🌀 nonlinear sciences

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

This paper investigates the robustness of fractal convergence boundaries in iterated quantum protocols involving CNOT gates and postselection, demonstrating that their structural dimensions remain stable under moderate preparation noise and weak gate errors while proposing a practical diagnostic test validated on real quantum hardware.

Original author: Attila Portik, Orsolya Kálmán, Tamás Kiss

Published 2026-11-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Attila Portik, Orsolya Kálmán, Tamás Kiss

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the rules of motion are usually strict and predictable. If you know the starting position of a particle, the laws of physics tell you exactly where it will go next, and if you run the process backward, you end up exactly where you started. This linear behavior is the bedrock of standard quantum mechanics. However, scientists have long been interested in what happens when they break this linearity by adding a specific kind of filter. Imagine a process where you take several quantum particles, mix them together, and then throw away every single attempt that does not produce a specific, rare result. By keeping only the successful outcomes and repeating this cycle over and over, the system stops behaving like a simple machine and starts acting like a complex, chaotic one. This creates a situation where tiny differences in how you start the process can lead to completely different endings, a phenomenon known as sensitive dependence on initial conditions.

This sensitivity often creates intricate, self-repeating patterns called fractals. In mathematics, these are shapes that look the same no matter how much you zoom in, with boundaries that are so jagged and complex they defy simple measurement. For years, researchers have wondered if these delicate fractal structures could survive the messy reality of actual quantum computers, where equipment is never perfect and particles are never perfectly isolated. If the noise of the real world smears out these patterns, the unique behaviors they enable might be lost. But if the patterns are robust, they could offer a new way to test how well quantum devices are working, turning the very complexity of the system into a diagnostic tool.

A team of researchers at the Wigner Research Centre for Physics in Budapest has now explored this question by running a series of repeated quantum experiments on a theoretical model and then testing the ideas on real hardware. They focused on a protocol where multiple identical quantum bits, or qubits, are linked together. In each round, they entangle these qubits, measure most of them, and keep only the rare instances where the measurements match a specific pattern. The one surviving qubit is then given a gentle push by a controlled operation before the whole process starts again. By repeating this cycle, they created a system that evolves in a nonlinear way, generating the complex fractal boundaries they wanted to study.

The researchers first mapped out the geometry of these boundaries for systems involving two to five qubits. They found that the edges between different possible outcomes are indeed fractal, forming a jagged landscape in the space of possible starting states. They measured the "roughness" of these edges, a property called the box-counting dimension, which tells you how much space the pattern fills. In the perfect, noise-free world of their calculations, these dimensions varied depending on the settings of the final push, creating a rich map of complexity. The team then asked a critical question: what happens when you introduce the inevitable errors of the real world?

They simulated two main types of errors. The first was preparation noise, where the qubits start out slightly mixed or imperfect, rather than in a pure, ideal state. The second was gate noise, where the operations used to link the qubits are slightly inaccurate. Surprisingly, they found that moderate levels of these errors did not destroy the fractal nature of the system. Even when the boundaries of the convergence regions visibly warped and shifted, the estimated roughness of the edges remained remarkably close to the ideal values. The fractal structure proved to be surprisingly resilient, persisting through a wide range of imperfections.

One of the most striking findings concerned the difference between perfect and imperfect starting states. When the qubits were mixed, the fractal boundary was no longer a flat, two-dimensional surface but a three-dimensional structure filling the space of possible states. The researchers discovered that the dimension of this three-dimensional boundary was consistently about one unit higher than the dimension of the corresponding two-dimensional boundary for perfect states. This suggests that the fractal geometry simply extends into the third dimension when the system is allowed to be imperfect, rather than collapsing or disappearing.

To prove that this could be done in practice, the team implemented a version of this protocol on two different quantum computers available through the cloud: the IQM Emerald and the Rigetti Cepheus-1-108Q. They chose specific starting points that were known to be stable or unstable in the ideal theory and ran the experiment multiple times. They developed a new way to judge the quality of the device by looking at two things at once: how often the experiment succeeded in passing the strict filters, and how close the final result was to the expected state. They combined these into a single score. As they increased the depth of the experiment, making the process more complex, the score naturally dropped, reflecting the accumulation of errors.

The results showed that while the devices could perform the task, the strict filtering required to see the fractal behavior made the process very difficult to complete many times. The number of successful runs dropped sharply as the experiment grew larger, a limitation imposed by the need for every single step to succeed. However, the data confirmed that the diagnostic method works: by watching how the success rate and the final state quality changed together, they could get a clear picture of the device's performance. The study concludes that while the delicate fractal patterns are sensitive to the environment, their fundamental geometric properties are robust enough to survive moderate noise, offering a promising new avenue for understanding and testing the complex dynamics of quantum systems.

