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arXiv:2608.24993

Mixed-State Symmetry-Protected Topology and Strong-to-Weak Spontaneous Symmetry Breaking in a Superconducting Qubit Array

Qian Cao, Zhen Bi, Kater W. Murch

quant-phcond-mat.mes-hallcond-mat.stat-mechSubmitted Aug 25, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

We experimentally investigate how symmetry-protected topological order in a one-dimensional cluster state is transformed by measurement and decoherence in a five-qubit superconducting array. We first characterize the state's nonlocal string order and show that controlled dephasing selectively suppresses one symmetry sector while leaving the other robust, consistent with average symmetry-protected topological order. We then measure one sublattice in a tunable basis and show that the remaining qubits are driven between a long-range-entangled GHZ state and a paramagnetic state. When the measurement record is discarded, the conventional long-range correlator vanishes while a nonlinear fidelity correlator remains finite, providing a finite-size signature of strong-to-weak spontaneous symmetry breaking. These experiments demonstrate how conditioning, averaging, and decoherence reveal distinct manifestations of order encoded in the same underlying cluster state, and establish a superconducting-circuit setting for probing mixed-state symmetry and topology.

Comments
9 pages, 3 figures

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We experimentally investigate how symmetry-protected topological order in a one-dimensional cluster state is transformed by measurement and decoherence in a five-qubit superconducting array. We first characterize the state's nonlocal string order and show that controlled dephasing selectively suppresses one symmetry sector while leaving the other robust, consistent with average symmetry-protected topological order. We then measure one sublattice in a tunable basis and show that the remaining qubits are driven between a long-range-entangled GHZ state and a paramagnetic state. When the measurement record is discarded, the conventional long-range correlator vanishes while a nonlinear fidelity correlator remains finite, providing a finite-size signature of strong-to-weak spontaneous symmetry breaking. These experiments demonstrate how conditioning, averaging, and decoherence reveal distinct manifestations of order encoded in the same underlying cluster state, and establish a superconducting-circuit setting for probing mixed-state symmetry and topology.
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We study the effect of subsystem decoherence to an extended cluster state which is a symmetry protected topological (SPT) phase. The model includes many subsystem Z2Z_2Z2​ symmetries. We report that subsystem decoherence induces local charge fluctuations, leading to a mixed SPT state in the unaffected subsystems. If we start from the extended cluster state, hierarchical mixed-state SPT phases emerge in response to step-by-step subsystem decoherences. These mixed-state SPT phases keep strong symmetries the symmetry of which is protecting symmetries for the initial cluster SPT. Moreover, these SPTs can be characterized by Rényi-2 string orders. Then, as the subsystems are progressively decohered, the hierarchy of mixed-state SPT phases terminates in a Z2Z_2Z2​ strong-to-weak spontaneous symmetry breaking (SWSSB) state on the final remaining subsystem, where a long-range entangled state appears, namely a glassy Greenberger-Horne-Zeilinger (GHZ) state. Our work demonstrates that decoherence is not merely a destructive process, but can induce and organize series of nontrivial mixed-states. This reveals a systematic route from mixed-state SPT order to SWSSB with the glassy GHZ-type long-range entanglement.
Abstract Symmetry-protected topological (SPT) phases are conventionally formulated for pure states protected by exact symmetries. In open quantum systems, however, decoherence generates mixed-state ensembles in which average symmetry can instead emerge only after averaging over microscopic trajectories. In this work, we realize decoherence-induced average SPT (ASPT) order on programmable quantum processors. Starting from a one-dimensional cluster SPT, we engineer sublattice-selective Pauli-ZZZ dephasing through ancilla-assisted quantum circuits. We diagnose the loss of strong symmetry through the decay of the symmetry charge, while the engineered channel preserves the corresponding symmetry at the ensemble level. Moreover, a two-replica R\'enyi-2 string correlator measured through destructive SWAP readout remains nontrivial, revealing the ASPT structure encoded at the density-matrix level. We further show that the engineered dephasing redistributes spectral weight among reduced-stabilizer sectors while preserving the characteristic twofold pairing of the half-chain entanglement spectrum. These results establish a gate-based route to engineering and probing ASPT order and demonstrate structured decoherence as a programmable resource for realizing mixed-state topological quantum matter.
Symmetry-protected topological (SPT) phases are conventionally formulated for pure states protected by exact symmetries. In open quantum systems, however, decoherence generates mixed-state ensembles in which average symmetry can instead emerge only after averaging over microscopic trajectories. In this work, we realize decoherence-induced averaged SPT (ASPT) order on programmable quantum processors. Starting from a one-dimensional cluster SPT, we engineer sublattice-selective Pauli-ZZZ dephasing through ancilla-assisted quantum circuits. We observe the resulting symmetry conversion through the decay of the symmetry charge. Moreover, a two-replica Renyi-2 string correlator measured through destructive SWAP readout remains nontrivial, revealing the ASPT structure encoded at the density-matrix level. We further show that the engineered dephasing redistributes spectral weight among reduced-stabilizer sectors while preserving the characteristic twofold pairing of the half-chain entanglement spectrum. These results establish a gate-based route to engineering and probing ASPT order and demonstrate structured decoherence as a programmable resource for realizing mixed-state topological quantum matter.
We explore the effect of decoherence on many-body mixed states from a purification perspective. Here, quantum channels map to unitary transformations acting on a purified state defined within an extended Hilbert space. First, we exploit this approach to analyze the phenomenon of spontaneous strong-to-weak symmetry breaking (sw-SSB) triggered by strongly symmetric local quantum channels. We find that sw-SSB for mixed states relates to proximity to a symmetry-protected topological (SPT) order in the purified state. Remarkably, the measurement-induced long-range order in the purified SPT state, as characterized by the R\'enyi-2 correlator, mirrors the ensuing long-range order in the mixed state due to sw-SSB. Moreover, we establish a 1-to-1 correspondence between various sw-SSB order parameters for the mixed state, and strange correlators in the purification, which signify the SPT order. This purification perspective is further extended to explore intrinsic mixed-state topological order and decoherent symmetry-protected topological phases.
Open quantum systems can exhibit inherently mixed quantum orders, such as strong-to-weak spontaneous symmetry breaking (SW-SSB), which have no analogue in pure states. It is generally expected that a finite-depth local quantum channel cannot induce a phase transition to SW-SSB in one spatial dimension. We circumvent this obstruction by introducing a long-range quantum channel, which implements two-body dephasing on pairs of sites with probability falling off according to a power-law in the site separation. Applying this channel to a one-parameter family of matrix product states interpolating between trivial and symmetry-protected topological (SPT) phases, we construct a phase diagram that hosts trivial, mixed SPT, and SW-SSB phases with respect to Rényi-2 diagnostics. Notably, the mixed SPT phase is distinct from previously studied decohered SPT phases: it hosts two logical qubits and exhibits nonvanishing Rényi-2 string order, despite vanishing conventional string order. Through a series of mappings to a classical statistical mechanics model, we establish a direct connection between the loss of logical information and the onset of SW-SSB in this model. Finally, we consider the feasibility of experimental realization of the phenomena we have studied. We emphasize that our family of mixed states can be tuned to regimes of high purity, substantially reducing measurement overhead and enabling their experimental realization and detection on near-term quantum hardware.