The first stage of
tackling this problem, was to investigate whether a stochastic process could be implemented in the
Quantum Computing paradigm.
A key tenet of Quantum Computing is that the quantum circuits are always invertible, ie. the starting
states of an operation can always be recovered from the final states. The reason for this is that
Quantum Mechanics does not permit information loss from a system. As an interesting aside, this implies
that quantum computing does not expend any energy, as energy consumption in computation is linked with
the reversibility of the computation. Trying to apply invertibility to a probabilistic system is
impossible however. The set of products of a unitary matrix is a cyclical group so that a
quantum system based on that set is also cyclical. Stochastic processes however, are not cyclical and
converge to a steady state. This implies that the transition matrix of a stochastic process is
not unitary, and cannot therefore be implemented on a quantum computer.
This conclusion leads me to believe that there is no easy way to implement Kalman Filtering on a
Quantum Computer. There is simply no easy way to ensure the key tenet of reversibility. A lot of research remains to be done at the doctorate level, before
it can be proven either way.