Efficient Simulation of Non-Markovian Path Integrals via Imaginary Time Evolution of an Effective Hamiltonian

Abstract

Accurately simulating the non-Markovian dynamics of open quantum systems remains a significant challenge. While the recently proposed time-evolving matrix product operator (TEMPO) algorithm based on path integrals successfully circumvents the exponential scaling associated with memory length, its reliance on layer-by-layer tensor contractions and compressions leads to steep scaling with respect to the system Hilbert space dimension. In this work, we introduce the effective Hamiltonian-based TEMPO (EH-TEMPO) algorithm, which reformulates the calculation of the Feynman-Vernon influence functional as an imaginary time evolution governed by an effective Hamiltonian. We demonstrate that this effective Hamiltonian admits a highly compact matrix product operator representation, enabling substantial compression with negligible loss of accuracy. Combining a one-shot global evolution with a backward retrieval approach, EH-TEMPO significantly reduces algorithmic complexity and is naturally suited for GPU acceleration. We benchmark the method against the process tensor TEMPO algorithm using the 7-site Fenna-Matthews-Olson complex model. The results demonstrate that EH-TEMPO achieves numerically exact accuracy with superior efficiency, delivering speedups of up to 17.5x on GPU hardware compared to standard CPU implementations.

Figures (7)

• Figure 1: Tensor network diagrams for the TEMPO algorithms. (a) The triangular tensor network representation of the influence functional used in the (PT-)TEMPO algorithm. (b) The core algorithm of EH-TEMPO, where the influence functional is generated via the time evolution governed by the effective Hamiltonian MPO ($\hat{H}_{\text{eff}}$). (c) The final tensor contraction to obtain the reduced density matrix, combining the influence functional MPS with the system propagators $U_S$ and the initial state $\rho_S(0)$. The white triangle represents $\delta_{ijk}$.
• Figure 2: Exciton population dynamics of the 7-site FMO complex at $T=77\,\text{K}$. The solid colored lines represent the results obtained via the EH-TEMPO algorithm (bond dimension $M_S=128$), while the dashed black lines denote the numerically exact HEOM reference. The two datasets are in excellent agreement, with the EH-TEMPO results accurately reproducing the detailed coherent oscillations.
• Figure 3: Time evolution of the cumulative error $\mathcal{E}(t)$ for the 7-site FMO complex. The performance of EH-TEMPO is compared against PT-TEMPO across three bond dimensions ($M_S=30, 60, 100$), represented by different colors. The line styles distinguish the algorithmic schemes: solid lines for EH-TEMPO with the backward retrieval scheme, dash-dotted lines for the independent scheme, and dashed lines for PT-TEMPO.
• Figure 4: Compressibility of the effective Hamiltonian MPO. (a) The maximum bond dimension of $\hat{H}_{\text{eff}}$ as a function of the SVD compression threshold $\eta$. The bond dimension decreases drastically from the exact value (1752) to compact sizes ($<100$) with non-zero thresholds. (b) The time-dependent calculation error introduced by compression, relative to the uncompressed case. The error remains negligible (order of $10^{-3}$) even for aggressive compression ($\eta=10^{-4}$).
• Figure 5: Comparison of the maximum bond dimension of the effective Hamiltonian MPO with various physical parameters. The x-axis labels represent the parameter pairs $(T/\text{K}, \omega_c/\text{fs}^{-1})$. The dashed line indicates the uncompressed bond dimension (1752), which is constant across these parameters for the given discretization. The colored bars show the significantly reduced bond dimensions after compression with thresholds $\eta=10^{-5}$, $10^{-6}$ and $10^{-7}$, highlighting the efficiency of the compression across different temperatures and cutoff frequencies.