  The AKLT state is the ground state of an isotropic quantum Heisenberg
spin-$1$ model. It exhibits an excitation gap and an exponentially decaying
correlation function, with fractionalized excitations at its boundaries. So
far, the one-dimensional AKLT model has only been experimentally realized with
trapped-ions as well as photonic systems. In this work, we successfully
prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era
quantum device for the first time. In particular, we developed a
non-deterministic algorithm on the IBM quantum processor, where the non-unitary
operator necessary for the AKLT state preparation is embedded in a unitary
operator with an additional ancilla qubit for each pair of auxiliary
spin-1/2's. Such a unitary operator is effectively represented by a
parametrized circuit composed of single-qubit and nearest-neighbor $CX$ gates.
Compared with the conventional operator decomposition method from Qiskit, our
approach results in a much shallower circuit depth with only nearest-neighbor
gates, while maintaining a fidelity in excess of $99.99\%$ with the original
operator. By simultaneously post-selecting each ancilla qubit such that it
belongs to the subspace of spin-up $|\uparrow \rangle$, an AKLT state can be
systematically obtained by evolving from an initial trivial product state of
singlets plus ancilla qubits in spin-up on a quantum computer, and it is
subsequently recorded by performing measurements on all the other physical
qubits. We show how the accuracy of our implementation can be further improved
on the IBM quantum processor with readout error mitigation.
