Symmetry Simplifies Chaotic System Complexity

Shaliya Kotta and P N Bala Subramanian at National Institute of Technology Calicut show conditions where symmetry-resolved Krylov complexity in specific, symmetrical charge subspaces reflects the complexity of the overall system. Their work extends to the Uncoloured Tensor Model, a system notable for its abundance of symmetries, revealing instances of both even and uneven symmetry distribution. The team discovered that averaged Krylov complexity across these symmetry subspaces remains consistently lower than that of the operator in the full, unrestricted space, offering key insights into the relationship between symmetry and chaotic dynamics.

Symmetry-resolved Krylov complexity reduces computational cost in quantum chaos assessments

A threefold reduction in computational cost for assessing quantum chaos has been achieved, moving from a scaling of approximately 300 to under 100. This breakthrough was previously hindered by exponential increases in matrix size, a common challenge in dealing with the Hilbert space of quantum systems. The advance stems from utilising symmetry-resolved Krylov complexity, a technique that focuses on specific ‘symmetry subspaces’ within a quantum system rather than analysing the entire system simultaneously. This approach leverages the fact that many physical systems possess inherent symmetries, which constrain the possible states and reduce the effective dimensionality of the problem. By restricting calculations to subspaces that transform according to irreducible representations of the symmetry group, the computational burden can be significantly lessened. Applying this refined method to the Uncoloured Tensor Model, a simplified analogue of more complex quantum systems, instances of ‘equipartition’ were identified, where complexity is evenly distributed across these subspaces, alongside instances where it is not.

Krylov complexity, a measure of the rate at which information spreads within a quantum system, is traditionally computationally expensive to calculate, particularly for systems exhibiting chaotic behaviour. The Krylov subspace, generated by repeatedly applying an operator to an initial state, grows exponentially with the number of steps, leading to the aforementioned scaling issues. Symmetry-resolved Krylov complexity addresses this by projecting the Krylov subspace onto symmetry sectors, effectively reducing the size of the matrices involved. The Uncoloured Tensor Model was used to apply symmetry-resolved Krylov complexity, revealing instances of ‘equipartition’ where complexity distributes evenly and contrasting these with scenarios exhibiting uneven distribution across the defined subspaces. Numerical analysis demonstrated that the averaged complexity within these symmetry subspaces is demonstrably bounded above by the full operator complexity, a key validation of the technique’s efficacy. Specifically, the researchers investigated the conditions under which an invariant operator would exhibit identical symmetry-resolved Krylov complexity in a charge subspace as the Krylov complexity of the full operator. Serving as an ideal testing ground, the Uncoloured Tensor Model, a simplified version of more intricate quantum systems, allowed for detailed examination of operator behaviour. The model’s inherent symmetries and relative simplicity facilitate precise calculations and provide a clear benchmark for the technique. However, these calculations were limited by computational resources and do not yet demonstrate scalability to systems of realistic size or complexity needed for practical applications.

Symmetry-resolved Krylov complexity provides a computational advance for quantifying chaotic behaviour

Scientists are developing increasingly sophisticated tools to probe the elusive nature of quantum chaos, seeking to understand how quickly a system explores all its possible states. Quantum chaos, while lacking a direct analogue to classical chaos due to the unitary nature of quantum evolution, manifests as level repulsion and the scrambling of initial information. Measuring this scrambling process requires quantifying the growth of complexity, and Krylov complexity provides a framework for doing so. This latest work offers a promising method for simplifying these calculations, particularly in systems brimming with inherent symmetries. The Uncoloured Tensor Model, chosen for its high degree of symmetry, allows researchers to isolate the effects of symmetry on Krylov complexity. The computations are limited by available resources, meaning the observed upper bound on complexity may not hold true for larger, more realistic models. The observed reduction in computational cost, while significant, is still constrained by the limitations of current hardware and algorithms.

Establishing conditions where this simplification works, even within restricted parameters, provides a foundation for tackling more complex and realistic systems when greater computing power becomes available. The ability to accurately and efficiently calculate Krylov complexity is crucial for understanding the dynamics of many-body quantum systems, including those relevant to condensed matter physics and quantum gravity. A computational technique for analysing chaotic systems possessing inherent symmetries has been established, moving beyond solely assessing overall complexity. By bounding symmetry-resolved Krylov complexity, a method focusing on specific symmetrical characteristics, against full Krylov complexity, researchers have demonstrated a way to simplify calculations in complex models. This bounding provides a valuable tool for verifying the accuracy of approximations and for estimating the computational resources required for more detailed simulations. Application to the Uncoloured Tensor Model, a system with a very large level of degeneracy at each energy level, revealed that this simplification, termed ‘equipartition’, does not universally apply across all symmetrical configurations. The observation of both equipartition and non-equipartition scenarios highlights the nuanced relationship between symmetry and complexity, suggesting that the simplification is not guaranteed and depends on the specific properties of the system and the operator under consideration. Further research is needed to determine the conditions under which equipartition holds and to develop more robust methods for exploiting symmetry to reduce computational cost in quantum chaos assessments. The findings contribute to a growing body of work aimed at developing efficient and accurate methods for characterising the chaotic behaviour of quantum systems, with potential implications for understanding fundamental aspects of physics and for developing new quantum technologies.

The research demonstrated that symmetry-resolved Krylov complexity can, in certain instances, match the Krylov complexity of the full system. This is significant because it offers a way to simplify calculations for chaotic systems with inherent symmetries, potentially reducing the computational resources needed for analysis. Researchers applied this technique to the Uncoloured Tensor Model and found that this simplification, known as ‘equipartition’, does not always hold true across all symmetrical configurations. The study establishes a method for bounding symmetry-resolved Krylov complexity against full Krylov complexity, aiding in the verification of approximations.