Quantum algorithm for one-quasiparticle excitations in the thermodynamic limit via cluster-additive block diagonalization

We propose a quantum algorithm for computing one-quasiparticle excitation energies in the thermodynamic limit by combining numerical linked-cluster expansions (NLCEs) and the variational quantum eigensolver (VQE). Our approach uses VQE to block diagonalize the cluster Hamiltonian through a single-unitary transformation. This unitary is then postprocessed using the projective cluster-additive transformation (PCAT) to ensure cluster additivity, a key requirement for NLCE convergence. We benchmark our method on the transverse-field Ising model (TFIM) in one and two dimensions, and with longitudinal field, computing one-quasiparticle dispersions in the high-field polarized phase. We compare two cost function classes, trace minimization and variance based, demonstrating their effectiveness with the Hamiltonian variational ansatz (HVA). For pure TFIM, ⌈ N / 2 ⌉ layers of HVA suffice: NLCE + VQE matches exact diagonalization. For TFIM with longitudinal field, where parity symmetry breaks and PCAT becomes essential, both ⌈ N / 2 ⌉ and N layers of HVA converge with increasing cluster size, with N layers providing improved accuracy. Our results establish PCAT as a cluster-additive framework that extends variational quantum algorithms to excited-state calculations in the thermodynamic limit via NLCE. While demonstrated with VQE, the PCAT postprocessing approach, which requires only low-energy eigenspace information, applies to any quantum eigenstate preparation method.