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1. Long-lived topological time-crystalline order on a quantum processor

   https://doi.org/10.1038/s41467-024-53077-9  

   Xiang, Liang; Jiang, Wenjie; Bao, Zehang; Song, Zixuan; Xu, Shibo; Wang, Ke; Chen, Jiachen; Jin, Feitong; Zhu, Xuhao; Zhu, Zitian; et al (December 2024, Nature Communications)

   Abstract Topologically ordered phases of matter elude Landau’s symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon—a prethermal topologically ordered time crystal—with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors. 

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2. Long-lived topological time-crystalline order on a quantum processor

   Xiang, Liang; Jiang, Wenjie; Bao, Zehang; Song, Zixuan; Xu, Shibo; Wang, Ke; Chen, Jiachen; Jin, Feitong; Zhu, Xuhao; Zhu, Zitian; et al (January 2024, arXiv)

   Topologically ordered phases of matter elude Landau's symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon -- a prethermal topologically ordered time crystal -- with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface-code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors. 

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3. Topological prethermal strong zero modes on superconducting processors

   https://doi.org/10.1038/s41586-025-09476-z  

   Jin, Feitong; Jiang, Si; Zhu, Xuhao; Bao, Zehang; Shen, Fanhao; Wang, Ke; Zhu, Zitian; Xu, Shibo; Song, Zixuan; Chen, Jiachen; et al (September 2025, Nature)

   Abstract Symmetry-protected topological phases^1–4cannot be described by any local order parameter and are beyond the conventional symmetry-breaking model⁵. They are characterized by topological boundary modes that remain stable under symmetry respecting perturbations^1–4,6–8. In clean, gapped systems without disorder, the stability of these edge modes is restricted to the zero-temperature manifold; at finite temperatures, interactions with mobile thermal excitations lead to their decay^9–11. Here we report the observation of a distinct type of topological edge mode^12–14, which is protected by emergent symmetries and persists across the entire spectrum, in an array of 100 programmable superconducting qubits. Through digital quantum simulation of a one-dimensional disorder-free stabilizer Hamiltonian, we observe robust long-lived topological edge modes over up to 30 cycles for a wide range of initial states. We show that the interaction between these edge modes and bulk excitations can be suppressed by dimerizing the stabilizer strength, leading to an emergent U(1) × U(1) symmetry in the prethermal regime of the system. Furthermore, we exploit these topological edge modes as logical qubits and prepare a logical Bell state, which exhibits persistent coherence, despite the system being disorder-free and at finite temperature. Our results establish a viable digital simulation approach^15–18to experimentally study topological matter at finite temperature and demonstrate a potential route to construct long-lived, robust boundary qubits in disorder-free systems. 

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4. Experimental Realization of Discrete Time Quasicrystals

   https://doi.org/10.1103/physrevx.15.011055  

   He, Guanghui; Ye, Bingtian; Gong, Ruotian; Yao, Changyu; Liu, Zhongyuan; Murch, Kater W; Yao, Norman Y; Zu, Chong (March 2025, Physical Review X)

   Floquet (periodically driven) systems can give rise to unique nonequilibrium phases of matter without equilibrium analogs. The most prominent example is the realization of discrete time crystals. An intriguing question emerges: What other novel phases can manifest when the constraint of time periodicity is relaxed? In this study, we explore quantum systems subjected to a quasiperiodic drive. Leveraging a strongly interacting spin ensemble in diamond, we identify the emergence of long-lived discrete time quasicrystals. Unlike conventional time crystals, time quasicrystals exhibit robust subharmonic responses at multiple incommensurate frequencies. Furthermore, we show that the multifrequency nature of the quasiperiodic drive allows for the formation of diverse patterns associated with different discrete time quasicrystalline phases. Our findings demonstrate the existence of nonequilibrium phases in quasi-Floquet settings, significantly broadening the catalog of novel phenomena in driven many-body quantum systems. Published by the American Physical Society2025 

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5. Floquet control of interactions and edge states in a programmable quantum simulator

   https://doi.org/10.1038/s41467-025-62897-2  

   Katz, Or; Feng, Lei; Porras, Diego; Monroe, Christopher (December 2025, Nature Communications)

   Abstract Quantum simulators based on trapped ions enable the study of spin systems and models with rich dynamical phenomena. The Su-Schrieffer-Heeger (SSH) model for fermions in one dimension is a canonical example that can support a topological insulator phase when couplings between sites are dimerized, featuring long-lived edge states. Here, we experimentally implement a spin-based variant of the SSH model using one-dimensional trapped-ion chains with tunable interaction range, realized in crystals containing up to 22 interacting spins. Using an array of individually focused laser beams, we apply site-specific, time-dependent Floquet fields to induce controlled bond dimerization. Under conditions that preserve inversion symmetry, we observe edge-state dynamics consistent with SSH-like behavior. We study the propagation and localization of spin excitations, as well as the evolution of highly excited configurations across different interaction regimes. These results demonstrate how precision Floquet engineering enables the exploration of complex spin models and dynamics, laying the groundwork for future preparation and characterization of topological and exotic phases of matter. 

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