Extracting Dynamical Maps of Non-Markovian Open Quantum Systems Using Tensor Networks and Thermofield Doubling

Concepts de base

This research paper presents a novel method for efficiently simulating and analyzing the dynamics of open quantum systems, particularly those exhibiting non-Markovian behavior, by combining tensor network techniques with thermofield doubling and the Choi-Jamiolkowski isomorphism.

Résumé

• Bibliographic Information: Strachan, D. J., Purkayastha, A., & Clark, S. R. (2024). Extracting Dynamical Maps of Non-Markovian Open Quantum Systems. arXiv preprint arXiv:2409.17051v2.
• Research Objective: To develop an efficient numerical method for simulating and analyzing the dynamics of open quantum systems strongly coupled to their environments, focusing on non-Markovian systems where standard perturbative approaches fail.
• Methodology: The researchers employ a combination of techniques within a tensor network framework:
  • Orthogonal polynomial mapping and thermofield doubling to represent the infinite bath with a finite purified chain.
  • Choi-Jamiolkowski isomorphism to reconstruct the dynamical map from a single pure state calculation of the system and its replica.
  • Calculation of the time-local propagator and analysis of its convergence to determine memory timescales.
• Key Findings:
  • The method accurately extracts the dynamical map and time-local propagator for interacting fermionic systems coupled to non-interacting Fermi baths.
  • It establishes a hierarchy of memory timescales associated with the bath correlations, dynamical map, and relaxation dynamics.
  • The approach offers significant speedup in determining the stationary state compared to direct long-time simulations, especially when the relaxation time is much larger than the bath memory time.
• Main Conclusions:
  • The combination of tensor networks, thermofield doubling, and the Choi-Jamiolkowski isomorphism provides a powerful tool for studying open quantum system dynamics beyond the limitations of perturbative methods.
  • The extracted dynamical maps and time-local propagators offer valuable insights into the non-Markovian behavior and relaxation processes in open quantum systems.
• Significance: This research contributes significantly to the field of open quantum systems by providing an efficient and accurate method for simulating and analyzing non-Markovian dynamics, which is crucial for understanding and controlling real-world quantum systems.
• Limitations and Future Research: The current study focuses on fermionic systems with non-interacting baths. Future research could explore extensions to bosonic baths and systems with more complex interactions. Additionally, investigating the applicability of this method to higher-dimensional systems would be of great interest.

Stats

Citations

"The most general description of quantum evolution up to a time τ is a completely positive tracing preserving map known as a dynamical map ˆΛ(τ)."
"By combining several techniques within a tensor network framework we directly and accurately extract ˆΛ(τ) for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths."
"Our numerical examples of interacting spinless Fermi chains and the single impurity Anderson model demonstrate regimes where τre ≫τm, where our approach can offer a significant speedup in determining the stationary state compared to directly simulating the long-time limit."

Idées clés tirées de

Extracting Dynamical Maps of Non-Markovian Open Quantum Systems

by David J. Str... à arxiv.org 10-28-2024

https://arxiv.org/pdf/2409.17051.pdf

Questions plus approfondies

How does the computational cost of this method scale with the system size and bath complexity compared to other existing techniques for simulating open quantum systems?

This method, combining Thermofield doubling, Choi-Jamiolkowski isomorphism, and the PReB formalism, offers several advantages in terms of computational cost compared to traditional open quantum system simulation techniques, particularly when dealing with long-time dynamics and large baths:
Advantages:

Reduced Entanglement Growth: By periodically refreshing the bath, the PReB approach limits the entanglement growth that plagues direct simulations, leading to smaller required bond dimensions (χ) in the MPS representation. This significantly reduces the computational cost, especially for long evolution times.
Finite Bath Representation: The method leverages the fact that the finite-time dynamics of a system coupled to an infinite bath can be accurately captured using a finite bath representation. This is in contrast to methods like HEOM, where the computational cost scales exponentially with increasing evolution time or hierarchy depth.
Single Pure State Calculation: The Choi-Jamiolkowski isomorphism allows for the complete reconstruction of the dynamical map from a single pure state simulation. This contrasts with directly simulating the open system dynamics for various initial states, which becomes computationally expensive.
Scaling:

System Size: The computational cost generally scales exponentially with the system size, similar to other tensor network methods. However, the reduced entanglement growth due to PReB can offer significant practical speedups.
Bath Complexity: The cost scales linearly with the bath memory time (τm), which dictates the required size of the finite bath representation. This is advantageous compared to methods where the cost scales with the total evolution time. However, complex spectral functions might require larger bath sizes, increasing the computational cost.
Comparison to other techniques:

