Massively parallel implementation and approaches to simulate quantum dynamics using Krylov subspace techniques

Abstract

We have developed an application and implemented parallel algorithms in order to provide a computational framework suitable for massively parallel supercomputers to study the unitary dynamics of quantum systems. We use renowned parallel libraries such as PETSc/SLEPc combined with high-performance computing approaches in order to overcome the large memory requirements to be able to study systems whose Hilbert space dimension comprises over 9 billion independent quantum states. Moreover, we provide descriptions on the parallel approach used for the three most important stages of the simulation: handling the Hilbert subspace basis, constructing a matrix representation for a generic Hamiltonian operator and the time evolution of the system by means of the Krylov subspace methods. We employ our setup to study the evolution of quasidisordered and clean many-body systems, focussing on the return probability and related dynamical exponents: the large system sizes accessible provide novel insights into their thermalization properties.

Many-Body Localization Dynamics from Gauge Invariance

“…where |e 1 is the first unit vector of the Krylov subspace. The approximation becomes an exact solution when m ≥ D, however, the method has been proven to be accurate even if m D for short enough time-steps [228,229]. For the particular case when m D, the much smaller matrix exponential exp(−itA m ) can be evaluated using standard numerical techniques, such as a Padè approximation with a scaling-andsquaring algorithm.…”

Evolution of Entanglement Spectra under Generic Quantum Dynamics

“…For our simulations we choose the parameters (h x /J, h z /J) = (0.9045, 0.809), corresponding to a nonintegrable regime in which thermalization is known to be fast [25,26] (in what follows we set J = 1 as the unit of energy for Hamiltonian dynamics). We use a Krylov-space method to efficiently time-evolve the state [27].…”

Asymmetric Transport in Long-Range Interacting Chiral Spin Chains

“…To treat the many-body problem and considering the interacting term ∆ = 0.0, we choose the exact diagonalizing (ED) method but note that ED is limited to small system size L ≈ 14 due to exponential growth of the Hilbert space dimension 2 L . However, to study the dynamic of many-body systems, the Krylov-space technique with exploiting the sparse structure of Hamiltonian can push limit bigger system size [26][27][28]. We notice for the long-range interaction the Hamiltonian matrix is not sparse as a short-range case.…”