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arXiv:2603.06325 · npj Quantum Information

Preparing 100-qubit symmetry-protected topological order on a digital quantum computer

George Pennington, Kevin C. Smith, James R. Garrison, Lachlan P. Lindoy, Jason Crain, Ben Jaderberg

quant-phSubmitted Mar 6, 2026• Announced Mar 9, 2026• 0 citations
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Abstract

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.

Journal reference
npj Quantum Inf 12, 141 (2026)
DOI
10.1038/s41534-026-01334-8

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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.
Quantum simulation on near-term quantum hardware is a topic of intense interest. The preparation of novel quantum states of matter provides a quantitative assessment of the capabilities of near-term digital quantum computers to implement circuits with structure of relevance to quantum simulation. Here, we conduct a benchmark study by realizing symmetry-protected topological (SPT) phases of a spin-1/2 Hamiltonian with next-nearest-neighbor hopping on up to 11 qubits on a programmable superconducting quantum processor using adiabatic state preparation. Using recompilation techniques to reduce the gate count to around 50 two-qubit gates, we observe clear signatures of the two distinct SPT phases, such as excitations localized to specific edges and finite string-order parameters. We identify a parasitic phase associated with the two-qubit gate as the dominant imperfection that limits the depth of the circuits, indicating a research topic of interest for future hardware development.
Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, a broad class of topological states, namely, symmetry-protected topological (SPT) states, can be identified by distinctive degeneracies in their entanglement spectrum. Here, we propose and realize two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The two protocols link quantum information processing to the classification of SPT phases of matter. They invoke the creation of a cluster state and are implemented on an IBM quantum computer. The experimental findings are compared to noisy simulations, allowing us to study the stability of topological states to perturbations and noise.
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.
It is a challenging problem to correctly characterize the symmetry-protected topological (SPT) phases in open quantum systems. As the measurement-based quantum computation (MBQC) utilizes non-trivial edge states of the SPT phases as the logical qubit, its computational power is closely tied to the non-trivial topological nature of the phases. In this paper, we propose to use the gate fidelity which is a measure of the computational power of the MBQC to identify the SPT phases in mixed-state settings. Specifically, we investigate the robustness of the Haldane phase by considering the MBQC on the Affleck-Kennedy-Lieb-Tasaki state subject to different types of noises. To illustrate how our criterion works, we analytically and numerically calculated the gate fidelity to find that its behavior depends crucially on whether the noises satisfy a certain symmetry condition with respect to the on-site Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ symmetry. In particular, the fidelity for the identity gate, which is given by the sum of the non-local string order parameters, plays an important role. Furthermore, we demonstrate that a stronger symmetry conditions are required to be able to perform other (e.g., the ZZZ-rotation gate) gates with high fidelity. By examining which unitary gates can be implemented with the MBQC on the decohered states, we can gain a useful insight into the richer structure of noisy SPT states that cannot be captured solely by the string order parameters.
Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.
1+1D symmetry-protected topological (SPT) phases with invertible symmetry can be characterized by the symmetric entanglers: globally symmetric finite-depth local unitary circuits that generate representative ground states from a product state. For non-invertible symmetries, however, distinct SPT phases, such as the three Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases, may not be connected by any symmetric entangler. We show that the three Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases can nevertheless be connected by \emph{symmetry-permuting} finite-depth local unitary circuits, realizing dualities predicted by topological quantum field theory (TQFT). We construct an explicit matrix product unitary connecting two of the Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases, and show that its action on string order parameters reproduces the anyon permutation predicted by the TQFT duality.