Statistics of Schmidt coefficients and the simulability of complex quantum systems

Abstract

We show that the transition from regular to chaotic spectral statistics in interacting many-body quantum systems has an unambiguous signature in the distribution of Schmidt coefficients dynamically generated from a generic initial state, and thus limits the efficiency of the time-dependent density-matrix renormalization-group algorithm. We investigate this mechanism on the tilted Bose-Hubbard model; however the emergence of universal spectral properties allows translation of our conclusions to generic many-body quantum systems.

Non‐equilibrium dynamics in dissipative Bose‐Hubbard chains

“…Also the time-dependent version of the density matrix renormalization group algorithm, see e.g. [19,50], is applicable safely only in one spatial dimension and not dynamically stable for strongly interacting bosons [51]. Another method which generalizes the so-called Mori projector [52] allows for the calculation of local properties of a closed or open lattice problem.…”

Subsystem dynamics under random Hamiltonian evolution

“…One interesting observation is that, because the convergence to the asymptotic invariant distribution is not monotonic, evolving for longer time can, somehow counter-intuitively, worsen the randomness of states. How the spectrum of the reduced density matrix changes with time has been numerically observed in a random protocol [26] and in a dynamical system [27].…”

A multiprecision C++ library for matrix-product-state simulation of quantum computing: Evaluation of numerical errors

“…The above result shows that we cannot truncate out any nonzero Schmidt coefficient during the simulation. This phenomenon should be related to the distribution of Schmidt coefficients if discussions of reference [9] apply to the present case. We show the distribution at the points we had the maximum Schmidt rank 12 in the simulation, in figure 9 (this is same for both of the points).…”

“…Besides the precision of basic operations, another factor of losing accuracy is the truncation of Schmidt coefficients. When many of Schmidt coefficients are nonnegligible at a certain step of a TDMPS simulation, imposing a threshold to the number of Schmidt coefficients for each splitting may truncate out important data affecting simulation results [9]. So far truncations of nonvanishing Schmidt coefficients have been uncommon 1 and the largest Schmidt rank and its upper bound in the absence of truncations have been of main concern when MPS and related data structures are used for handling quantum and/or classical computational problems [5,11,12,6,13,14,15,16,17].…”