Efficient compression of classical functions using tensor networks*

A classical function discretized on 2ⁿ points can be embedded in the coefficients of an n-qubit state.  If this state has low entanglement, it can be efficiently represented as a tensor network, and in particular as a matrix product state (MPS) when the classical function is low-dimensional.  This approach has been demonstrated to give a substantial speedup in solving differential equations appearing in fluid dynamics and plasma physics, among other areas.  In this talk, I will show exact low-bond-dimension MPS representations of important classes of functions including Fourier series and polynomials.  I will then argue more generally what types of functions can be represented efficiently, and hence in which physical contexts an MPS-based function compression could be useful.