Band structure of two-dimensional graphene and of one-dimensional nanoribbons via variational quantum algorithms: from energies to wavefunction quality

Simulating quantum many-body systems classically encounters exponential bottlenecks, which quantum devices circumvent by natively encoding wavefunctions. In the Noisy Intermediate-Scale Quantum (NISQ) era, however, hardware noise constrains simulation accuracy. As a stepping stone toward strongly correlated materials, we compute the band structure of 2D monolayer graphene and 1D nanoribbons (zigzag and armchair) using a single variational workflow in the $p_z$ tight-binding approximation. Non-interacting ground states also provide essential initial reference states for subspace and warm-started variational methods to solve interacting spectra. By mapping the Bloch Hamiltonian to Pauli operators, we extract the ground band via the Variational Quantum Eigensolver (VQE) and excited subbands via Variational Quantum Deflation (VQD), validating our approach from ideal simulations to execution on physical IBM Quantum processors. In ideal simulations, the workflow reproduces exact diagonalization results with high precision, whereas hardware noise introduces energy deviations of a few hundredths of the hopping parameter. While qubit counts scale logarithmically with ribbon width, fixed-depth ansatz accuracy degrades with Hilbert space dimension. Crucially, deepening the ansatz improves ideal simulation accuracy by over three orders of magnitude but worsens energy estimates under noise, proving that practical circuit depth is bounded by hardware noise rather than expressivity. Finally, we complement energy evaluations with a wavefunction-level diagnostic measuring subspace containment. This diagnostic captures degenerate manifolds and exposes optimization failure modes disguised by energy agreement alone. Operating directly on NISQ hardware, this validated workflow establishes a controlled baseline for studying topological and strongly correlated phases in realistic materials.