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arXiv:2609.07831

Realization of decoherence-induced averaged symmetry-protected topological phases on quantum processors

Ruizhe Shen, Xue-Jia Yu, Ching Hua Lee

quant-phSubmitted Sep 7, 2026• Announced Sep 9, 2026• 0 citations
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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 averaged SPT (ASPT) order on programmable quantum processors. Starting from a one-dimensional cluster SPT, we engineer sublattice-selective Pauli-$Z$ 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.

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23 pages and 7 figures

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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.
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.
Edge bit in an average symmetry protected topological (ASPT) mixed state is studied. The state is protected by one strong Z2Z_2Z2​ and one weak (average) Z2Z_2Z2​ symmetries. As analogous objects of pure symmetry protected topological (SPT) states, the ASPT possesses edge bits. In particular, the analogous operator response exists, that is, symmetry fractionalization. The fractionalization preserves the presence of the ASPT in the bulk, and the fractionalized edge operators acting on the edge bits of the ASPT. %analogous to the ones in the pure SPTs. In this work, based on the cluster model and by employing Choi mapping, we discuss generic features of the edge bits and numerically clarify the behavior of the edge bits and their robustness for varying decoherence and perturbative interactions. By using an operator-space mutual information (OSMI), we track the flow of quantum correlations between the two edges. Remarkably, even in the ASPT regime, a finite portion of the initial edge-to-edge correlation survives.
The fate of average symmetry-protected topological (ASPT) states based on the one-dimensional cluster state is investigated by using the Choi mapping, namely, the doubled Hilbert-space formalism. By introducing two effective spins, τττ and ηηη, on each doubled rung, we obtain a transparent representation of the ASPT parent Hamiltonian and systematically identify the conserved stabilizers and the corresponding orders. We further study symmetry-preserving operations that deform the ASPT by combining qualitative effective-Hamiltonian and systematic parent-Hamiltonian analyses with numerical matrix-product-state filtering calculations. We find that although the ASPT remains robust over a broad range of decoherence strengths, strong-to-weak spontaneous symmetry breaking (SWSSB) becomes exact in the maximal-decoherence limit. In addition, we consider symmetry preserving no-click postselection, which removes the exact conservation law supporting the Rényi-2 string order of the ASPT and drives the system toward weak-symmetry spontaneous symmetry breaking (WSSB). Again, the ASPT remains robust over a broad range of postselection strengths, with WSSB becoming pronounced only near the projective limit and exact at the limit itself.
Symmetry-protected topological (SPT) phases extend the Landau paradigm of quantum matter by admitting distinct symmetry-preserving phases that lack any local order parameter. Demonstrating these phases at scale on programmable quantum processors is a key milestone in using quantum hardware to probe emergent many-body phenomena, yet it is impeded by the circuit depth normally required to capture non-trivial entanglement. Here we use a tensor network based approximate quantum compiling (AQC) protocol to construct shallow quantum circuits (18-39 CNOT depth), which prepare 100-site ground states of the spin-1/2 bond-alternating Heisenberg chain in both SPT phases, to 97.9-99.0% fidelity. Upon executing the circuits on IBM quantum hardware, the resulting states exhibit all defining signatures of SPT order including non-local string order for strings of up to length 20, characteristic degeneracies in the entanglement spectrum and clear evidence of symmetry-protected edge modes. The simultaneous observation of these independent diagnostics establishes current quantum computers as versatile platforms for large-scale studies of symmetry-protected quantum matter. More broadly, our results establish a practical foundation for probing non-equilibrium quench dynamics of such systems in regimes that challenge classical computational methods.
Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra.
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.