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arXiv:2605.28599

Thermodynamic-limit dispersion relations on trapped-ion quantum hardware

Lucas Marti, Sumeet, Stefan Wolf, K. P. Schmidt, Michael J. Hartmann

quant-phcond-mat.str-elSubmitted May 27, 2026• Announced May 28, 2026• 0 citations
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Abstract

We run a numerical linked-cluster expansion with a quantum algorithm (NLCE+QA), computing ground-state energies and one quasi-particle dispersions in the thermodynamic limit using a 20-qubit trapped-ion quantum processing unit (QPU). The NLCE+QA framework extracts thermodynamic-limit properties from small-cluster calculations, making it naturally suited for near-term quantum devices. Projector-based block-diagonalization schemes such as projective cluster-additive transformation (PCAT) are essential to NLCE+QA, and they involve matrix inversion and square root operations that amplify measurement noise. A central question is therefore whether current hardware can provide expectation values that are accurate enough to withstand non-linear classical post-processing. We explore this challenge for the transverse-field Ising model (TFIM) in one dimension, on a ladder geometry, as well as in a longitudinal field in one dimension. For the quantum algorithm, we consider adiabatic state preparation (ASP), as well as a variational quantum eigensolver (VQE) trained on a classical device. The final expectation values are obtained from the QPU, using a novel alternative to the Hadamard test that we name the CX-test. We explore the regimes currently attainable on quantum devices and comment on the improvements needed for quantum computers to achieve results beyond classical reach.

Comments
15+6 pages, 7+11 figures

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We run a numerical linked-cluster expansion with a quantum algorithm (NLCE+QA), computing ground-state energies and one quasi-particle dispersions in the thermodynamic limit using a 20-qubit trapped-ion quantum processing unit (QPU). The NLCE+QA framework extracts thermodynamic-limit properties from small-cluster calculations, making it naturally suited for near-term quantum devices. Projector-based block-diagonalization schemes such as projective cluster-additive transformation (PCAT) are essential to NLCE+QA, and they involve matrix inversion and square root operations that amplify measurement noise. A central question is therefore whether current hardware can provide expectation values that are accurate enough to withstand non-linear classical post-processing. We explore this challenge for the transverse-field Ising model (TFIM) in one dimension, on a ladder geometry, as well as in a longitudinal field in one dimension. For the quantum algorithm, we consider adiabatic state preparation (ASP), as well as a variational quantum eigensolver (VQE) trained on a classical device. The final expectation values are obtained from the QPU, using a novel alternative to the Hadamard test that we name the CX-test. We explore the regimes currently attainable on quantum devices and comment on the improvements needed for quantum computers to achieve results beyond classical reach.
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Quantum computers are an ideal platform to study the ground state properties of strongly correlated systems due to the limitation of classical computing techniques particularly for systems exhibiting quantum phase transitions. While the error rates of Noisy Intermediate-Scale Quantum (NISQ) computers are still high, simulating strongly correlated systems on such devices and extracting information of possible phases may be within reach. The frustrated transverse-field Ising model (TFIM) is such a system with multiple ordered magnetic phases. In this study, we simulate a two-dimensional frustrated TFIM with next-nearest-neighbor spin-exchange interactions at zero temperature. The competition between the nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic coupling gives rise to frustration in the system. Moreover, the presence of quantum fluctuations makes the ground-state phase profile even richer. We use the Variational Quantum Eigensolver (VQE) to compute the phases on a square lattice with periodic boundary conditions for a system of 16 sites (qubits). The trained VQE circuits are compared to exact diagonalization, allowing us to extract error measures of VQE. We focus on the ground-state phase transitions of this model, where VQE succeeds in finding the dominant magnetic phases. The optimized VQE circuits are then executed on the Quantinuum H1-1 trapped-ion quantum computer without using any error mitigation techniques. Our experiments show near perfect recovery of the magnetic phases of the frustrated model through ground-state energy, the energy derivative, and the spin correlation functions. Thus, we show that the trapped-ion quantum processor is able to achieve reliable simulations of a strongly correlated system within the limitations of the VQE approach.
Calculating the equilibrium properties of condensed-matter systems is one of the promising applications of near-term quantum computing. Recently, hybrid quantum-classical time-series algorithms have been proposed to efficiently extract these properties from a measurement of the Loschmidt amplitude ⟨ ψ | e − i H ^ t | ψ ⟩ from initial states | ψ ⟩ and a time evolution under the Hamiltonian H ^ up to short times t . In this work, we study the operation of this algorithm on a present-day quantum computer. Specifically, we measure the Loschmidt amplitude for the Fermi-Hubbard model on a 16 -site ladder geometry (32 orbitals) on the Quantinuum H2-1 trapped-ion device. We assess the effect of noise on the Loschmidt amplitude and implement algorithm-specific error-mitigation techniques. By using a thus-motivated error model, we numerically analyze the influence of noise on the full operation of the quantum-classical algorithm by measuring expectation values of local observables at finite energies. Finally, we estimate the resources needed for scaling up the algorithm. Published by the American Physical Society 2024
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