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.
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.
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.
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.
The dynamics of an ensemble of identically prepared two-qubit systems is investigated which is subjected to the iteratively applied measurements and conditional selection of a typical entanglement purification protocol. The resulting dynamics exhibits strong sensitivity to initial conditions. For one class of initial states two types of islands characterize the asymptotic limit. They correspond to a separable and a fully entangled two-qubit state, respectively, and their boundaries form fractal-like structures. In the presence of incoherent noise an additional stable asymptotic cycle appears.
We analyze the consequences of iterative measurement-induced nonlinearity on the dynamical behavior of qubits. We present a one-qubit scheme where the equation governing the time evolution is a complex-valued nonlinear map with one complex parameter. In contrast to the usual notion of quantum chaos, exponential sensitivity to the initial state occurs here. We calculate analytically the Lyapunov exponent based on the overlap of quantum states, and find that it is positive. We present a few illustrative examples of the emerging dynamics.