Geometry of Krylov Complexity (2109.03824v2)

Abstract: We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues towards computation of Lanczos coefficients and sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy and capacity of entanglement.

• The paper introduces a geometric framework linking operator growth to classical phase space motion via generalized coherent states.
• The paper demystifies Lanczos coefficients by showing their role in driving exponential Krylov complexity through symmetry operations.
• The paper applies its model to systems like SYK, SU(2), and 2D CFTs, highlighting potential links to chaos theory and holographic complexity.

Insights on "Geometry of Krylov Complexity"

The paper "Geometry of Krylov Complexity" by Pawel Caputa, Javier M. Magan, and Dimitrios Patramanis presents an innovative approach to understanding operator growth and Krylov complexity in quantum many-body systems. Situated within the broader context of quantum chaos and complexity, this work provides a geometric perspective by linking the dynamics of operators to trajectories in a classical phase space. Below, we summarize and critically analyze the key contributions and implications of the paper.

Geometric Approach to Operator Growth

A central contribution of the paper is the mapping of operator growth to a classical motion in phase space, facilitated by symmetries inherent in quantum systems. This geometric representation posits that the evolution of operators, under a specified Liouvillian, corresponds to the motion of a classical particle in the geometry of generalized coherent states. By leveraging this connection, the paper addresses operator growth through the lens of information geometry, where Krylov complexity—the measure of "spread" of an initially simple operator over time—relates to the volume spanned by the corresponding geodesic paths in this space.

Lanczos Coefficients and Krylov Complexity

The paper effectively demystifies the computational framework surrounding Lanczos coefficients, integral to Krylov subspace methods. These coefficients are revealed to have a direct relationship with the action of ladder operators associated with symmetry groups, such as SU(2) and SL(2,R). The geometric insights provided allow these coefficients to be extracted more intuitively than through direct computation, thereby offering a universal approach to determining the rate of exponential complexity growth, characterized by a Lyapunov-like exponent. The connection with generalized coherent states offers not only conceptual clarity but also computational tractability when analyzing systems with varying degrees of complexity.

Case Studies and Generalization

Through detailed paradigms like the Sachdev-Ye-Kitaev (SYK) model, SU(2), Heisenberg-Weyl, and two-dimensional conformal field theories (CFTs), the authors illustrate the robustness and applicability of their geometric framework. For instance, the SL(2,R) example maps operator growth to hyperbolic geodesics, where the Krylov complexity asymptotically grows with the hyperbolic volume, linking to theoretical constructs such as holographic complexity.

Particularly notable is the treatment of composite operator growth in two-dimensional CFTs, extending the Krylov complexity framework beyond single operator dynamics and hinting at potential connections between chaos theory and out-of-time-order correlators. Such generalizations underscore the theoretical richness of Krylov complexity across a variety of symmetric and holographic systems.

Implications and Future Directions

The theoretical contributions of the paper have broad implications for both fundamental research and practical computations in quantum physics. The insights gained from the geometric interpretation of Krylov complexity enrich our understanding of quantum chaos through a novel dimensional facet—linking complexity growth with the intrinsic curvature of the information manifold. This provides a physically motivated bridge to the burgeoning field of quantum computational complexity, particularly in the context of emerging holography and quantum information paradigms.

Future work might explore applications of these geometric principles to non-integrable and truly chaotic systems, providing a clearer picture of how such symmetries might generalize or adapt in less structured quantum environments. The exploration of coherent states in various gauge theories or the incorporation of stringy corrections in holographic models could offer rich avenues for extending these foundational constructs.

Additionally, the experimental realization of these theoretical constructs, particularly through setups inspired by quantum optics, remains a promising frontier that could validate and further elucidate the geometric properties underlying Krylov complexity.

In conclusion, "Geometry of Krylov Complexity" contributes significantly to the understanding of quantum operator dynamics by bridging elegant geometric concepts with rigorous computational methods, promising new insights in theoretical physics and beyond.