Section: New Results

Adding a single state memory optimally accelerates symmetric linear maps

Participants: Alain Sarlette

The results of this section are to be published in IEEE Trans. Automatic Control [24] .

This work is exploring the context and benefits of so-called “non-Markovian” dynamics, where the dynamics implied by hidden variables modifies the behavior of an iterative procedure. Such mechanisms appear in both classical and quantum systems, and one of our future goals is to better characterize the benefits of engineered non-Markovianity in terms of stabilizing power in very constrained systems. The precise setting here is a discrete-time linear map, which is unknown except for a lower and upper bound on its eigenvalues. By adding one memory slot to each coordinate, this map can be accelerated quadratically. We prove that by adding more memory slots, this cannot be further improved. This is reminiscent of the acceleration of random walks by lifting them or by quantizing them, which we are currently exploring.