Ergodic dynamics in iterated quantum protocols

TL;DR

This work investigates measurement-induced nonlinear dynamics from iterated quantum protocols that combine entangling gates, post-selection, and a parametrized single-qubit unitary to realize ergodic and quasi-ergodic evolution in a single qubit. For pure inputs, the Lattès map $f(z)=\frac{z^2+i}{1+i z^2}$ on the Bloch (Riemann) sphere produces global chaos with time- and ensemble-averages coinciding, and a positive Lyapunov exponent $\lambda=\frac{\ln 2}{2}$. When mixed states are included, all interior states are attracted to the maximally mixed state, though many trajectories exhibit transient purification, giving rise to a quasi-ergodic regime under realistic noise. The authors further identify a broad ergodic-like region in the complex parameter plane for the final unitary, showing robust angular mixing and positive Lyapunov exponents even away from the Lattès point, while some parameters admit purification of mixed states, demonstrating a coexistence of chaotic mixing and purification with potential applications to noise-robust state preparation and benchmarking.

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.

Figures (12)

• Figure 1: Comparison of time-averaged and ensemble-averaged distributions on the Bloch sphere for the case of the Lattès map $f(z)$, illustrating the ergodic properties of the iterated map. (a) Time-averaged density: visitation frequency of a forward orbit averaged over 100 randomly chosen initial pure states, each iterated for $10^7$ steps. The Bloch sphere is discretized into a $500 \times 500$ equal-area grid, and the color indicates the average number of visits per cell. (b) Ensemble-averaged density: distribution after $100$ iterations of $10^7$ initial states sampled uniformly from a small region within the Bloch sphere. This process was repeated for 100 randomly chosen regions (of small, equal area), and the resulting histograms were averaged over these trials. Both distributions are visualized using the same grid and color scale.
• Figure 2: Purity of the states along the trajectories of $100$ random initial states sampled uniformly from the spherical surface with initial purity $P_0 = 0.95$. Blue curves show the purity of individual trajectories under the iterated protocol, and the red curve highlights a representative example. The trajectories display a characteristic behaviour, the purity of the state repeatedly increases and decreases over successive iterations, while its overall value gradually declines. Although transient purification occurs, all trajectories ultimately approach the maximally mixed state.
• Figure 3: Fraction of initial states whose purity increases relative to the previous iteration. Plot (a) shows the fraction for initial purity $P_0 = 0.95$, sampled from $10^6$ random states on the corresponding spherical surface. Plot (b) displays the fraction of states that purify at each step, shown as a function of both the initial purity and the iteration number. Transient purification is most pronounced for states with high initial purity, while it is weak or absent for more mixed initial states. In all cases, the effect disappears as trajectories approach the maximally mixed state.
• Figure 4: Fraction of mixed initial states whose purity becomes higher than their initial value $P_0$ as a function of the iteration number. Panel (a) shows the case $P_0 = 0.95$, panel (b) displays the dependence on both $P_0$ and the iteration number.
• Figure 5: (a) Average purity of random initial states under the iterated nonlinear protocol, as a function of the number of steps and the initial purity. (b) The fraction of states whose purity remains above the purity of the maximally mixed state during the evolution. The white contour marks the $10\%$ level.