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.

The paper asks whether fractal convergence boundaries in iterated, postselected qubit protocols survive preparation and gate errors. Each round combines $n$ identical qubits using CNOTs, postselects $n-1$ measurement outcomes, and applies a parametrized single-qubit unitary. Pure-state evolution follows the rational map $f_p(z)=(z^n-\bar p)/(1+pz^n)$; mixed states evolve nonlinearly within the Bloch ball. For $n=2$–$5$, numerical box-counting surveys map Julia-set dimensions across the complex parameter plane using backward iteration, with $10^7$ samples per parameter. Forward basin classification finds that initial purities $P=0.99$ and $0.98$ largely preserve the pure-state dimension profiles where boundaries remain resolved. The three-dimensional boundary between pure-cycle phases has estimated dimension approximately equal to the pure-state Julia-set dimension plus one. Weak coherent errors at $\varepsilon=0.01$ and global depolarization at $\eta=10^{-3}$ generally leave estimated dimensions close to ideal values. An exact conjugacy shows that the modeled $ZZ$ error is equivalent to ideal dynamics at $p e^{-2i\varepsilon}$. A separate hardware demonstration on IQM Emerald and Rigetti Cepheus-1-108Q tests fixed reference states through depths $k=1$–$5$, combining conditional fidelity with acceptance statistics rather than measuring fractal geometry. Postselection becomes restrictive: one reference has ideal depth-five acceptance $3.4\times10^{-8}$. The dimension results are finite-resolution estimates; regression errors exclude sampling and fitting-range systematics. Unresolved convergence can hide boundaries, and backward sampling can underestimate dimensions. The dimension-plus-one relation is numerical, not a theorem. Gate-noise models are effective rather than device-calibrated, and the hardware tests do not isolate error sources.