Tree tensor network state approach for solving hierarchical equations of motion.

Area of Science:

• Quantum mechanics
• Computational physics
• Open quantum systems

Background:

• The hierarchical equations of motion (HEOM) is a numerically exact method for simulating open quantum systems.
• HEOM relies on an exponential expansion of bath correlation functions, mapping the environment into effective bath modes.
• This allows for efficient computation at finite temperatures by transforming the system into a Schrödinger-like equation with a non-Hermitian super-Hamiltonian.

Purpose of the Study:

• To explore the representation of the HEOM extended wave function using tree tensor network states (TTNS).
• To develop and apply a time propagation algorithm based on the time-dependent variational principle for the HEOM+TTNS approach.
• To assess the computational efficiency and accuracy of the proposed method compared to conventional HEOM.

Main Methods:

• Representing the extended wave function and super-Hamiltonian within the HEOM framework as tree tensor network states (TTNS) and operators, respectively.
• Implementing a time propagation algorithm utilizing the time-dependent variational principle.
• Performing benchmark calculations using the spin-boson model with a slow-relaxing bath.

Main Results:

• The HEOM+TTNS approach yields results consistent with the conventional HEOM method.
• The proposed method significantly accelerates the computation of open quantum system dynamics.
• Simulations using TTNS demonstrated a fourfold speedup compared to one-dimensional matrix product state decomposition schemes.

Conclusions:

• The integration of tree tensor network states offers a computationally efficient and accurate alternative for the hierarchical equations of motion method.
• The HEOM+TTNS approach provides a promising direction for advancing the simulation of complex open quantum systems.
• This work highlights the potential of tensor network methods in accelerating quantum dynamics calculations.