HEOM: While HEOM can be very accurate, it suffers from exponential scaling with evolution time and becomes computationally intractable for long-time dynamics, especially in the presence of strong system-bath coupling.
TEMPO: TEMPO, while efficient for short times, can become computationally expensive for long-time dynamics due to the growth of the process tensor.
Perturbative methods: While computationally cheaper, these methods are limited to weak system-bath coupling and Markovian regimes, which might not be applicable in many realistic scenarios.
In summary: This method offers a computationally advantageous approach for simulating open quantum systems, particularly for long-time dynamics and systems with a clear separation between bath memory time and relaxation time (τm << τre). However, the computational cost still scales exponentially with system size, and bath complexity influences the required bath size.

Could the assumption of a unique steady state be relaxed to study systems exhibiting multiple steady states or limit cycles, and how would that affect the extraction of the dynamical map?

Relaxing the assumption of a unique steady state, while desirable for studying a broader class of open quantum systems, poses significant challenges to the presented method:
Challenges:

PReB Formalism Relies on Steady State: The PReB method fundamentally relies on the existence of a unique steady state to refresh the bath. With multiple steady states or limit cycles, the system might not reach a well-defined state within a single PReB cycle, making bath refreshment ambiguous.
Dynamical Map Becomes Time-Dependent: In systems with multiple steady states, the effective dynamical map describing the evolution over a PReB cycle would depend on the initial state and potentially the entire history of the system. This time-dependence would make the extracted dynamical map less useful for predicting long-time behavior.
Memory Time Ambiguity: The concept of a single bath memory time (τm) becomes ill-defined with multiple steady states. The system might exhibit different relaxation timescales depending on the initial state and the specific steady state it evolves towards.
Potential Modifications:

Modified PReB: One could explore modifications to the PReB algorithm, potentially involving multiple bath refreshments based on different steady states or using a different criterion for bath replacement.
Ensemble Averages: Instead of a single dynamical map, one might extract an ensemble of maps corresponding to different initial states and their respective long-time behavior.
Time-Dependent Dynamical Maps: Developing methods to extract and utilize time-dependent dynamical maps could be necessary to capture the system's complex evolution in the presence of multiple steady states.
In conclusion: Relaxing the unique steady-state assumption introduces significant complexities in defining and extracting a meaningful dynamical map using the presented method. Further research and modifications to the PReB algorithm and the dynamical map extraction process are required to extend its applicability to such systems.

What are the potential implications of this research for developing more robust and efficient quantum technologies, such as quantum computers or sensors, that are inevitably subject to environmental interactions?

This research, focusing on efficiently extracting dynamical maps for open quantum systems, holds significant potential implications for developing more robust and efficient quantum technologies:
1. Improved Quantum Error Correction:

Accurate Noise Modeling: By accurately capturing the non-Markovian dynamics induced by the environment, this method can lead to more realistic noise models for quantum computers. This is crucial for developing and optimizing quantum error correction codes tailored to specific noise channels.
Optimized Error Mitigation Strategies: Understanding the dynamical map allows for the development of targeted error mitigation strategies that exploit the specific structure of the noise process.
2. Enhanced Quantum Control:

Open-Loop Control Optimization: Knowing the dynamical map allows for optimizing open-loop control pulses that account for the environmental interactions, leading to more robust and accurate quantum gate operations.
Tailored System-Environment Coupling: The insights gained from the dynamical map can guide the design of quantum devices with engineered system-environment interactions, potentially exploiting the environment for beneficial purposes like dissipative quantum computing.
3. Efficient Quantum Sensor Design:

Optimized Sensitivity and Coherence: By understanding the environmental interactions through the dynamical map, one can design quantum sensors with optimized sensitivity and coherence times, even in the presence of noise.
Novel Sensing Modalities: The extracted dynamical map can reveal subtle features of the environment, potentially enabling the development of novel sensing modalities based on the specific system-environment interaction.
4. Accelerated Material Design and Discovery:

Realistic Open System Simulations: This method enables more efficient simulations of open quantum systems, crucial for understanding the role of the environment in material properties and chemical reactions.
Dissipative Quantum Simulations: The ability to simulate open system dynamics accurately paves the way for dissipative quantum simulations, which can provide insights into complex quantum phenomena relevant for material design and drug discovery.
In summary: This research, by providing a powerful tool for characterizing open quantum systems, can significantly impact the development of more robust and efficient quantum technologies. From improved error correction and control to enhanced sensor design and material discovery, the ability to efficiently extract and analyze dynamical maps holds immense potential for advancing the field of quantum technologies.