Información - Quantum Computing - # Krylov Complexity in Quantum Many-Body Systems

Complexity Enriched Dynamical Phases for Fermions on Regular Graphs: An Analysis of Entanglement and Krylov Complexity

Q: Could the observed differences in Krylov complexity be attributed to the specific choice of initial states and operators, rather than representing a fundamental distinction between the systems?

Answer: While the choice of initial states and operators can influence the transient behavior of Krylov complexity, the fundamental distinctions observed in the context stem primarily from the graph structure itself. Here's why: Long-Time Behavior: Krylov complexity, as a measure of operator growth, is often most informative in its long-time behavior. In the provided examples, the scaling laws for Krylov complexity are derived from its behavior as time, and potentially system size, tends to infinity. This long-time limit often washes out the specific details of the initial state or operator. Operator Spreading as the Dominant Factor: The core mechanism driving the observed differences is the way connectivity affects operator spreading. A highly connected graph allows for rapid information propagation regardless of the initial operator's location. While the initial choice might influence how quickly the operator spreads initially, the overall reach of the operator, and hence the Krylov complexity, is ultimately constrained by the graph's structure. Analogy to Classical Diffusion: Consider the analogy of classical diffusion on a graph. The initial distribution of a substance might influence its early spread, but the long-time diffusion behavior is dictated by the graph's connectivity. Similarly, Krylov complexity captures the "diffusion" of operator information, with connectivity playing the defining role. Exceptions and Further Considerations: Localized Systems: In systems exhibiting many-body localization, where information propagation is strongly suppressed, the choice of initial state might have a more pronounced and lasting effect on Krylov complexity. Specific Operator Choices: While generic local operators are likely to reveal the fundamental distinctions arising from graph structure, carefully chosen operators with support tailored to exploit specific graph features could potentially obscure these differences.

Conceptos Básicos

While entanglement entropy fails to differentiate between dynamical phases in fermionic systems on regular graphs with varying degrees, Krylov complexity provides a more sensitive measure, revealing distinct complexity phases.

Resumen

Bibliographic Information: Xia, W., Zou, J., & Li, X. (2024). Complexity enriched dynamical phases for fermions on graphs. arXiv preprint arXiv:2404.08055v2.
Research Objective: This study investigates whether Krylov complexity can distinguish dynamical phases in fermionic systems on regular graphs, particularly when entanglement entropy measures fail to do so.
Methodology: The researchers numerically analyze the entanglement entropy and Krylov complexity of free and interacting fermion models on regular graphs with degrees d=2 and d=3. They also develop an analytical theory to calculate the Krylov dimension, which serves as a proxy for Krylov complexity.
Key Findings:
- Entanglement entropy exhibits volume law scaling (S~N) for both d=2 and d=3, irrespective of the presence of interactions, suggesting a single dynamical phase.
- Krylov complexity, however, shows distinct scaling behaviors for d=2 and d=3 in both free and interacting cases, indicating different complexity phases.
- The developed analytical theory for Krylov dimension accurately predicts the scaling behavior observed in numerical simulations.
- Out-of-time-order correlators (OTOCs) are proposed as an experimentally accessible measure to probe these distinct complexity phases.
Main Conclusions: Krylov complexity acts as a more sensitive diagnostic tool compared to entanglement entropy for characterizing dynamical phases in quantum many-body systems, particularly in the context of regular graphs.
Significance: This research highlights the importance of exploring complexity measures beyond entanglement to gain a deeper understanding of quantum many-body dynamics and phase transitions.
Limitations and Future Research: The numerical simulations for interacting fermions are limited to small system sizes due to computational constraints. Further investigations into the potential phase transitions between dynamical phases with different complexities are warranted.

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Estadísticas

Entanglement entropy scales as S ~ N for both d = 2 and d = 3 regular graphs.
For free fermions, the Krylov dimension scales as D ~ N for d = 2 and D ~ N^2 for d = 3.
For interacting fermions, the Krylov dimension scales as D ~ 4^(N^α) for d = 2 (0.38 ≤ α ≤ 0.59) and D ~ 4^N for d = 3.
In the interacting case with d = 3, the Lyapunov exponent extracted from OTOC is λL = 0.31(2).

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Ideas clave extraídas de

Complexity enriched dynamical phases for fermions on graphs

by Wei Xia, Jie... a las arxiv.org 10-15-2024

https://arxiv.org/pdf/2404.08055.pdf

Complexity enriched dynamical phases for fermions on graphs

Consultas más profundas

How does the connectivity of graphs, beyond regular graphs, influence the Krylov complexity and the emergence of distinct dynamical phases?

Answer:
The connectivity of graphs plays a crucial role in shaping the Krylov complexity and the emergence of distinct dynamical phases. While the provided context focuses on regular graphs, extending the analysis to more general graph structures reveals a richer landscape.

