Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians

Abstract

In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states type tensor approximation that allows to represent N-d tensors generated by the grid representation of d-dimensional functions and operators with log-volume complexity, O(d log N), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and the spatial parts of the solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For the class of "analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0 is the error bound and q is a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), where N_t is the number of sampling points in time. The latter allows efficient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute the dynamics and the spectrum in the time-dependent molecular Schrödinger equation.