paces: Parallelized Application of Co-Evolving Subspaces, a method for computing quantum dynamics on GPUs

TL;DR

The method is built from the ground up as a parallel algorithm for graphics processing units and is applicable to arbitrary Hamiltonians that are sparse in a given basis and can be extended to open quantum system dynamics and/or time-dependent generators.

Abstract

An efficient method of computing the dynamics of a pure quantum state under the time-dependent Schrödinger equation is described: At each timestep, a restricted subspace of the potentially infinite-dimensional total Hilbert space is systematically and naturally constructed via the image of repeated applications of the Hamiltonian operator, and the time evolution is computed exactly within said subspace. The subspace is dynamically recomputed at each timestep such that it co-evolves with the state vector. We benchmark the method using the Holstein model and compare the formal information content of its representation to the matrix-product state formalism. The method is built from the ground up as a parallel algorithm for graphics processing units and is applicable to arbitrary Hamiltonians that are sparse in a given basis. It can be extended to open quantum system dynamics and/or time-dependent generators.

Figures (3)

• Figure 1: The basic idea of paces: Construct an effective Hilbert space, evolve exactly within this effective Hilbert space, truncate and repeat. Each circle represents a basis state. Dark blue: the effective Hilbert subspace; light blue: non-zero components of the state vector; yellow highlighting: basis states affected by a change.
• Figure 2: The distribution of the expansion coefficients for a paces Holstein model at several times after the start of the evolution from a fully localized Franck--Condon excitation, in a lin-log plot (left) and a log-log plot (right). We see that the vast majority of basis states contribute negligibly to the total wavefunction, and the distribution (for the most part) seems to be in between an exponential and a power law, spreading out over time. The hatched areas apply to the fat-tailed $t = 50/\omega$ state and show that less than half of the coefficients account for 99.99% of the total weight. The dashed line shows a $1/n^2$ power law that is speculated to be attained here (see section \ref{['sec:benchmarking']}). Parameters: $L=25$, $g=4\hbar\omega$, $J = \hbar\omega$.
• Figure 3: The time evolution of a perturbed quantum harmonic oscillator (QHO), $H = (\hbar\omega)\left[ a^\dag a + 2.5\left(a + a^\dag\right)\right]$, starting from an initial vacuum state $\ket{0}$. The dark blue line is the average occupation number, and the shaded areas mark the basis states required to capture 50%, 90% or 99% of the weights $\abs{\braket{n}{\psi(t)}}^2$ of the state.