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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. Enhancing detection of topological order by local error correction

   https://doi.org/10.1038/s41467-024-45584-6  

   Cong, Iris; Maskara, Nishad; Tran, Minh C; Pichler, Hannes; Semeghini, Giulia; Yelin, Susanne F; Choi, Soonwon; Lukin, Mikhail D (December 2024, Nature Communications)

   Abstract The exploration of topologically-ordered states of matter is a long-standing goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as long-range entanglement, emergent gauge fields and non-local correlations, and can aid in realization of scalable fault-tolerant quantum computation. However, these same features also make creation, detection, and characterization of topologically-ordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a paradigm for quantifying topological states—locally error-corrected decoration (LED)—by combining methods of error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations. We subsequently apply it to an experimental realization, providing new insights into a quantum spin liquid created on a Rydberg-atom simulator. Finally, we extend LED to generic topological phases, including those with non-abelian order. 

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5. Topology meets time-reversal symmetry breaking in FeSe1−xTex superconductors

   https://doi.org/10.1038/s41467-025-61651-y  

   Roppongi, Masaki; Cai, Yipeng; Ogawa, Koki; Liu, Supeng; Zhao, Guoqiang; Oudah, Mohamed; Fujii, Takenori; Imamura, Kumpei; Fang, Shengjie; Ishihara, Kota; et al (December 2025, Nature Communications)

   Time-reversal symmetry breaking (TRSB) in magnetic topological insulators induces a Dirac gap in the topological surface state (TSS), leading to exotic phenomena such as the quantum anomalous Hall effect. Yet, the interplay between TRSB and topology in superconductors remains underexplored due to limited suitable materials. Here we employ zero-fieldmuon spin relaxation (μSR) as a sensitive probe of TRSB to map out the electronic phase diagrams of iron-chalcogenide superconductors FeSe1−xTex. For the Te composition x = 0.64 with the highest superconducting transition temperature Tc = 14.5K, which is known to host a TSS and Majorana zero modes within vortices, we detect spontaneous magnetic fields below Tc distinct from a magnetic order. This signifies a TRSB superconducting state in the bulk, revealing the convergence of unconventional TRSB superconductivity with topologically nontrivial electronic structures in FeSe1−xTex. Given the relatively high Tc and the tunability of the Fermi level through chemical substitution, iron-chalcogenide superconductors offer an intriguing platform for investigating the synergy between topological superconductivity and TRSB. 

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