Technical Summary: Persistence of Fractal Structures in Iterated Quantum Protocols

Problem Statement
Iterated nonlinear quantum protocols, which combine entangling gates with measurement and postselection, induce effective nonlinear dynamics on qubit states. For pure states, these dynamics are described by rational maps on the complex plane, where the boundaries between different asymptotic convergence regions form fractal structures known as Julia sets. While the sensitivity of these fractal boundaries to initial conditions is a hallmark of such systems, it remains unclear how robust these geometric structures are against the inevitable imperfections of experimental quantum hardware. Specifically, the persistence of fractal dimensions under preparation noise (mixed initial states) and gate errors (stochastic and coherent) has not been systematically characterized across a broad family of protocols.

Methodology
The authors investigate a family of iterated protocols involving $n$ identical qubits ( $n=2, 3, 4, 5$ ). In each round, $n$ qubits undergo a CNOT network controlled by the first qubit; the remaining $n-1$ qubits are measured, and the protocol proceeds only if all outcomes are $|0\rangle$ (postselection). A parametrized single-qubit unitary $U_p$ is then applied to the survivor.

The study employs the following analytical and numerical approaches:

Ideal Dynamics: The protocol is mapped to a rational function $f_p(z)$ on the Riemann sphere. The authors analyze the Julia sets $J(f_p)$ for varying complex parameters $p$ , calculating their box-counting dimensions ( $D_B$ ) over a $100 \times 100$ grid in the complex plane.
Preparation Noise: The analysis is extended to mixed initial states within the Bloch ball. The authors define "quasi-Julia sets" as the boundaries between distinct convergence phases in the three-dimensional Bloch space. They estimate dimensions on constant-purity spherical sections and for the full 3D structure.
Gate Noise: Two error models are applied to the ideal evolution:
- Depolarizing Noise: An effective $n$ -qubit depolarizing channel acting after the CNOT network but before postselection.
- Coherent Errors: Fixed unitary interactions (specifically $ZZ$ , $XX$ , and $YY$ types) acting on the qubit pairs.
Experimental Diagnostic: A fixed-point diagnostic is proposed to test hardware implementation. This involves selecting a reference state $| \psi(z_0) \rangle$ that is a fixed point of the ideal map, then measuring the conditional fidelity and acceptance probability of the protocol. A combined score $Q_k$ is defined to assess performance.

Key Results

Fractal Dimension and Symmetry: For ideal protocols ( $n=2$ to $5$), the box-counting dimension of the Julia sets varies with the parameter $p$ , ranging from $D_B \approx 1$ (at $p=0$ ) to approximately $1.8\text{--}1.9$ in complex regions. The dimension diagrams exhibit symmetries related to the protocol order $n$ and the parameter $p$ , including rotational and reflection symmetries.
Robustness to Preparation Noise: Moderate preparation noise (mixed states with purity $P \approx 0.98\text{--}0.99$ ) preserves the main parameter-dependent trends of the boundary dimensions. The study finds that the box-counting dimension of the three-dimensional quasi-Julia boundary in the Bloch ball is approximately equal to the dimension of the pure-state Julia set plus one ( $D_B^{(3D)} \approx D_B^{(2D)} + 1$ ). A "purity-driven phase transition" is observed where the fractal structure disappears below a critical purity, leaving a smooth boundary.
Robustness to Gate Noise:
- Coherent Errors: Weak coherent errors (e.g., $ZZ$ interactions) deform the convergence regions but leave the estimated fractal dimensions consistent with ideal values within numerical uncertainty. The $ZZ$ error model is shown to be exactly conjugate to the ideal protocol at a rotated parameter, explaining the preservation of dimension trends.
- Depolarizing Noise: Weak depolarizing noise also preserves the estimated dimensions despite visible deformations in the convergence basins. Stronger noise leads to lower dimension estimates and significant basin deformation.
Hardware Demonstration: The proposed diagnostic was implemented on IQM Emerald and Rigetti Cepheus-1-108Q devices using the quadratic ( $n=2$ ) protocol. The study demonstrates that the combined score $Q_k$ (fidelity $\times$ acceptance statistics) decreases with tree depth, reflecting accumulated errors. The results align with stochastic noise simulations, though hardware-specific factors like routing overhead and readout errors contribute to deviations.

Significance and Claims
The paper claims to establish that the geometric signatures of nonlinear quantum dynamics—specifically the fractal dimensions of convergence boundaries—are robust against moderate levels of both preparation and gate imperfections. This suggests that fractal structures can serve as reliable indicators of protocol behavior even in noisy intermediate-scale quantum (NISQ) devices.

The authors propose a practical diagnostic tool that combines conditional fidelity with acceptance statistics to benchmark these nonlinear protocols. They emphasize that this test characterizes the specific implementation of the protocol family (preparation, entangling gates, measurement, and final unitary) rather than general computational performance. The work provides a framework for understanding how noise affects the sensitivity of quantum protocols to initial conditions and offers a method to quantify the trade-off between sensitivity (via repelling fixed points) and resource efficiency (acceptance probability) in experimental settings.

The study remains modest regarding universal invariance, noting that while dimensions are preserved for weak errors, exact invariance is not established, and changes may occur below the accessible numerical resolution. The findings are presented as a foundation for further experimental investigation into the stability of nonlinear quantum dynamics.

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