Resonances, Recurrence Times and Steady States in Monitored Noisy Qubit Systems

Abstract

We study non-equilibrium steady states and recurrence times in noisy, stroboscopically monitored qubit systems using complete measurements. In the noiseless limit, recurrence times are integer-quantized, with dips to lower integers when sampling approaches revival conditions associated with ergodicity breaking. Using an IBM quantum platform, we find that quantization is robust when sampling far from revivals, but breaks down dramatically near revivals: even weak noise produces large deviations and can invert the expected dips into pronounced peaks. To explain this behavior, we formulate a statistical-physics model of monitored noisy circuits in which monitoring drives an effective infinite-temperature steady state while thermal-like relaxation competes to favor a low-temperature limit. We show that the sampling time tunes a crossover between these regimes, near revivals stabilizing low-temperature behavior, and far from revivals restoring infinite-temperature behavior -- with noise strength and detuning acting as coupled small parameters near resonance.

Figures (4)

• Figure 1: Mean recurrence time $\langle n \rangle$ for states $|0\rangle$ and $|1\rangle$ versus sampling time $\tau$ in a single IBM qubit system. Without noise, $\langle n \rangle = 2$ except at simple revival resonances, where $\langle n \rangle = 1$, as shown by the ideal black solid line. Near resonances, noise effects are pronounced, even if the noise per cycle is weak. Far from resonances the recurrence time for the $|0\rangle$ and $|1\rangle$ states are identical and given by the integer $2$ as expected from noise-free theory and time reversal symmetry. At resonances, we observe time reversal symmetry breaking where quantization of the mean recurrence time is clearly invalid. The excited state $|1\rangle$ has a smaller population in a thermodynamics sense and thus a longer recurrence time than the ground state $|0\rangle$. Theory from Eq. (\ref{['eq02']}) matches the data quantitatively. A least-squares fit gives $p_{0\to1}=0.0012$ and $p_{1\to0}=0.0086$, with $R^2=0.941$. Error bars indicate the statistical uncertainty of the measured recurrence times. The experimental setup is described in the Methods section, and additional details of the data analysis are provided in Supplementary Materials S2.
• Figure 2: Mean recurrence time $\langle n \rangle$ versus sampling time $\tau$ for a two-qubit system with complete measurements. In the noiseless case, $\langle n \rangle = 4$ except for two classes of resonances: $\tau = \pi/2 + k\pi$ and $\tau = k\pi$. Weak noise removes the first class, while the second ($\tau = k\pi$) survives with modified dips, peaks, and asymmetry between basis states (see subtitles), reflecting noise sensitivity near resonances and negligible effects far from them. At resonance noise effects are amplified, with $|00\rangle$$(|11\rangle)$ showing pronounced dips (peaks) due to its high (low) occupation probability, consistent with Kac's lemma. The dashed gray line indicates the reference value $4$. A joint least-squares fit of the weak-noise theory to all four panels yields $p_{1\to0,a}=0.0189$, $p_{0\to1,a}=0.0026$, $p_{0\to1,b}=0.0022$, and $p_{1\to0,b}=0.0188$, with $R^2=0.939$. Detailed analysis of the experimental data is presented in the Supplementary Materials S2.
• Figure 3: Steady-state probabilities grouped by the Hamming weight of the computational basis states of the Hypercube model. (A–C) Results from the first perturbation theory [Eq. (\ref{['eq05']}, \ref{['hyperI']})], shown as orange dotted lines, are compared with exact numerical solutions (blue solid lines) for noise-induced transition probabilities $p_{1\to0}=0.05$ and $p_{0\to1}=0.01$. Excellent agreement is observed for generic sampling times $\tau$, except in the vicinity of $\tau=0,\pi,2\pi$, where noise-induced effects become non-perturbative. These special sampling times correspond to near-revival conditions with $G(\tau)\approx\mathcal{I}$, where the first perturbative expansion [Eq. (\ref{['eq05']})] breaks down. (D–F) Zoomed-in view near $\tau=\pi$. The red dotted lines show the predictions of the second perturbation theory [Eq. (\ref{['eq06']}, \ref{['hyperII']})], which treats both the noise strength and the deviation from $\tau_R$ as small parameters. In this resonant regime, the second perturbative approach [Eq. (\ref{['eq06']})] accurately captures the steady state and restores agreement with the exact numerical results (blue solid lines).
• Figure 4: Examples of detection times for initial state $|\psi_{in}\rangle=|1\rangle$ with "threading" measurement protocol. Each trial consists of up to $T_T = 1000$ measurement attempts, a limitation that was imposed by the hardware. If the target state is detected, like in the first two records, the detection step is directly recorded. If detection fails, a new measurement sequence is initiated using the final state of the previous sequence and is randomly selected from the pool of trials. This process is iterated until we successfully detect the target state. In this way, the practical limitation of a finite number of measurements (1000 here) is bypassed, allowing us to observe new phenomena related to the recurrence time. The mean detection time $\langle n \rangle$ is obtained by averaging over all successful events. In our experiments, we collected 40,000 measurement sequences, each of length 1000, on the IBM quantum processor ibm_fez.