Chaos, Fractals, and Ergodicity in Nonlinear Quantum Protocols

Abstract

Iterated nonlinear quantum protocols provide a straightforward way of realizing nonlinear dynamics in quantum systems through measurement and postselection. We investigate a family of protocols in which the order of the induced nonlinearity is controlled by the number of qubits involved in a single step. By examining the resulting rational maps and applying a set of numerical diagnostics, we identify extended parameter regions that exhibit ergodic-like dynamics for protocol orders $n=3,4,$ and $5$. We further characterize the associated chaotic behavior through the distribution of Lyapunov exponents. Our results show that ergodic-like dynamics persists beyond the quadratic case, while the geometry of the corresponding parameter regions and the typical degree of chaoticity depend strongly on the order of the nonlinear transformation.

The paper asks whether globally chaotic, ergodic-like dynamics previously observed in quadratic postselected quantum protocols persists at higher nonlinear orders. Each step uses $n$ identically prepared qubits, a CNOT network, postselection, and a single-qubit unitary, inducing the pure-state rational map $f_p(z)=(z^n-\bar p)/(1+pz^n)$. The study numerically examines $n=3,4,5$ and compares them with $n=2$. Parameters are classified as ergodic-like only when four diagnostics agree: no attracting cycles detected from either critical orbit, stable long-time visitation distributions, agreement between trajectory and evolved-ensemble distributions, and positive Lyapunov exponents approximately independent of initial state. Tests use trajectories of up to $10^7$ iterations, cycle searches through period $500$, and parameter grids up to $2000\times2000$. Connected ergodic-like regions are reported for every order. Within $\operatorname{Re}p,\operatorname{Im}p\in[0,3]$, their estimated area fractions are $8.7\%$, $35\%$, $5.2\%$, and $13.3\%$ for $n=2,3,4,5$, respectively. The geometry of these regions and distributions of normalized spherical Lyapunov exponents, $2\lambda/\ln n$, vary substantially with order. Quadratic exponents mostly lie around $0.7$–$0.95$; higher orders generally show smaller, more heterogeneous values. Unlike the quadratic case, $p=i$ is not a Lattès point at higher orders and fails the ergodic-like tests for odd orders. The conclusions are numerical signatures rather than rigorous proofs of ergodicity. The analysis assumes ideal operations and pure states, excluding preparation noise and implementation imperfections.