Persistence of Fractal Structures in Iterated Quantum Protocols: Robustness to Preparation and Gate Errors

Abstract

Repeatedly combining entangling gates on an ensemble of qubits with measurement and postselection on a part of them induces a nonlinear evolution of the unmeasured qubits' state, whose fractal convergence boundaries make the outcome sensitive to the initial state. We study a family of iterated protocols in which $n$ identical qubits are coupled by CNOT gates, $n-1$ of them are postselected, and a parametrized single-qubit unitary acts on the survivor. For orders $n=2$ to $5$, we map the box-counting dimension of the pure-state Julia sets over the complex parameter plane and identify the symmetries that constrain it. Allowing mixed initial states, we find that moderate preparation noise preserves the main parameter-dependent trends of the boundary dimension over much of the studied range, and that the three-dimensional quasi-Julia boundary in the Bloch ball has a dimension that is approximately equal to the dimension of the pure-state Julia set plus one. We further model gate imperfections by an effective depolarizing channel and by fixed coherent interactions. Weak errors leave the estimated dimensions consistent with their ideal values even where the convergence regions visibly deform, and $ZZ$ errors are shown to be exactly conjugate to the ideal protocol at a rotated parameter. Finally, we analyze the postselection and storage cost of an implementation and propose a diagnostic test combining conditional fidelity with acceptance statistics, which we demonstrate on IQM Emerald and Rigetti Cepheus-1-108Q.