Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis, including rank-structured solvers and preconditioners based on "quantum-inspired" algorithms such as DMRG. However, the theory of QTT approximation is not fully understood.

We advance this theory from the point of view of multiscale polynomial interpolation. This perspective clarifies why QTT ranks decay with increasing depth, quantitatively controls QTT rank in terms of smoothness of the target function, and explains why certain functions with sharp features and poor quantitative smoothness can still be well approximated by QTTs. The perspective also motivates new practical algorithms for the construction of QTTs.

Finally, we leverage the perspective of multiscale interpolation to offer the first direct construction of the fast Fourier transform (FFT) as a QTT operator, equipped with a priori compression guarantees.

Michael Lindsey is an Assistant Professor whose research focuses on computational methods at the intersection of numerical linear algebra, optimization, and randomization. His work addresses high-dimensional scientific computing challenges, particularly in quantum many-body problems relevant to quantum chemistry and condensed matter physics, as well as various issues in applied probability. His approaches leverage techniques such as semidefinite relaxation, Monte Carlo sampling, and optimization over parametric function classes like tensor networks and neural networks.