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Quantum many-body systems display rich phase structure in their low-temperature equilibrium states1. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases2-8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)7,9-15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order7,16,17. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors.

The thermal, acoustic, and dielectric properties of glasses below 1 K are dictated by the interaction of two-level systems (TLS) with strain and electric fields. In a previous paper, we proposed a modified TLS model to quantitatively account for the universally small phonon scattering in glasses at low temperatures. A key ingredient of this model was a wide distribution of couplings between TLS and phonons, contrary to the standard model which assumes a single averaged value is sufficient. In this paper, we expand on this view and include couplings to strain as well as electric fields. We then compare our theoretical results to measurements obtained using superconducting qubits. We find that the predictions of the modified TLS model are more consistent with experiments than those of the standard model. For the distribution of couplings between TLS and the strain field, there is a better agreement with experiments if we include a random distribution of local strains. Such a distribution of local strains is consistent with those found from molecular dynamics simulations.