Understanding dynamics and quantum chaos through Krylov space

In everyday life, we often associate chaos with randomness, disorder, or unpredictability—phenomena that appear to lack any discernible pattern. However, from a physics standpoint, understanding chaos requires a more rigorous and precise mathematical framework. In classical physics, chaos is often perceived through its sensitive dependence on initial conditions. Small perturbations in an initial state of a system can lead to vastly different outcomes over time, a behavior commonly known as “butterfly effect”, and typically analyzed within the framework of phase space trajectories, and its detailed topological properties.

In contrast, chaos in the quantum realm presents unique challenges. The notion of sharp trajectories in phase space clashes with Heisenberg’s uncertainty principle in quantum mechanics, and initial perturbations cannot be treated in a way that mirrors classical intuition. As a result, quantum chaos requires distinct formulations, relying on diagnostic tools like spectral statistics, out-of-time-order correlators, and entanglement measures. However, the relationships between these different probes are not always clear, and a unified understanding remains an open area of research.

In recent years, significant progress has been made in understanding quantum chaos through the lens of operator growth, where localized quantum information encoded in simple operators spreads across a system—a process known as information scrambling. Such phenomena is crucial in understanding the thermalization of a system. Krylov space, a subspace of the operator Hilbert space, provides an elegant framework to describe such operator growth. By decomposing operator dynamics using an orthonormal basis, it traces how simple operators evolve into increasingly complex ones—quantified by a complexity measure in Krylov space. A parallel formulation exists for the evolution of quantum states, offering a complementary perspective.

Importantly, to make these ideas applicable to realistic physical scenarios, one must consider open quantum systems—systems that interact with their environment. In such contexts, the dynamics become richer, requiring more generic theoretical and computational techniques. Furthermore, efficient quantum control protocols often leverage the structured Hilbert space, with Krylov subspace methods providing computationally efficient frameworks. These methods facilitate guiding systems along desired adiabatic trajectories, reducing runtime and mitigating decoherence effects, as often required for quantum technologies. These developments form the core focus of this review article.