Cooling down trees for finite-temperature quantum dynamics: Purification within ML-MCTDH

Simulating multidimensional quantum systems at finite temperature is inherently challenging as the system is no longer described by a single, pure-state wavefunction but by a density operator, squaring an already exponential scaling of the Hilbert space. Building upon the compact wavefunction ansatz of the multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH) method, we present a new scheme for simulating finite-temperature quantum dynamics based on purification. Here, a density operator is mapped to a single ML-MCTDH wavefunction in an enlarged Hilbert space, comprising physical and auxiliary degrees of freedom. In the key step, one obtains a pure-state representation of the canonical density operator via imaginary time-propagation of the infinite-temperature state. The most important observation is that this state can be exactly decomposed as a Hartree product of maximally entangled combined modes, each combined mode consisting of a physical degree of freedom and its auxiliary counterpart. Through dynamically pruning the node ranks of the ML-tree during the "cool down" stage yields a compact finite-temperature wavefunction, thus accelerating the real-time propagation and enabling finite-temperature simulations of multidimensional, correlated molecular systems. Our method circumvents both intensive statistical sampling and costly tensor-decomposition of the full density-operator, while being broadly applicable to model Hamiltonians and general ab initio potential energy surfaces alike. Two applications of the method are presented: benchmark results on the thermal ground-state of H2O, as well as temperature-dependent infrared absorption spectra of the more challenging, floppy H3O2- anion.