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arxiv: 2605.28599 · v1 · pith:OUYEI3ZRnew · submitted 2026-05-27 · 🪐 quant-ph · cond-mat.str-el

Thermodynamic-limit dispersion relations on trapped-ion quantum hardware

Lucas Marti , Sumeet , Stefan Wolf , K. P. Schmidt , Michael J. Hartmann This is my paper

Pith reviewed 2026-06-29 12:16 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords numerical linked-cluster expansionquantum algorithmstrapped-ion quantum processorthermodynamic limittransverse-field Ising modelprojective cluster-additive transformationadiabatic state preparationvariational quantum eigensolver
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The pith

A 20-qubit trapped-ion processor combined with linked-cluster expansion computes ground-state energies and quasi-particle dispersions in the thermodynamic limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates a hybrid numerical linked-cluster expansion with quantum algorithms (NLCE+QA) that extracts thermodynamic-limit properties from calculations performed on small clusters. This is applied to the transverse-field Ising model in one dimension, on a ladder geometry, and with an added longitudinal field, using a 20-qubit trapped-ion device. State preparation is done via adiabatic methods or variational optimization trained classically, with measurements taken through a custom CX-test. The work tests whether hardware expectation values can tolerate the matrix inversion and square-root steps in projective cluster-additive transformation (PCAT) post-processing, and identifies the regimes now reachable along with the improvements still required.

Core claim

NLCE+QA extracts thermodynamic-limit ground-state energies and one-quasi-particle dispersions from small-cluster calculations on a 20-qubit trapped-ion QPU. Projector-based block-diagonalization via PCAT is performed after adiabatic state preparation or VQE, with final expectation values obtained from the device using the CX-test. The method is shown to function for the transverse-field Ising model in one dimension, on ladders, and in a longitudinal field, establishing that current hardware noise levels permit the required non-linear classical post-processing for these cases.

What carries the argument

The NLCE+QA framework, which combines quantum state preparation and measurement on small clusters with classical linked-cluster expansion and PCAT block-diagonalization to reach thermodynamic-limit quantities.

If this is right

  • Thermodynamic-limit ground-state energies and quasi-particle dispersions become accessible from small-cluster quantum calculations without simulating the full infinite system.
  • The NLCE+QA approach succeeds for the transverse-field Ising model in one dimension, on ladder geometries, and with longitudinal fields.
  • A CX-test provides a practical alternative to the Hadamard test for extracting the required expectation values on the trapped-ion device.
  • Current noise levels on 20-qubit trapped-ion hardware allow the non-linear post-processing steps in PCAT for the models examined.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If noise tolerance improves, the same hybrid expansion could be applied to models whose classical linked-cluster expansions become prohibitive at larger cluster sizes.
  • Better error mitigation on future devices would permit larger clusters inside the expansion and therefore tighter control over extrapolation errors.
  • The demonstrated noise resilience of the post-processing pipeline indicates that hybrid quantum-classical schemes of this type may reach classically inaccessible regimes sooner than full quantum simulation of large lattices.

Load-bearing premise

Expectation values supplied by the quantum processor remain accurate enough after the matrix inversion and square-root operations of the projective cluster-additive transformation.

What would settle it

A computed ground-state energy or dispersion relation for the transverse-field Ising model that deviates from the known exact thermodynamic-limit value by an amount traceable to amplified measurement noise in the PCAT step would show the hardware is not yet accurate enough.

Figures

Figures reproduced from arXiv: 2605.28599 by K. P. Schmidt, Lucas Marti, Michael J. Hartmann, Stefan Wolf, Sumeet.

Figure 1
Figure 1. Figure 1: A layer of the HVA Ansatz, for a 3-qubit TFIM [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The CX-test. In the second register, the controlled [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Interplay between shot noise and depolarization noise when computing the dispersion curve, using noisy simulation, [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Number of unique circuits to compute the full [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 1QP Dispersion for the one-dimensional TFIM [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 1QP dispersion for the one-dimensional TFIM [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean dispersion error, |ωNLCE+QA(k) − ωNLCE+ED(k)| averaged over k, as a function of J/h. ASP chain: NLCE+ASP for a TFIM chain with Nmax = 5 at J/h = 0.3, 0.8. VQE chain: NLCE+VQE for a TFIM chain with Nmax = 5 at J/h = 0.3, 0.8 and at the critical point at J/h = 1. VQE LF chain: NLCE+VQE for the TFIM+LF chain with Nmax = 5 at hl = 0.1 and J/h = 0.5. VQE ladder: NLCE+VQE for the TFIM on a ladder geometry a… view at source ↗
Figure 8
Figure 8. Figure 8: 1QP dispersion obtained using NLCE+VQE for the [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 1QP dispersion for the one-dimensional TFIM [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: 1QP dispersion relation for the one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Noisy simulations for multiple shot counts and gate-level depolarization rate, as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: 1QP dispersion of the quasi one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Normalized moduli of the elements of the [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: Convergence of the mean error-on-the-mean [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The Hadamard test. The first register, denoted by [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Same setting as in Fig [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces NLCE+QA, combining numerical linked-cluster expansions with quantum algorithms to extract thermodynamic-limit ground-state energies and single-quasiparticle dispersions for the transverse-field Ising model (TFIM) on 1D, ladder, and longitudinal-field geometries. Computations use a 20-qubit trapped-ion QPU with adiabatic state preparation (ASP) or classically trained VQE, PCAT block-diagonalization, and a novel CX-test for expectation values; the central question addressed is whether current hardware expectation values remain usable after the noise-amplifying matrix inversion and square-root steps inside PCAT.

