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

TL;DR

This work develops a hybrid quantum-classical method to compute one quasi-particle excitations in the thermodynamic limit by marrying numerical linked-cluster expansions (NLCEs) with the variational quantum eigensolver (VQE). A single-unitary transformation from VQE decouples the ground-state and 1QP subspace on each cluster, and the projective cluster-additive transformation (PCAT) post-process ensures cluster additivity, enabling NLCE convergence for excited states. Benchmarking on TFIM in 1D and 2D, with and without a longitudinal field, shows that NLCE+VQE with Hamiltonian variational ansatz (HVA) layers of roughly $\lceil N/2\rceil$ can reproduce exact diagonalization results for pure TFIM and approach ED/series benchmarks when a longitudinal field is present, with PCAT necessary for additivity in the symmetry-broken case. The approach generalizes beyond VQE, as PCAT relies only on low-energy eigenspace information, suggesting applicability to other eigenstate preparation methods and providing a practical pathway toward excited-state calculations in the thermodynamic limit on near-term and fault-tolerant quantum hardware.

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.

Figures (14)

• Figure 1: Workflow of the NLCE+VQE approach for computing 1QP excitation energies in the thermodynamic limit. The six-step procedure combines VQE cluster solving with PCAT post-processing and NLCE embedding to obtain the dispersion $\omega(\vec{k})$ in the thermodynamic limit.
• Figure 2: Diagrammatic view of a block-diagonal Hamiltonian. The ground-state ($0$QP) block is shown in dark green and the $1$QP block is shown in dark blue. The light shaded blocks are off-diagonal terms that are minimized during block-diagonalization.
• Figure 3: Rectangular graph expansion for the square lattice, showing clusters of extension $L_m \times L_n$ with $L_m \cdot L_n \leq N_{\mathrm{max}}$. Some examples for the clusters are shown.
• Figure 4: Comparison of Schrieffer-Wolff (SW) and PCAT transformations on disconnected clusters. The SW transformation allows unphysical hopping of a dressed quasi-particle (yellow) between clusters $A$ and $B$ that share no bonds. PCAT forbids such hopping, ensuring cluster additivity.
• Figure 5: Quantum circuit representation of one layer (green box) of the HVA for the one-dimensional TFIM+LF model. The blue, orange, and red blocks implement $e^{\mathrm{i}\sum_{\langle\nu,\mu\rangle}\theta^{XX}_{\nu,\mu}X_{\nu} X_{\mu}}$, $e^{\mathrm{i}\sum_{\nu}\theta^{X}_{\nu} X_{\nu}}$, and $e^{\mathrm{i}\sum_{\nu}\theta^{Z}_{\nu} Z_{\nu}}$, respectively, showing the XX-X-Z ordering within each layer.