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Tensor-network (TN) states are efficient parametric representations of ground states of local quantum Hamiltonians extensively used in numerical simulations. Employing TN Ansatz states directly on a quantum simulator can potentially offer an exponential computational advantage over purely numerical simulation. We implement a quantum-encoded TN Ansatz state using a variational quantum eigensolver on an ion-trap quantum computer that approximates the ground states of the extended Su-Schrieffer-Heeger model. The generated states are characterized by estimating the topological invariants, verifying their topological order. Our TN encoding as a trapped-ion circuit employs only single-site optical pulses—the native operations naturally available on the platform. We reduce nearest-neighbor crosstalk by selecting different magnetic sublevels with well-separated transition frequencies to encode the qubits in neighboring ions.