Significance. If the hardware-derived values prove stable under PCAT post-processing, the work would constitute a concrete experimental demonstration that near-term QPUs can contribute to thermodynamic-limit many-body calculations that are otherwise limited by cluster size. The explicit focus on error amplification in projector schemes and the introduction of the CX-test are positive steps toward reproducible quantum-assisted NLCE.

major comments (2)
  1. [Abstract / PCAT section] Abstract and § on PCAT: the central claim that QPU expectation values survive the non-linear operations requires a quantitative error budget showing how shot noise and gate errors propagate through matrix inversion and square-root steps into the linked-cluster series; without explicit stability tests or bounds on the extrapolated energies/dispersions, the thermodynamic-limit results rest on an unverified assumption.
  2. [Results (TFIM ladder / longitudinal field)] Results for TFIM ladder and longitudinal-field cases: the manuscript must include direct comparisons of QPU+PCAT dispersions against exact or high-precision classical benchmarks, with error bars from finite shots; absent these, the claim that hardware values remain usable after non-linear post-processing cannot be assessed.
minor comments (2)
  1. [Methods / CX-test] Clarify the precise definition and circuit implementation of the CX-test relative to the standard Hadamard test, including any additional assumptions on state preparation.
  2. [NLCE setup] Provide the cluster sizes used in the NLCE and the truncation order of the linked-cluster series for each geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback on the need for explicit error analysis. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / PCAT section] Abstract and § on PCAT: the central claim that QPU expectation values survive the non-linear operations requires a quantitative error budget showing how shot noise and gate errors propagate through matrix inversion and square-root steps into the linked-cluster series; without explicit stability tests or bounds on the extrapolated energies/dispersions, the thermodynamic-limit results rest on an unverified assumption.

    Authors: We agree that a quantitative error budget would strengthen the central claim regarding stability under PCAT post-processing. In the revised manuscript we will add an appendix containing both analytical estimates of error propagation through the matrix inversion and square-root steps and numerical simulations that inject realistic shot noise and gate errors into the expectation values before PCAT. These will be used to bound the uncertainty in the final linked-cluster coefficients and extrapolated thermodynamic-limit energies and dispersions. revision: yes

  2. Referee: [Results (TFIM ladder / longitudinal field)] Results for TFIM ladder and longitudinal-field cases: the manuscript must include direct comparisons of QPU+PCAT dispersions against exact or high-precision classical benchmarks, with error bars from finite shots; absent these, the claim that hardware values remain usable after non-linear post-processing cannot be assessed.

    Authors: We will add the requested direct comparisons in the revised manuscript. For the ladder and longitudinal-field geometries we will present QPU+PCAT dispersions side-by-side with exact or high-precision classical benchmarks (obtained from exact diagonalization on small clusters and classical NLCE extrapolations), together with error bars obtained by propagating the finite-shot statistical uncertainties through the PCAT operations and the subsequent linked-cluster extrapolation. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental hardware demonstration of NLCE+QA

full rationale

The paper presents an empirical demonstration of running numerical linked-cluster expansion with quantum algorithms (NLCE+QA) on a 20-qubit trapped-ion QPU to extract thermodynamic-limit ground-state energies and dispersions for the TFIM. The workflow applies established projector-based methods (PCAT) and state-preparation techniques (ASP, classically-trained VQE) plus a new CX-test for measurements; none of these steps reduce a claimed result to its own inputs by construction, rename a fitted quantity as a prediction, or rest on a load-bearing self-citation chain. The central claim is the feasibility of the hardware pipeline under realistic noise, which is an external, falsifiable assertion rather than a self-referential definition. No equations or sections exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5765 in / 1101 out tokens · 31306 ms · 2026-06-29T12:16:50.463881+00:00 · methodology

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Reference graph

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    TFIM chain with longitudinal field The TFIM with longitudinal field (TFIM+LF), whose Hamiltonian is given in Eq. (24), is the first benchmark requiring the full PCAT correction. The longitudinal field breaks theZ 2 parity symmetry, coupling the ground state to the 1QP sector and producing non-vanishing overlaps⟨Φ [0]|Ψ[1] i ⟩ ̸= 0. As discussed in Sec. II...

  55. [55]

    The Hamiltonian for this model is given by Eq

    TFIM on a ladder Finally, we turn to the TFIM on a two-leg ladder, the simplest two-dimensional geometry accessible with our cluster sizes. The Hamiltonian for this model is given by Eq. (24) forh l = 0, with the nearest-neighbor cou- plings along both rungs and legs of the ladder. Unlike the one-dimensional TFIM, the ladder does not have an exact analyti...

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    14 the matrix elements computed using the CX-test, that is the ˜Omatrix elements

    Matrix elements To give a sense of the difficulty in inverting the ma- trices of all the models that we have studied so far, we show in Fig. 14 the matrix elements computed using the CX-test, that is the ˜Omatrix elements. As we get closer to criticality, the matrix gets denser and further away from an easy inversion, since it grows out from the diag- ona...