arXiv subjects

Hendrik J R Van Zyl

Publications and source records attributed to Hendrik J R Van Zyl.

4 recordsLinked to original sources

arXiv2026

Krylov Complexity and Entropy of Composite Quantum Systems

We investigate the behaviour of Krylov complexity as the number of degrees of freedom of a quantum system increases. Specifically, we show that for tensor products of commuting subsystems, the leading excess complexity is controlled by a weighted variance of the subsystem Lanczos coefficients. This provides a quantitative measure of dynamical heterogeneity and a precise characterization of synchronous evolution. We also show that Krylov entropy is able to disambiguate the information associated with subsystems composition from that generated by dynamical mismatch, and that the same Lanczos variance controls the leading mismatch contribution to both complexity and the entropy. Our results establish excess complexity and Krylov entropy as complementary probes of the emergence of collective dynamics in large quantum systems.

hep-th↗

arXiv2026

A Quantum Computational Perspective on Spread Complexity

We establish a direct connection between spread complexity and quantum circuit complexity by demonstrating that spread complexity emerges as a limiting case of a circuit complexity framework built from two fundamental operations: time-evolution and superposition. Our approach leverages a computational setup where unitary gates and beam-splitting operations generate target states, with the minimal cost of synthesis yielding a complexity measure that converges to spread complexity in the infinitesimal time-evolution limit. This perspective not only provides a physical interpretation of spread complexity but also offers computational advantages, particularly in scenarios where traditional methods like the Lanczos algorithm fail. We illustrate our framework with an explicit SU(2) example and discuss broader applications, including cases where return amplitudes are non-perturbative or divergent

hep-th↗

arXiv2026

Krylov complexity, path integrals, and instantons

Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity $K(t)$ for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for $K(t)$, the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.

hep-th↗

arXiv2023

Complexity and Operator Growth for Quantum Systems in Dynamic Equilibrium

Krylov complexity is a measure of operator growth in quantum systems, based on the number of orthogonal basis vectors needed to approximate the time evolution of an operator. In this paper, we study the Krylov complexity of a $\mathsf{PT}$-symmetric system of oscillators, which exhibits two phase transitions that separate a dissipative state, a Rabi-oscillation state, and an ultra-strongly coupled regime. We use a generalization of the $su(1,1)$ algebra associated to the Bateman oscillator to describe the Hamiltonian of the coupled system, and construct a set of coherent states associated with this algebra. We compute the Krylov (spread) complexity using these coherent states, and find that it can distinguish between the $\mathsf{PT}$-symmetric and $\mathsf{PT}$ symmetry-broken phases. We also show that the Krylov complexity reveals the ill-defined nature of the vacuum of the Bateman oscillator, which is a special case of our system. Our results demonstrate the utility of Krylov complexity as a tool to probe the properties and transitions of $\mathsf{PT}$-symmetric systems.

hep-th↗