Connectivity and Operator Spreading:  The key lies in how connectivity influences the spreading of operators under Heisenberg evolution. In highly connected graphs, an initial local operator can quickly spread its information across the entire system. This rapid information propagation leads to a larger Krylov space, as more basis operators are needed to capture the evolving operator's complexity. Conversely, in sparsely connected graphs, operator spreading is hindered, resulting in smaller Krylov spaces and slower complexity growth.

Examples Beyond Regular Graphs:

Trees:  Tree graphs, with their hierarchical structure and no loops, would exhibit limited operator spreading. This suggests a slower Krylov complexity growth compared to more connected graphs.
Random Graphs: Random graphs, with their statistical distribution of connections, can display a range of Krylov complexity behaviors depending on their connection probability.  Highly connected random graphs would resemble regular graphs with high degree, while sparsely connected ones would exhibit slower complexity growth.
Small-World Networks: These networks, characterized by high clustering and short path lengths, would likely exhibit rapid operator spreading and potentially faster Krylov complexity growth due to their efficient information flow.

Dynamical Phases:  Different graph structures, with their varying connectivity patterns, can lead to the emergence of distinct dynamical phases characterized by their Krylov complexity scaling.  For instance, a phase transition might be observed by tuning the connectivity of a random graph, transitioning from a phase of slow complexity growth to one of rapid growth.

Experimental Implications: Exploring Krylov complexity in diverse graph structures could be achieved in systems like Rydberg atom arrays, where connectivity can be engineered. This opens avenues for probing the interplay between graph topology and quantum dynamics.

Could the observed differences in Krylov complexity be attributed to the specific choice of initial states and operators, rather than representing a fundamental distinction between the systems?

Answer:
While the choice of initial states and operators can influence the transient behavior of Krylov complexity, the fundamental distinctions observed in the context stem primarily from the graph structure itself. Here's why:

Long-Time Behavior: Krylov complexity, as a measure of operator growth, is often most informative in its long-time behavior. In the provided examples, the scaling laws for Krylov complexity are derived from its behavior as time, and potentially system size, tends to infinity. This long-time limit often washes out the specific details of the initial state or operator.

Operator Spreading as the Dominant Factor: The core mechanism driving the observed differences is the way connectivity affects operator spreading.  A highly connected graph allows for rapid information propagation regardless of the initial operator's location.  While the initial choice might influence how quickly the operator spreads initially, the overall reach of the operator, and hence the Krylov complexity, is ultimately constrained by the graph's structure.

Analogy to Classical Diffusion: Consider the analogy of classical diffusion on a graph. The initial distribution of a substance might influence its early spread, but the long-time diffusion behavior is dictated by the graph's connectivity. Similarly, Krylov complexity captures the "diffusion" of operator information, with connectivity playing the defining role.

Exceptions and Further Considerations:

Localized Systems: In systems exhibiting many-body localization, where information propagation is strongly suppressed, the choice of initial state might have a more pronounced and lasting effect on Krylov complexity.
Specific Operator Choices:  While generic local operators are likely to reveal the fundamental distinctions arising from graph structure, carefully chosen operators with support tailored to exploit specific graph features could potentially obscure these differences.

If we consider Krylov complexity as a measure of information scrambling, what are the implications of different scrambling behaviors on the potential for quantum information processing in these systems?

Answer:
Viewing Krylov complexity as a measure of information scrambling provides valuable insights into the capabilities and limitations of different graph structures for quantum information processing tasks:

Fast Scrambling and Quantum Advantage: Systems exhibiting rapid Krylov complexity growth, indicative of fast scrambling, are generally desirable for certain quantum algorithms.

Quantum Simulation:  Fast scrambling can enable efficient simulation of complex quantum systems, as it allows for the rapid spread and processing of quantum information.
Quantum Chaos and Error Correction:  The chaotic dynamics associated with fast scrambling can be harnessed for error correction, where errors are quickly dispersed throughout the system, making them easier to correct.

Slow Scrambling and Information Protection: Systems with slow Krylov complexity growth, implying slow scrambling, can be advantageous for tasks requiring information protection and storage.

Quantum Memory: The slow scrambling ensures that quantum information remains localized and protected from decoherence for longer periods.
Adiabatic Quantum Computing:  In adiabatic quantum computing, slow and controlled evolution is crucial, and systems with slow scrambling can provide a more stable platform.

Graph Engineering for Tailored Scrambling: The ability to engineer connectivity in systems like Rydberg atom arrays offers exciting possibilities for tailoring scrambling behavior to specific quantum information processing needs.

Dynamically Tunable Scrambling:  By dynamically changing the graph structure, one could potentially switch between fast and slow scrambling regimes, enabling hybrid quantum processing architectures.

Complexity and Resource Requirements: It's important to note that faster scrambling, while potentially beneficial, often comes at the cost of increased complexity and resource requirements for control and manipulation.

Beyond Krylov Complexity: While Krylov complexity provides a valuable lens for understanding scrambling, a comprehensive assessment of a system's quantum information processing capabilities would require considering other factors like decoherence times, controllability, and the specific requirements of the desired quantum algorithms.