Krylov Complexity in Dual Quantum Systems: Exploring the Impact of Boundary Conditions on Operator Growth

Core Concepts

While dual quantum systems share similar spectra, the growth of operators, as measured by Krylov complexity, can differ significantly due to boundary effects and the non-local nature of dual operators.

Abstract

This research paper investigates the relationship between Krylov complexity and duality in quantum systems, focusing on the transverse field Ising model (TFIM) and its fermionic dual, the Kitaev chain.

Research Objective:
The study aims to understand how duality transformations, specifically the Jordan-Wigner transformation, affect the growth of operators in dual quantum systems, as measured by Krylov complexity.

Methodology:
The authors employ Krylov complexity as a tool to quantify operator growth in the TFIM and the Kitaev chain. They analyze the complexity of various operators, including single and multi-site fermionic operators, under both open and periodic boundary conditions.

Key Findings:

• In systems with open boundary conditions, the duality mapping reorganizes the space of operators, but each operator in one system has a dual counterpart with identical Krylov complexity in the other.
• The operator dynamics in the open boundary case are constrained by the quadratic nature of the Kitaev chain Hamiltonian, leading to relatively small Krylov complexity values.
• With periodic boundary conditions, the boundary term in the dual Hamiltonian allows operators that mix parity sectors to access a much larger space of operators, resulting in significantly higher Krylov complexity.
• The authors observe a dramatic increase in complexity for parity-mixing operators in periodic systems, particularly for single fermion operators, as the system size increases.

Main Conclusions:
The study demonstrates that while dual quantum systems share similar spectra, the growth of operators, as measured by Krylov complexity, can differ significantly due to boundary effects and the non-local nature of dual operators. The findings highlight the importance of considering both locality and boundary conditions when analyzing operator growth and complexity in dual quantum systems.

Significance:
This research provides valuable insights into the relationship between Krylov complexity, duality, and operator growth in quantum systems. It contributes to a deeper understanding of the dynamics of quantum systems and has implications for the study of integrable systems and quantum chaos.

Limitations and Future Research:
The study focuses on a specific duality transformation and a limited set of operators. Future research could explore these ideas in more complex dualities, higher-dimensional systems, and different operator types. Investigating the implications of these findings for quantum chaos and holography is another promising direction.

Stats

The complexity of a single fermionic operator in a periodic Ising chain with L = 5 is an order of magnitude larger than in the open boundary condition.
The largest Krylov subspace in the Kitaev chain has a dimension bounded by (2L)!/(L!)^2.

Quotes

Key Insights Distilled From

On Complexity and Duality

by Jeff Murugan... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02546.pdf

Deeper Inquiries

How do these findings concerning Krylov complexity and duality manifest in the context of quantum algorithms and their efficiency?

The findings of this paper, particularly the dependence of Krylov complexity on boundary conditions and operator locality in dual theories, could have significant implications for designing and understanding the efficiency of quantum algorithms. Here's how:

Resource Estimation: Krylov complexity provides a way to quantify the spread of quantum information and the difficulty of simulating a quantum system on a classical computer. The observation that non-local operators in one dual theory can exhibit complexity growth comparable to local operators in the other suggests that seemingly complex operations in one representation might be efficiently implementable in the dual representation. This could lead to more efficient quantum algorithms by choosing representations where the desired operations have lower complexity.
Algorithm Design: The duality between the Ising model and the Kitaev chain, connected by the Jordan-Wigner transformation, is a prime example. Algorithms relying on simulating fermionic systems (like quantum chemistry simulations) could potentially be mapped to equivalent spin systems where certain operations might be easier to implement. The insights from Krylov complexity could guide this mapping and the choice of optimal representations for specific quantum algorithms.
Understanding Algorithm Complexity: Krylov complexity can be used to analyze the complexity of existing quantum algorithms. By mapping algorithms to different representations and studying the Krylov complexity of the involved operators, we can gain insights into the resources required for their execution. This understanding can lead to optimizations or identify bottlenecks in current algorithms, paving the way for more efficient implementations.
However, it's important to note:

Overhead of Mapping: While duality mappings offer potential advantages, the mapping itself introduces overhead. The Jordan-Wigner transformation, for instance, involves strings of operators that grow with system size. This overhead needs careful consideration when evaluating the overall efficiency gain from exploiting duality.
Beyond Integrable Systems: The specific examples in the paper focus on integrable models. Extending these insights to more general quantum algorithms dealing with non-integrable systems, where efficient classical simulation is often impossible, requires further investigation.

Could the observed dependence of Krylov complexity on boundary conditions be exploited to engineer quantum systems with specific dynamical properties?

The paper's findings suggest that boundary conditions can dramatically influence the Krylov complexity of operators, especially those that mix parity sectors. This sensitivity to boundary conditions opens up intriguing possibilities for engineering quantum systems with tailored dynamical properties:

Controlling Information Spreading: By carefully choosing boundary conditions, one could potentially control the rate and extent of quantum information scrambling within a system. For instance, periodic boundary conditions could be used to enhance the spread of information, while open boundary conditions might restrict it to specific regions. This control could be valuable in quantum information processing tasks like error correction and quantum communication.
Engineering Effective Hamiltonians: Boundary conditions can be viewed as modifying the effective Hamiltonian of the system. By manipulating these conditions, one could engineer Hamiltonians with desired properties, such as specific energy spectra or entanglement structures. This could be particularly relevant in condensed matter physics, where simulating materials with exotic properties is a significant challenge.
Topological Protection: The connection between boundary conditions and topological order, as highlighted in the context of the Kitaev chain, suggests that manipulating boundaries could be a route to engineer systems with robust topological properties. These properties are of great interest for fault-tolerant quantum computation.
However, practical challenges exist:

Experimental Realization: Implementing and dynamically controlling boundary conditions in real-world quantum systems can be experimentally demanding.
Scalability: The effects of boundary conditions might become less pronounced in larger systems, where bulk properties dominate.

What are the implications of these findings for our understanding of the relationship between quantum information scrambling and spacetime geometry in the context of holography?

The paper's exploration of Krylov complexity in dual theories, particularly the role of non-locality and boundary conditions, could offer intriguing hints for the holographic duality between quantum information and spacetime geometry:

Complexity and Spatial Geometry: The observation that non-local operators in one dual theory can exhibit complexity growth similar to local operators in the other suggests a nuanced relationship between complexity and spatial locality. In the holographic context, this might imply that complexity, often associated with the growth of wormholes in the bulk spacetime, is not solely determined by local properties on the boundary theory. Non-local correlations could play a crucial role.
Boundary Conditions and Bulk Dynamics: The sensitivity of Krylov complexity to boundary conditions hints at a potential connection between boundary modifications and the dynamics of the holographic bulk. Changing boundary conditions in the field theory could correspond to altering the geometry or introducing new degrees of freedom in the bulk. This aligns with the general holographic principle, where boundary data encodes information about the bulk.
Operator Growth and Black Hole Physics: The study of operator growth, as quantified by Krylov complexity, is closely related to information scrambling, a key feature of black holes. Understanding how operator growth behaves in dual theories, especially under different boundary conditions, could provide insights into the scrambling process and information paradox in black hole physics.
However, it's crucial to acknowledge:

Toy Models vs. Holography: The paper focuses on relatively simple, integrable models. Extrapolating these findings to the complexities of holographic duality, which typically involves strongly coupled quantum field theories and quantum gravity, requires significant caution.
Beyond Krylov Complexity: While Krylov complexity is a valuable tool, it's just one measure of complexity. Exploring other complexity measures in the context of holography and investigating their relationship with spacetime geometry is crucial for a more complete understanding.