Solving the Gross-Pitaevskii Equation with Quantic Tensor Trains: Ground States and Nonlinear Dynamics

TL;DR

The paper introduces a quantic tensor-train (QTT) framework to solve the nonlinear Gross-Pitaevskii equation for Bose-Einstein condensates, leveraging TDVP and gradient-descent optimization with nonlinear term compression to achieve exponentially fine grids at linear cost. It demonstrates ground-state computations and real-time dynamics, including vortex lattices in rotating BECs, with bond-dimension saturation ensuring stable long-time evolution. The results show significant efficiency and accuracy advantages over traditional grid-based methods, highlighting QTT as a powerful tool for nonlinear quantum simulations. The work suggests potential integrations with adaptive meshes and implicit time integration to broaden applicability to continuum nonlinear quantum problems.

Abstract

We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. By adapting time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE's nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost. We benchmark ground states and dynamics of BECs--including vortex lattice formation and breathing modes--demonstrating superior performance over conventional grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.

Figures (14)

• Figure 1: Schematic diagrams of tensor contractions and QTT structures. (a) The tensor contractions for computing the effective Hamiltonian. (b) QTT wavefunctions for $\psi^2$ and its compressed QTT form $\psi_\chi^2$. (c) Tensor contractions involved in computing the gradient of the interaction energy, shown for both the exact calculation and the approximation using $\psi_\chi^2$.
• Figure 2: Absolute errors in the gradient of the interaction energy (black curve) and in the slope along the descent direction (red curve) as functions of the number of tensors updated in $\psi_\chi^2$ after updating a tensor in $\psi$.
• Figure 3: Absolute fidelity errors in the QTT of ground-state wavefunctions for (a) a BEC in a harmonic and (b) a BEC with vortices, plotted as functions of bond dimension $D$. The inset in panel (a) shows the density profile of the corresponding ground-state wavefunction. The density profiles of the vortex states can be found in Fig. \ref{['fig:2d_diffvor']}.
• Figure 4: CPU times for a single ITE step using the QTT and conventional finite-difference methods, as functions of the number of qubits (equivalently, the number of grid points). Results are shown in (a) linear and (b) logarithmic scales to highlight the different scaling behaviors of the two approaches.
• Figure 5: Convergence of the chemical potential $\mu$ computed using gradient descent (GD) and TDVP within the QTT framework, shown as functions of (a) CPU time and (b) optimization steps, for a BEC in a harmonic trap. Different colors correspond to different bond dimensions of the QTT wavefunctions. Panel (c) shows the evolution of the bond dimension of $\psi_\chi^2(\mathbf{r})$ during the optimization.