Fig. 2: Subharmonic behavior.

Performance of the device in the discrete time-crystalline regime. The results are compared to numerical simulations, where we use a two-dimensional tensor network state (2dTNS), see Section II B of Supplementary Information. Noise recovery is performed using the renormalization methods described in Methods and Supplementary Information, Section III. Where present, error bars indicate statistical errors corresponding to five standard deviations (5σ), estimated via a nonparametric bootstrap procedure with 200 resamples. The shaded regions represent the standard error of the mean used as the measure of finite-size error (see Methods). a The order parameter Δ(t), defined in Eq. (5), is plotted as a function of the number of Floquet cycles, illustrating the subharmonic response characteristic of the discrete time-crystalline regime. The data shown include raw device measurements (orange squares), classical simulations (green diamonds), and noise-recovered data (empty circles). The right panel presents the Fourier transform of the order parameter as a function of the frequency ω in units of the drive frequency ω_D. The components for ω > ω_D/2 represent the original spectrum folded into this band. b The spin-spin correlation parameter χ(t), defined in Eq. (6), is shown. The notations are consistent with (a). c The evolution of Hamming distance distributions, defined in Eq. (S.22), is presented on a logarithmic scale for raw data (top) and mitigated data (bottom) in a 2 × 2 system. The mitigated distribution converges to a steady-state value. d An illustration of noise filtering through a comparison of Hamming distance distributions for raw device data (orange squares), 2dTNS classical simulations (green diamonds), and noise-filtered results (blue circles), exemplified at Floquet cycle 25, revealing small skewness and sizable broadening.