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1. 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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2. Digital quantum simulation of Floquet symmetry-protected topological phases

   https://doi.org/10.1038/s41586-022-04854-3  

   Zhang, Xu; Jiang, Wenjie; Deng, Jinfeng; Wang, Ke; Chen, Jiachen; Zhang, Pengfei; Ren, Wenhui; Dong, Hang; Xu, Shibo; Gao, Yu; et al (July 2022, Nature)

   Abstract Quantum many-body systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals 1–8 , in which time-translational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions 9,10 , solid-state spin systems 11–15 , ultracold atoms 16,17 and superconducting qubits 18–20 . Here we report the observation of a distinct type of non-equilibrium state of matter, Floquet symmetry-protected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust long-lived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetry-protected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic non-equilibrium phases of matter with current noisy intermediate-scale quantum processors 21 . 

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3. Ground state degeneracy on torus in a family of ZN toric code

   https://doi.org/10.1063/5.0134010  

   Watanabe, Haruki; Cheng, Meng; Fuji, Yohei (May 2023, Journal of Mathematical Physics)

   Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species. 

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4. Probing Topological Entanglement on Large Scales

   https://doi.org/10.1103/mdsf-wrbj  

   Ott, Robert; Zache, Torsten V; Maskara, Nishad; Lukin, Mikhail D; Zoller, Peter; Pichler, Hannes (August 2025, Physical Review Letters)

   Topologically ordered quantum matter exhibits intriguing long-range patterns of entanglement, which reveal themselves in subsystem entropies. However, measuring such entropies, which can be used to certify topological order, on large partitions is challenging and becomes practically unfeasible for large systems. We propose a protocol based on local adiabatic deformations of the Hamiltonian which extracts the universal features of long-range topological entanglement from measurements on small subsystems of finite size, trading an exponential number of measurements against a polynomial-time evolution. Our protocol is general and readily applicable to various quantum simulation architectures. We apply our method to various string-net models representing both Abelian and non-Abelian topologically ordered phases and illustrate its application to neutral atom tweezer arrays with numerical simulations. 

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5. 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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