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arxiv: 2511.21867 · v2 · submitted 2025-11-26 · 🪐 quant-ph · cs.DS· physics.chem-ph

Recognition: 2 theorem links

· Lean Theorem

Accuracy and resource advantages of quantum eigenvalue estimation with non-Hermitian transcorrelated electronic Hamiltonians

Alexey Uvarov , Artur F. Izmaylov
Authors on Pith no claims yet

Pith reviewed 2026-05-17 04:14 UTC · model grok-4.3

classification 🪐 quant-ph cs.DSphysics.chem-ph
keywords quantum eigenvalue estimationtranscorrelated Hamiltoniannon-Hermitian operatorselectronic structureT-gate countsecond-row atomsquantum resource estimation
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The pith

QEVE on transcorrelated Hamiltonians places T-gate costs between standard qubitization on cc-pVTZ and cc-pVQZ bases for second-row atoms.

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

The paper examines the resource costs of a quantum eigenvalue estimation algorithm tailored to non-Hermitian transcorrelated electronic Hamiltonians for atoms from Li to Ne. With the xTC approximation, the T-gate count in the minimal STO-6G basis sits between the costs of conventional qubitization applied to the same systems in the larger cc-pVTZ and cc-pVQZ bases. Accuracy of the resulting energies exceeds the cc-pVQZ level for Li and Be but drops below the cc-pVDZ level for O, F, and Ne. The work therefore tests whether removing the Coulomb cusp through a non-Hermitian transformation can trade basis-set size for algorithmic overhead in a way that lowers overall quantum resources.

Core claim

With the xTC approximation the T-gate count of QEVE in the minimal STO-6G basis lies between the counts of standard qubitization in the cc-pVTZ and cc-pVQZ bases, while the transcorrelated energy is more accurate than cc-pVQZ for Li and Be yet less accurate than cc-pVDZ for O, F, and Ne.

What carries the argument

The quantum eigenvalue estimation (QEVE) algorithm for non-Hermitian Hamiltonians with real spectra, applied after the transcorrelated transformation that removes the Coulomb cusp from the wave function.

If this is right

  • The combination allows target accuracies that would otherwise require larger bases to be reached with fewer quantum gates.
  • Non-Hermitian operators become practical inputs for quantum phase estimation in chemistry once a suitable eigenvalue algorithm is used.
  • Accuracy that remains competitive only for the lightest atoms indicates that basis-set reduction via transcorrelation works best when the cusp removal dominates the error.

Where Pith is reading between the lines

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

  • The same resource comparison could be repeated for small molecules to check whether the T-gate advantage survives when nuclear geometry and multiple centers are added.
  • If the accuracy drop for heavier atoms is tied to the xTC approximation, replacing it with a more faithful transcorrelated form might restore the accuracy edge without raising the gate count.
  • The reported T-gate numbers provide a concrete benchmark that other non-Hermitian quantum algorithms can be measured against on the same atomic test set.

Load-bearing premise

The xTC approximation captures the main benefits of the full transcorrelated transformation without introducing large uncontrolled errors, and QEVE incurs no hidden overheads beyond the reported T-gate counts on these Hamiltonians.

What would settle it

A direct count of T gates required to run QEVE on an xTC Hamiltonian in STO-6G that exceeds the T-gate count of standard qubitization on the same atom in cc-pVTZ would falsify the claimed resource advantage.

read the original abstract

In electronic structure calculations, the transcorrelated method consists in transforming the Hamiltonian so as to remove the Coulomb cusp in its eigenfunctions. As a result, the wavefunction can be described more accurately without increasing the size of the basis set. However, the transcorrelated Hamiltonian is non-Hermitian and non-normal, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation algorithm (QEVE) was proposed for non-Hermitian Hamiltonians with real spectra [FOCS 65, 1051 (2024)]. Although the asymptotic scaling of this algorithm with the desired accuracy is shown to be optimal, the constant factor in its complexity scaling has not been analyzed. Here we investigate the cost of QEVE applied to transcorrelated electronic Hamiltonians of second-row atoms and compare it to the cost of applying standard qubitization to non-transcorrelated Hamiltonians. We find that, with the xTC approximation, the T gate count of QEVE in the minimal STO-6G basis is between those of standard qubitization in the cc-pVTZ and cc-pVQZ bases. The accuracy of the transcorrelated energy differs between systems: for Li and Be, it is more accurate than the cc-pVQZ energy, while for larger atoms, the error gradually increases, exceeding the cc-pVDZ level for O, F, and Ne.

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 / 1 minor

Summary. The manuscript investigates the cost of applying the recently proposed quantum eigenvalue estimation (QEVE) algorithm to non-Hermitian transcorrelated electronic Hamiltonians for second-row atoms, using the xTC approximation in the minimal STO-6G basis. It compares the resulting T-gate counts to those obtained by applying standard qubitization to the corresponding non-transcorrelated Hamiltonians in the larger cc-pVTZ and cc-pVQZ bases. The central numerical finding is that the T-gate count for QEVE+xTC in STO-6G lies between the qubitization costs for cc-pVTZ and cc-pVQZ, while the transcorrelated energies are more accurate than cc-pVQZ for Li and Be but show progressively larger errors that exceed the cc-pVDZ level for O, F, and Ne.

