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Chaos An Interdisciplinary Journal of Nonlinear Science

Ergodic dynamics in iterated quantum protocols

Attila Portik, Orsolya Kálmán, Tamás Kiss

Published May 1, 2026• 0 citations
Publisher

Abstract

We study measurement-induced nonlinear dynamics generated by an iterated quantum protocol combining an entangling gate, a single-qubit rotation, and post-selection. For pure single-qubit inputs, a particular choice of the single-qubit unitary yields globally chaotic, strongly mixing dynamics that explores the entire Bloch sphere, providing a physical realization of ergodic behavior in a complex map. We extend the analysis to realistic, noisy preparation by considering mixed initial states and the induced nonlinear evolution inside the Bloch sphere. Numerical results show that the maximally mixed state is an attractor for mixed inputs, although many trajectories exhibit transient increases in purity before ultimately converging. To quantify robustness against noise, we introduce a practical notion of quasi-ergodicity: ensembles prepared in a small angular patch at fixed purity rapidly spread to cover all directions, while the purity gradually decreases toward its minimal value. By varying the final single-qubit gate, we identify a broad family of protocols that remain ergodic-like for pure states, supported by consistent diagnostics, including the absence of attracting cycles, agreement of time and ensemble statistics, rapid spreading from localized regions, and exponential sensitivity to initial conditions. Away from the special globally mixing case, the mixed-state dynamics can change qualitatively: for most ergodic-like parameters, a finite subset of noisy inputs is driven toward purification rather than complete mixing, demonstrating the coexistence of statistical mixing and purification within a single iterated protocol.

DOI
10.1063/5.0320372

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We study measurement-induced nonlinear dynamics generated by an iterated quantum protocol combining an entangling gate, a single-qubit rotation, and post-selection. For pure single-qubit inputs, a particular choice of the single-qubit unitary yields globally chaotic, strongly mixing dynamics that explores the entire Bloch sphere, providing a physical realization of ergodic behavior in a complex map. We extend the analysis to realistic, noisy preparation by considering mixed initial states and the induced nonlinear evolution inside the Bloch sphere. Numerical results show that the maximally mixed state is an attractor for mixed inputs, although many trajectories exhibit transient increases in purity before ultimately converging. To quantify robustness against noise, we introduce a practical notion of quasi-ergodicity: ensembles prepared in a small angular patch at fixed purity rapidly spread to cover all directions, while the purity gradually decreases toward its minimal value. By varying the final single-qubit gate, we identify a broad family of protocols that remain ergodic-like for pure states, supported by consistent diagnostics, including the absence of attracting cycles, agreement of time and ensemble statistics, rapid spreading from localized regions, and exponential sensitivity to initial conditions. Away from the special globally mixing case, the mixed-state dynamics can change qualitatively: for most ergodic-like parameters, a finite subset of noisy inputs is driven toward purification rather than complete mixing, demonstrating the coexistence of statistical mixing and purification within a single iterated protocol.
We study measurement-induced nonlinear dynamics generated by an iterated quantum protocol combining an entangling gate, a single-qubit rotation, and post-selection. For pure single-qubit inputs, a particular choice of the single-qubit unitary yields globally chaotic, strongly mixing dynamics that explores the entire Bloch sphere, providing a physical realization of ergodic behavior in a complex map. We extend the analysis to realistic, noisy preparation by considering mixed initial states and the induced nonlinear evolution inside the Bloch sphere. Numerical results show that the maximally mixed state is an attractor for mixed inputs, although many trajectories exhibit transient increases in purity before ultimately converging. To quantify robustness against noise, we introduce a practical notion of quasi-ergodicity: ensembles prepared in a small angular patch at fixed purity rapidly spread to cover all directions, while the purity gradually decreases toward its minimal value. By varying the final single-qubit gate, we identify a broad family of protocols that remain ergodic-like for pure states, supported by consistent diagnostics including the absence of attracting cycles, agreement of time and ensemble statistics, rapid spreading from localized regions, and exponential sensitivity to initial conditions. Away from the special globally mixing case, the mixed-state dynamics can change qualitatively: for most ergodic-like parameters, a finite subset of noisy inputs is driven toward purification rather than complete mixing, demonstrating the coexistence of statistical mixing and purification within a single iterated protocol.
Abstract Iterated quantum protocols with measurement-based selection lead to deterministic chaos for the evolving pure state representing an ensemble of qubits. Deterministic chaos for the pure quantum state may lead to ergodic evolution in the sense that initial states from any small area on the Bloch sphere will cover the whole sphere after a finite number of iterations. We realize two steps of an ergodic protocol in a photonic experiment, where initial qubit states are encoded in the polarization and path degrees of freedom of down-converted photons stemming from a parametric process. We numerically analyze the effect of noise on the time evolution and show that the protocol, described by a Lattès map, remains quasi-ergodic for any initial state if the initial noise is small. Tomographic reconstruction of the quantum states throughout the evolution is consistent with simulations and thus demonstrates ergodicity of the quantum dynamics.
Quantum information processing exploits all the features quantum mechanics offers. Among them there is the possibility to induce nonlinear maps on a quantum system by involving two or more identical copies of the given system in the same state. Such maps play a central role in distillation protocols used for quantum key distribution. We determine that such protocols may exhibit sensitive, quasi-chaotic evolution not only for pure initial states but also for mixed states, i.e., the complex dynamical behavior is not destroyed by small initial uncertainty. We show that the appearance of sensitive, complex dynamics associated with a fractal structure in the parameter space of the system has the character of a phase transition. The purity of the initial state plays the role of the control parameter, and the dimension of the fractal structure is independent of the purity value after passing the phase transition point. The critical purity coincides with the purity of a repelling fixed point of the dynamics, and we show that all the pre-images of states from the close neighborhood of pure chaotic initial states have purity larger than this. Initial states from this set can be considered as quasi-chaotic.
One of the simplest possible quantum circuits, consisting of a cnot gate, a Hadamard gate, and a measurement on one of the outputs is known to lead to chaotic dynamics when applied iteratively on an ensemble of equally prepared qubits. The evolution of pure initial quantum states is characterized by a fractal (in the space of states), formed by the border of different convergence regions. We examine how the ideal evolution is distorted in the presence of both coherent error and incoherent initial noise, which are typical imperfections in current implementations of quantum computers. It is known that under the influence of initial noise only, the fractal is preserved, moreover, its dimension remains constant below a critical noise level. We systematically analyze the effect of coherent Hadamard gate errors by determining fixed points and cycles of the evolution. We combine analytic and numerical methods to explore to what extent the dynamics is altered by coherent errors in the presence of preparation noise as well. We show that the main features of the dynamics, and especially the fractal borders, are robust against the discussed noise, they will only be slightly distorted. We identify a range of error parameters, for which the characteristic properties of the dynamics are not significantly altered. Hence, our results allow to identify reliable regimes of operation of iterative protocols.
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.
Complex chaos is specified by an iterated mapping on complex numbers. It has recently been found in the dynamics of qubits where each time step is conditioned on a measurement result on part of the system. We analyse the simplest case of one qubit dynamics with one complex parameter in some detail. We point out that two attractive cycles can exist and provide examples how the fractal like Julia set divides the areas of corresponding initial states. We show how to determine the set of parameters for which one, two or no stable fixed cycles exists and provide the numerically calculated images of the sets. The results can be relevant for the quantum state purification protocol based on the similar dynamics of two or more qubits and in general for any protocol based on conditioned nonlinear dynamics where truly chaotic behavior may occur.