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arXiv:2501.18461 · Nature

Probing non-equilibrium topological order on a quantum processor

Melissa Will, Tyler A. Cochran, Eliott N. Rosenberg, B. Jobst, Norhan M. Eassa, P. Roushan, Michael Knap, A. Gammon-Smith, Frank Pollmann

Published Sep 10, 2025• 20 citations
arXivHTMLPDF

Abstract

Abstract Out-of-equilibrium phases in many-body systems constitute a new paradigm in quantum matter—they exhibit dynamical properties that may otherwise be forbidden by equilibrium thermodynamics. Among these non-equilibrium phases are periodically driven (Floquet) systems1–5, which are generically difficult to simulate classically because of their high entanglement. Here we realize a Floquet topologically ordered state theoretically proposed in ref. 6, on an array of superconducting qubits. We image the characteristic dynamics of its chiral edge modes and characterize its emergent anyonic excitations. Devising an interferometric algorithm allows us to introduce and measure a bulk topological invariant to probe the dynamical transmutation of anyons for system sizes up to 58 qubits. Our work demonstrates that quantum processors can provide key insights into the thus-far largely unexplored landscape of highly entangled non-equilibrium phases of matter.

DOI
10.1038/s41586-025-09456-3

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Chiral Parafermions in Floquet ZN\mathbb{Z}_NZN​ Topological Order
Floquet topological order (FTO) extends topology beyond equilibrium, giving rise to phenomena, such as the dynamical transmutation of anyons. Here, we realize ZN\mathbb{Z}_NZN​ FTO in a generalized Kitaev honeycomb model with clock degrees of freedom using a three-step periodic drive. The driven system admits an emergent description in terms of constrained parafermions coupled to conserved ZN\mathbb{Z}_NZN​ gauge fields. At the boundary, the drive generates chiral parafermionic Floquet edge modes. We resolve their propagation through correlation functions, characterize their fractional quantum-information transport through the radical chiral unitary index, and use their dynamics to probe exchange statistics. In the bulk, we explore the dynamical anyon transmutation with a non-equilibrium loop order parameter. At a Floquet sweet spot, where the drive becomes Clifford, these phenomena can be accessed using stabilizer simulations. Short-time matrix-product-state simulations away from this point show that the chiral edge response persists at accessible times. Our results establish ZN\mathbb{Z}_NZN​ Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.
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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.
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.
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.
The interplay between topology and quantum entanglement holds profound implications for fundamental physics and the advancement of quantum technologies. Topological phases of matter are characterized by robust properties impervious to local perturbations, while entanglement serves as a critical resource for quantum information processing. Floquet engineering, the manipulation of quantum systems through periodic driving, offers a powerful non-equilibrium pathway to realize novel topological states that may not exist in static systems. This paper explores the theoretical and experimental avenues for realizing topological quantum entanglement within Floquet-engineered lattices. We review the foundational concepts of Floquet topological phases and various measures of entanglement, such as entanglement entropy and entanglement spectrum. We then delve into the methodologies for designing such systems across diverse platforms, including ultracold atoms in optical lattices, photonic waveguide arrays, and superconducting circuits. The discussion highlights how Floquet driving can induce robust entanglement in topological edge and bulk states, paving the way for fault-tolerant quantum information storage and processing. We propose specific protocols for generating and detecting topologically-protected entangled states, considering the challenges associated with heating and decoherence in periodically driven systems. Ultimately, this work emphasizes the potential of Floquet engineering as a versatile tool for synthesizing exotic entangled topological states, fostering new frontiers in quantum science and engineering.