Significance. If the T-gate counts are correctly derived and the xTC approximation faithfully captures the benefits of the full transcorrelated transformation without uncontrolled errors, the work would provide concrete evidence that non-Hermitian quantum algorithms combined with transcorrelation can reduce the effective basis-set size and thereby lower gate counts for quantum chemistry simulations on early fault-tolerant devices. The explicit numerical comparisons for a sequence of atoms constitute a useful benchmark that could guide further algorithm development, provided the accuracy-resource trade-off is analyzed at fixed chemical accuracy.

major comments (2)
  1. [Abstract / Results] Abstract and results section: The headline resource claim states that the T-gate count of QEVE in the minimal STO-6G basis with xTC lies between those of standard qubitization in cc-pVTZ and cc-pVQZ. However, the accuracy data show that transcorrelated errors remain below the cc-pVQZ level only for Li and Be and exceed the cc-pVDZ level for O, F, and Ne. Because the comparison is not performed at equivalent accuracy, the reported T-count advantage is not uniform across the atom series; for the heavier atoms a larger basis would likely be required to reach chemically relevant accuracy, which would increase the effective T-count and weaken the central claim.
  2. [Methods] Methods / resource estimation: The manuscript reports specific T-gate counts for QEVE on the xTC Hamiltonians but does not provide the explicit derivation, including how the block-encoding costs, phase estimation precision, and any overheads from the non-Hermitian structure were obtained or verified against full transcorrelated results. Without these details the quantitative resource-accuracy comparison cannot be independently assessed.
minor comments (1)
  1. [Introduction / Methods] The definition and implementation details of the xTC approximation should be stated more explicitly, including any additional approximations introduced relative to the full transcorrelated Hamiltonian.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment point by point below, providing clarifications and indicating revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: [Abstract / Results] Abstract and results section: The headline resource claim states that the T-gate count of QEVE in the minimal STO-6G basis with xTC lies between those of standard qubitization in cc-pVTZ and cc-pVQZ. However, the accuracy data show that transcorrelated errors remain below the cc-pVQZ level only for Li and Be and exceed the cc-pVDZ level for O, F, and Ne. Because the comparison is not performed at equivalent accuracy, the reported T-count advantage is not uniform across the atom series; for the heavier atoms a larger basis would likely be required to reach chemically relevant accuracy, which would increase the effective T-count and weaken the central claim.

    Authors: We agree that the accuracy of the xTC energies is not uniform and varies across the series, which is already stated explicitly in the manuscript. The central resource claim concerns the T-gate count required for the QEVE+xTC calculation in the STO-6G basis, presented alongside the corresponding accuracy for each atom; it does not assert that chemical accuracy is achieved uniformly or that the T-count advantage holds after basis enlargement. To address the concern, we have added a dedicated paragraph in the Results section that discusses the accuracy-resource trade-off at fixed chemical accuracy and notes that, for O, F, and Ne, reaching cc-pVDZ-level accuracy with xTC would require a modestly enlarged basis whose T-count impact can be estimated from the existing scaling data. This revision makes the limitations for heavier atoms more prominent while preserving the reported numerical comparison for the minimal-basis case. revision: partial

  2. Referee: [Methods] Methods / resource estimation: The manuscript reports specific T-gate counts for QEVE on the xTC Hamiltonians but does not provide the explicit derivation, including how the block-encoding costs, phase estimation precision, and any overheads from the non-Hermitian structure were obtained or verified against full transcorrelated results. Without these details the quantitative resource-accuracy comparison cannot be independently assessed.

    Authors: We thank the referee for highlighting the need for greater transparency. The revised Methods section now contains the explicit step-by-step derivation of the T-gate counts: the block-encoding cost for the non-Hermitian xTC operator (including the additional terms arising from the similarity transformation), the choice of phase-estimation precision sufficient for the target energy accuracy, and the overhead factor associated with the non-normal spectrum. We also include a short subsection comparing the xTC resource estimates to those obtained from a limited set of full transcorrelated calculations on smaller systems, confirming that the xTC approximation introduces only controlled additional cost. These additions enable independent verification of the reported numbers. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct numerical gate-count and energy comparisons

full rationale

The paper reports explicit T-gate counts obtained by applying the QEVE algorithm to xTC Hamiltonians in the STO-6G basis and compares those counts to standard qubitization costs in cc-pVTZ and cc-pVQZ bases. Accuracy is assessed by direct subtraction of transcorrelated energies from reference values in progressively larger bases. These are empirical measurements and tabulated comparisons; no derivation step equates an output to a fitted input, renames a known result, or reduces the central claim to a self-citation chain. The single citation to the prior QEVE proposal supplies the algorithm definition but is not invoked as a uniqueness theorem that forces the reported resource ordering.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the xTC approximation and the assumption that QEVE applies without extra costs to these non-normal operators; no new entities are postulated.

axioms (2)
  • domain assumption Transcorrelated Hamiltonians possess real spectra suitable for QEVE
    Invoked when applying the non-Hermitian eigenvalue estimation algorithm to the transformed electronic Hamiltonians.
  • standard math Standard qubitization costs scale as reported for the chosen basis sets
    Used as the baseline for resource comparison.

pith-pipeline@v0.9.0 · 5565 in / 1391 out tokens · 71812 ms · 2026-05-17T04:14:18.644969+00:00 · methodology

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