Symmetry-protected topological order in a 100-site spin chain on a digital quantum computer

Abstract

Symmetry-protected topological (SPT) systems extend the Landau paradigm of quantum matter by admitting distinct phases that lack local order parameters, making them challenging to characterise using conventional experimental probes. While material platforms provide important evidence for SPT order through indicative signatures such as boundary states, experimental access is typically limited to such partial probes rather than fully diagnostic measurements of the underlying quantum state. Programmable quantum processors offer a powerful complementary approach, enabling direct access to non-local properties of many-body wavefunctions. However, preparing such states on digital hardware at scales sufficient to avoid finite-size effects is typically limited by the circuit depth 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) that prepare 100-site ground states of the spin-1/2 bond-alternating Heisenberg chain across distinct SPT phases with 97.9–99.0% fidelity. Executing these circuits on IBM quantum hardware, we directly extract multiple non-local diagnostics of SPT order, including string order for lengths up to 20, persisting well beyond the decay of conventional two-point correlations, characteristic features of the entanglement spectrum, and symmetry-protected edge modes. The simultaneous observation of these independent diagnostics provides a scalable and programmable approach to preparing and characterising SPT phases on quantum processors. More broadly, this establishes digital quantum devices as flexible platforms for studying complex quantum matter and provides a practical foundation for exploring non-equilibrium dynamics in regimes that challenge classical computational methods.