[Paper Review] Sublinear scaling in non-Markovian open quantum systems simulations

Moritz Cygorek, Jonathan Keeling+2 more|arXiv (Cornell University)|Apr 11, 2023

Tensor decomposition and applications99 references4 citations

TL;DR

This paper presents a numerically exact algorithm for simulating non-Markovian open quantum systems with sublinear scaling by exploiting self-similarity in tensor networks representing Gaussian environments. By using a divide-and-conquer strategy based on repeatable blocks in process tensors, the method achieves O(n log n) scaling for infinite memory and O(n_c log n_c) for truncated memory, enabling simulations over millions of time steps in minutes—demonstrated on quantum dot fluorescence, superradiance, and strong coupling regimes.

ABSTRACT

While several numerical techniques are available for predicting the dynamics of non-Markovian open quantum systems, most struggle with simulations for very long memory and propagation times, e.g., due to superlinear scaling with the number of time steps $n$. Here, we introduce a numerically exact algorithm to calculate process tensors -- compact representations of environmental influences -- which provides a scaling advantage over previous algorithms by leveraging self-similarity of the tensor networks that represent Gaussian environments. Based on a divide-and-conquer strategy, our approach requires only $\mathcal{O}(n\log n)$ singular value decompositions for environments with infinite memory. Where the memory can be truncated after $n_c$ time steps, a scaling $\mathcal{O}(n_c\log n_c)$ is found, which is independent of $n$. This improved scaling is enabled by identifying process tensors with repeatable blocks. To demonstrate the power and utility of our approach we provide three examples. (1) We calculate the fluorescence spectra of a quantum dot under both strong driving and strong dot-phonon couplings, a task requiring simulations over millions of time steps, which we are able to perform in minutes. (2) We efficiently find process tensors describing superradiance of multiple emitters. (3) We explore the limits of our algorithm by considering coherence decay with a very strongly coupled environment. The algorithm we present here not only significantly extends the scope of numerically exact techniques to open quantum systems with long memory times, but also has fundamental implications for simulation complexity.

Motivation & Objective

• Address the computational bottleneck in simulating non-Markovian open quantum systems with long memory times and extended propagation durations.
• Overcome the superlinear scaling (O(n²) or worse) of existing process tensor algorithms that limit simulation feasibility.
• Develop a numerically exact method that maintains accuracy while drastically reducing computational cost for Gaussian environments.
• Enable practical simulations of complex quantum dynamics such as strong coupling, superradiance, and long-time coherence decay.
• Establish a foundation for scalable tensor network-based simulations of open quantum systems with fundamental implications for simulation complexity.

Proposed method

• Leverage self-similarity in tensor network representations of environmental influences to identify repeatable blocks in process tensors.
• Implement a divide-and-conquer algorithm that recursively decomposes the process tensor using singular value decompositions (SVDs), reducing the number of required operations.
• Apply the algorithm to environments with Gaussian statistics, such as spin-boson models, where the bath correlation function is exponentially decaying.
• For infinite memory, the method scales as O(n log n) in the number of time steps n, while for truncated memory after n_c steps, it scales as O(n_c log n_c), independent of total propagation time.
• Use the structure of the process tensor to avoid recomputing identical or similar blocks, exploiting recursive patterns in the tensor network.
• Integrate the algorithm with existing tensor network techniques like MPOs (matrix product operators) and process tensor methods for efficient dynamics propagation.

Experimental results

Research questions

• RQ1 Can a numerically exact algorithm for non-Markovian open quantum systems achieve sublinear scaling in computational cost with respect to time steps?
• RQ2 How can self-similarity in tensor network representations of environmental memory be exploited to reduce the number of required SVD operations?
• RQ3 To what extent can this method simulate long-memory and long-propagation-time dynamics that are intractable with standard process tensor techniques?
• RQ4 Can the algorithm be applied to physically relevant models such as quantum dots with strong dot-phonon coupling and superradiant emitter arrays?
• RQ5 What is the practical performance gain in terms of computation time for simulations involving millions of time steps?

Key findings

• The algorithm achieves O(n log n) scaling for environments with infinite memory and O(n_c log n_c) for truncated memory, where n_c is the memory cutoff, representing a significant improvement over standard O(n²) or worse scaling.
• Simulations of quantum dot fluorescence under strong driving and strong dot-phonon coupling were performed over millions of time steps in minutes, demonstrating practical feasibility.
• The method efficiently computes process tensors for superradiant emitter systems, enabling the study of collective emission dynamics with high accuracy.
• Coherence decay in a strongly coupled environment was simulated successfully, showing the method's robustness in challenging parameter regimes.
• Despite matrix dimension growth with time steps, the observed computation time scales quasi-linearly or sublinearly in practice across a wide range of parameters.
• The algorithm enables numerically exact simulations of non-Markovian dynamics that were previously computationally infeasible due to scaling limitations.