quantum Kalman filter is a recursive Bayesian filter used to predict and estimate the state of a quantum system, and was first introduced by Viacheslav Belavkin. It is described in his work (Belavkin, 1999). Another interesting work is the quantum extended Kalman filter. However, much of the robots and autonomous agents that we work with currently live in the classical regime and cannot be treated as quantum systems. Thus, the classical Kalman filters, EKF, and etc. will suffice. However, robots and autonomous agents with underlying dynamics in the quantum regime should use the quantum Kalman filter.

Recently, I thought about trying to run a classical Kalman filter or a particle filter on a quantum computer, but I believe that this will not be more efficient than simply running the filter on a classical computer due to the fact that both the system observation and underlying model will both be governed by classical dynamics. The only part of any such algorithm where things can be sped up by a quantum computer is when matrix inversion needs to be done. The algorithm responsible for a quantum speed up in matrix inversion was first invented by Harrow, Hassidim, and Lloyd, and improved upon by Childs, Kothari, and Somma . However, if there exists a way to "Groverize" the problem, or to map the classical system to a state of quantum superpositions, then maybe there exists a speed-up for Kalman filters through quantum computing. One naive idea to do this could be to map a vector such as \((x, y, \theta)\) to the phases of qubits, and then representing the classical actions through some type of isomorphic manipulations in the phase of the qubit, and extracting the result through quantum phase estimation. However, some immediate issues come up such as the numerical stability of the algorithm, since \(x, y\) are real variables getting mapped onto \([0, 2\pi]\), and the question of whether representing the classical actions of the "update" and "observation" step with corresponding manipulations of the phase is efficient on a quantum computer, or even possible.

Thus, in my opinion it isn't too productive to directly look for speedups in recursive Bayesian filtering algorithms through quantum computing. However, quantum computers have been promising towards realizing improvements in reinforcement learning such as a quadratic speed up in training time, and ability to use the underlying Hilbert space of a quantum computer for gradient descent.

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