[Paper Review] Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems

Meng Xu, Yaming Yan+3 more|arXiv (Cornell University)|Feb 8, 2022

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper introduces Free-Pole HEOM (FP-HEOM), a novel method that overcomes the exponential resource scaling of hierarchical equations of motion (HEOM) at low temperatures by using optimized rational decomposition to cluster Matsubara poles. It enables accurate, efficient, and stable simulations of open quantum systems down to T=0, even for structured reservoirs and long-time dynamics, as demonstrated by quantitatively verifying the Shiba relation in the subohmic spin-boson model.

ABSTRACT

The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems that are embedded in thermal environments. However, its applicability is restricted to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy and long-time stability. Moreover, the technique can directly be implemented in alternative approaches such as Green's function, stochastic, and pseudo-mode formulations. As one highly non-trivial application, for the sub-ohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

Motivation & Objective

• To extend the applicability of the hierarchical equations of motion (HEOM) to arbitrary temperatures, especially the zero-temperature limit, where prior methods fail due to exponential resource scaling.
• To address the fundamental challenge of scale-free quantum noise at low frequencies, which causes an explosion in the number of Matsubara poles in standard HEOM formulations.
• To develop a numerically stable and computationally efficient framework for simulating long-time dynamics and quantum phase transitions in open quantum systems with structured reservoirs.
• To enable high-accuracy simulations of strongly correlated, non-Markovian dynamics in quantum technologies, such as quantum sensing and quantum thermodynamics, at ultralow temperatures.
• To provide a generalizable technique that can be integrated into other approaches like Green’s functions, HOPS, and pseudomode formulations to enhance their performance.

Proposed method

• The method replaces the standard Matsubara pole representation with an optimized rational decomposition of the reservoir noise power spectrum in frequency space.
• It treats auxiliary quantities, including the analytic pole structure, as adjustable parameters under tight accuracy constraints on observable quantities like the reduced density matrix.
• By focusing on the real-time correlation function C(t) and its real-frequency spectrum, it avoids ill-posed numerical analytic continuation issues.
• The approach uses a barycentric representation to efficiently represent the noise correlation function with a moderate number of quasimodes, even at T=0.
• The resulting FP-HEOM framework maintains linear computational complexity in the number of poles K, enabling long-time simulations with high stability.
• The method is implemented via a systematic clustering of higher-order Matsubara poles, effectively emulating an optimized rational approximation of the spectral function.

Experimental results

Research questions

• RQ1 Can the HEOM formalism be extended to arbitrary temperatures, including T=0, without exponential resource scaling?
• RQ2 How can quantum noise at low temperatures, dominated by zero-point fluctuations, be efficiently represented in open quantum system simulations?
• RQ3 Can the FP-HEOM method accurately capture long-time dynamics and quantum phase transitions in strongly coupled, subohmic spin-boson models?
• RQ4 To what extent can the optimized pole clustering approach be generalized and applied to other open system methods such as Green’s functions or HOPS?
• RQ5 Does the FP-HEOM method quantitatively reproduce exact theoretical predictions, such as the Shiba relation, in the zero-temperature limit?

Key findings

• The FP-HEOM method enables stable and accurate simulations of open quantum systems down to T=0, overcoming the exponential resource scaling that plagues standard HEOM at low temperatures.
• The method successfully captures the quantum phase transition from delocalized to localized behavior in the subohmic spin-boson model, with a transition point at s ≈ 0.3.
• For s = 1/2 and α = 0.4 at T=0, FP-HEOM results for long-time dynamics agree quantitatively with highly accurate but computationally expensive TD-DMRG and ML-MCTDH simulations, validating its accuracy.
• The FP-HEOM achieves this accuracy with only 21 quasimodes, demonstrating a substantial reduction in computational cost compared to full Hilbert space methods.
• The method quantitatively verifies the Shiba relation for the long-time decay of correlation functions in the subohmic spin-boson model at T=0, confirming its theoretical consistency.
• The pole-clustering technique is generalizable and can be readily implemented into other approaches such as Green’s functions, HOPS, and pseudomode formulations to improve their efficiency and accuracy.