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Science

Programmable higher-order nonequilibrium topological phases on a superconducting quantum processor

Haoran Qian, Ming Gong, Jia-Hui Zhang, Shaojun Guo, Chen Zha, Fusheng Chen, Yangsen Ye, Yulin Wu, Sirui Cao, Chong Ying, Qingling Zhu, He-Liang Huang, Youwei Zhao, Shaowei Li, Jiale Yu, Daojin Fan, Dachao Wu, Hong Te Su, Hui Deng, Hao Rong, Yuan Li, Kaili Zhang, Tung-Hsun Chung, Futian Liang, Jin Lin, Yu Xu, Cheng Guo, Na Li, Kai Qi Yan, Feifan Su, Gang Wu, Yong-Heng Huo, Cheng-Zhi Peng, Chao‐Yang Lu, Feng Mei, Suotang Jia, Xiaobo Zhu, Jian-Wei Pan

Published Nov 27, 2025• 11 citations
Publisher

Abstract

Topological phases of matter are of both fundamental and practical interest. In this study, we implemented both equilibrium and nonequilibrium higher-order topological phases using a two-dimensional programmable superconducting quantum processor. Quantum programming of nonequilibrium higher-order topological phases was achieved by constructing quantum circuits comprising >50 cycles of Floquet operators on a six-by-six qubit array. Additionally, we introduce a universal approach based on measuring the dynamics of chiral density to identify distinct nonequilibrium higher-order topological features, including Floquet corner topological invariants and π-energy topological corner modes. Our work may enable the use of programmable quantum processors to explore exotic higher-order nonequilibrium topological phases of matter.

DOI
10.1126/science.adp6802

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Topological phases of matter are of both fundamental and practical interest. In this study, we implemented both equilibrium and nonequilibrium higher-order topological phases using a two-dimensional programmable superconducting quantum processor. Quantum programming of nonequilibrium higher-order topological phases was achieved by constructing quantum circuits comprising >50 cycles of Floquet operators on a six-by-six qubit array. Additionally, we introduce a universal approach based on measuring the dynamics of chiral density to identify distinct nonequilibrium higher-order topological features, including Floquet corner topological invariants and π-energy topological corner modes. Our work may enable the use of programmable quantum processors to explore exotic higher-order nonequilibrium topological phases of matter.
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High-order topological phases of matter refer to the systems of n-dimensional bulk with the topology of m-th order, exhibiting (n-m)-dimensional boundary modes and can be characterized by topological pumping. Here, we experimentally demonstrate two types of second-order topological pumps, forming four 0-dimensional corner localized states on a 4×4 square lattice array of 16 superconducting qubits. The initial ground state of the system at half-filling, as a product of four identical entangled 4-qubit states, is prepared using an adiabatic scheme. During the pumping procedure, we adiabatically modulate the superlattice Bose-Hubbard Hamiltonian by precisely controlling both the hopping strengths and on-site potentials. At the half pumping period, the system evolves to a corner-localized state in a quadrupole configuration. The robustness of the second-order topological pump is also investigated by introducing different on-site disorder. Our Letter studies the topological properties of high-order topological phases from the dynamical transport picture using superconducting qubits, which would inspire further research on high-order topological phases.
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 .
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.