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

TL;DR

This paper introduces a quantum algorithm combining VQE and NLCEs to compute quasi-particle excitations in the thermodynamic limit, with a novel post-processing step ensuring cluster additivity for improved convergence.

Contribution

The authors develop a cluster-additive post-processing method (PCAT) that extends variational quantum algorithms to excited states in the thermodynamic limit, demonstrated with VQE and NLCEs.

Findings

Successfully computed quasi-particle dispersions in the transverse-field Ising model.

Matched exact diagonalization results with fewer layers in pure TFIM.

Improved convergence with increased cluster size and layers in models with symmetry breaking.

Abstract

We propose a quantum algorithm for computing one quasi-particle 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 post-processed 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 quasi-particle 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, $\lceil N/2 \rceil$ layers suffice: NLCE+VQE matches exact diagonalization. For TFIM with longitudinal field, where parity symmetry breaks and PCAT becomes essential, both $\lceil N/2 \rceil$ and $N$ layers 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 post-processing approach, which requires only low-energy eigenspace information, applies to any quantum eigenstate preparation method.