Variational Quantum Eigensolver Simulation of a 1D Topological Insulator: Ansatz Benchmarking and Spectral Targeting on the SSH Chain

<p dir="ltr">The Su-Schrieffer-Heeger (SSH) chain provides a particularly stringent setting for variational quantum simulation because exact solvability, symmetry-protected topology, and boundary localized zero-mode physics coexist in a single analytically controlled lattice Hamiltonian. In this work, we use the open SSH chain as a one-particle benchmark for the Variational Quantum Eigensolver (VQE) and develop a two-part variational treatment that separately addresses its low-lying bulk spectrum and its topological near-zero sector. For the bulk benchmark, we restrict the SSH model to the one-particle sector, where the Hamiltonian reduces to an N times N hopping matrix that is encoded as a qubit operator for variational simulation. We compare three shallow variational families: a problem-aligned fermionic ansatz, a hardware-efficient ansatz, and a two-local baseline. Across the reported system sizes, the fermionic ansatz reproduces the benchmarked exact-diagonalization levels most faithfully, while the remaining circuits exhibit larger and more persistent deviations, particularly for the second reported level. These results demonstrate (in a controlled topological setting) that shallow variational accuracy depends strongly on whether the circuit manifold reflects the fermionic and lattice structure of the target Hamiltonian. We then turn to the central topological feature of the open SSH chain, namely the finite-size near-zero pair associated with exponentially localized edge states. Since these states are not extremal eigenvalues of the original Hamiltonian, they are not isolated naturally by VQE. To access this sector, we construct a folded-spectrum workflow based on the squared Hamiltonian, use variational minimization and deflation to identify the corresponding low-|E| subspace, and recover the signed near-zero energies by explicit orthonormalization and projection onto the original SSH Hamiltonian. The reconstructed pair agrees with the results of exact diagonalization and reproduces the expected exponential suppression of the finite-size splitting with increasing chain length. Taken together, the results establish a variational route to both the benchmarked bulk spectrum and the topological near-zero sector of a canonical one-dimensional topological Hamiltonian. More broadly, they show that in variational quantum simulation of topological systems, accuracy depends not only on ansatz compatibility with the Hamiltonian structure, but also on whether the chosen variational objective isolates the spectral feature of interest.</p>