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

Nitish Gupta, Prahlad Madhavan Veervalli, Saksham Aryan, Abhijeet Kumar, Ardra Muriyankandathil, Anjani Tiwari

Published Jul 17, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1364/opticaopen.33005459

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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.
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.
We study the effects of electron-electron interactions on the charge excitation spectrum of the spinful Su-Schrieffer-Heeger (SSH) model, a prototype of a one-dimensional bulk obstructed topological insulator. In view of recent progress in the fabrication of dopant-based quantum simulators we focus on experimentally detectable signatures of interacting topology in finite lattices. To this end we use Lanczos-based exact diagonalization to calculate the single-particle spectral function in real space which generalizes the local density of states to interacting systems. Its spatial and spectral resolution allows for the direct investigation and identification of edge states. By studying the noninteracting limit, we demonstrate that the topological in-gap states on the boundary are robust against both finite-size effects as well as random bond and onsite disorder which suggests the feasibility of simulating the SSH model in engineered dopant arrays in silicon. While edge excitations become zero-energy spinlike for any finite interaction strength, our analysis of the spectral function shows that the single-particle charge excitations are gapped out on the boundary. Despite the loss of topological protection, we find that these edge excitations are quasiparticlelike as long as they remain within the bulk gap. Above a critical interaction strength of ${U}_{c}\ensuremath{\approx}5t$ these quasiparticles on the boundary lose their coherence which is explained by the merging of edge and bulk states. This is in contrast to the many-body edge excitations which survive the limit of strong coupling, as established in the literature. Our findings show that for moderate repulsive interactions the nontrivial phase of the interacting SSH model can be detected through remnant signatures of topological single-particle states using single-particle local measurement techniques such as scanning tunneling spectroscopy.
We investigate the use of variational quantum algorithms to prepare and characterize fractional quantum Hall states on near-term quantum processors. Focusing on the ν=1/3ν=1/3ν=1/3 Laughlin phase described by the V1V_1V1​ Haldane pseudopotential, we formulate the lowest-Landau-level problem in second quantization, and implement particle-number-preserving variational circuits combined with the variational quantum eigensolver (VQE) and variational quantum deflation (VQD). We benchmark the approach in two complementary geometries: Haldane sphere and torus shape. On the Haldane sphere, the target state is a unique zero-energy Laughlin ground state, providing a controlled test of the variational workflow and of excited-state reconstruction. On the torus, the problem retains the genuinely two-dimensional periodic character of the quantum Hall liquid and exhibits the threefold topological ground-state degeneracy expected for the ν=1/3ν=1/3ν=1/3 fractional filling factor. This feature makes the torus a more demanding benchmark than the quasi-one-dimensional cylinder or thin-torus limits commonly exploited in state-preparation quantum protocols. We benchmark the hardware-optimized variational states against exact diagonalization using energy estimates, error-mitigated observables, and subspace-containment diagnostics. Our results show that hybrid quantum algorithms can approximately reconstruct the low-energy structure of small fractional quantum Hall systems, including the topological ground-state manifold on the torus. Beyond serving as a benchmark for quantum hardware, this geometry-resolved approach provides a route toward quantum simulations of fractional Chern insulators and strongly correlated topological phases in realistic two-dimensional materials.
We propose an efficient circuit structure of variational quantum circuit Ans\"atze used for the variational quantum eigensolver (VQE) algorithm in calculating gapped topological phases on the currently feasible noisy intermediate-scale quantum computers. An efficient circuit Ansatz should include two layers: the initialization layer and the variational layer. In the initialization layer, a fixed-depth circuit state with a compatible entanglement structure to the target topological phase is constructed. The circuit state is further adjusted subsequently to capture the details of the local correlations, which is dictated with the Hamiltonian, in the parametrized variational layer. Based on this strategy, we design a circuit Ansatz to investigate the symmetry-protected topological Haldane phase in a nonexactly solvable alternating spin-12\frac{1}{2}21​ Heisenberg chain by VQE calculations. Main characterizations of the Haldane phase, including the long-ranged string order, the fourfold nearly degenerate ground states associated with four different localized edge-mode patterns for the system with open boundaries, and the twofold degeneracy of the entanglement spectrum, are all observed for the optimized shallow circuit state with only one-depth variational layer both in numerical simulations and on real quantum computers. We further demonstrate that the computational capacity (i.e., expressibility) of this quantum circuit Ansatz is determined not by the system size but only by the intrinsic correlation length of the system, thus implying that the scalable VQE calculation is possible.
Variational quantum algorithms are among the most promising approaches for simulating interacting quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. However, the practical success of variational quantum eigensolvers (VQE) critically depends on the structure of the chosen variational ansatz. In this work, we investigate the ground-state properties of the one-dimensional antiferromagnetic Heisenberg spin-1/2 chain using both generic hardware-efficient ansatz and physics-informed, symmetry-preserving variational circuits. We benchmark variational results against exact diagonalization and noiseless simulations, and subsequently validate the approach on real IQM Garnet quantum hardware. Our results demonstrate that incorporating physical symmetries into the circuit design leads to significantly improved energy estimates, enhanced robustness against hardware noise, and clearer convergence behavior when compared to hardware-efficient ansatz under identical resource constraints. These findings highlight the importance of problem specific ansatz construction for reliable quantum simulations in the NISQ era.
We propose a momentum-space based variational quantum eigensolver (VQE) framework for simulating quasiparticle excitations in interacting quantum many-body systems on near-term quantum devices. Leveraging translational invariance and other symmetries of the Hamiltonian, we reconstruct the momentum-resolved quasiparticle excitation spectrum through targeted simulation of low-lying excited states using VQE. We construct a translationally symmetric variational ansatz designed to evolve a free-fermion particle-hole excited state with definite momentum qqq to an excited state of the interacting system at the same momentum, employing a fermionic fast Fourier transform (FFFT) circuit coupled to a Hamiltonian Variational Ansatz (HVA) circuit. Even though the particle number is not explicitly conserved in the variational ansatz, the correct quasiparticle state is reached by energetic optimization. We benchmark the performance of the proposed VQE implementation on the XXZ Hamiltonian, which maps onto the Tomonaga-Luttinger liquid in the fermionic representation. Our numerical results show that VQE can capture the low-lying excitation spectrum of the bosonic quasiparticle/two-spinon dispersion of this model at various interaction strengths. We estimate the renormalized velocity of the quasiparticles by calculating the slope of the dispersion near zero momentum using the VQE-optimized energies at different system sizes, and demonstrate that it closely matches theoretical results obtained from Bethe ansatz. Finally, we highlight extensions of our proposed VQE implementation to simulate quasiparticles in other interacting quantum many-body systems.