Ground-State Energy Estimation of HeH$^{+}$, ArH$^{+}$, and H$_2$O via Sample-Based Quantum Diagonalization

Chae-Hyun Yoon; Jubin Park; MinKyu Lee; Myung-Ki Cheoun

REVIEW 3 major objections 5 minor 32 references

Ground-State Energy Estimation of HeH$^{+}$, ArH$^{+}$, and H$_2$O via Sample-Based Quantum Diagonalization

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper reports that sample-based quantum diagonalization on a superconducting quantum processor reproduces the CCSD potential-energy curves of HeH+, ArH+, and H2O near equilibrium, with deviations of 0.00, 2.51, and 6.34 mHa from…

desk verdict A candid, useful SQD feasibility study on three small molecules; treat the reported deviations as single-run demonstrations, because the H2O 6.34 mHa gap likely reflects the truncated ansatz, not hardware noise. read the letter →

arxiv 2608.06415 v1 pith:YP2WBCSA submitted 2026-08-05 physics.chem-ph nucl-th

Jubin Park , Chae-Hyun Yoon , Minkyu Lee , Myung-Ki Cheoun This is my paper · ORCID

classification physics.chem-phnucl-th

keywords sample-basedquantumdiagonalizationground-stateenergyLUCJansatzchemistryNISQdevicespotential-energycurvesmolecularionswatermolecule

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that sample-based quantum diagonalization (SQD), a hybrid workflow that builds a low-energy determinant subspace from bitstrings sampled on quantum hardware and then diagonalizes the Hamiltonian classically inside that subspace, can compute chemically meaningful ground-state energies for small molecules. Using the cc-pVDZ basis, the SQD curves for HeH+, ArH+, and H2O closely follow the same-basis CCSD references near equilibrium, and at the adopted equilibrium geometries the SQD energies differ from the exact same-active-space CASCI energies by 0.00, 2.51, and 6.34 mHa, respectively. The H2O case uses a 46-qubit active space and is presented as evidence that the workflow extends beyond diatomic ions to a polyatomic benchmark. The residual error grows with active-space size, so the paper presents the result as a proof of principle for feasibility rather than as a demonstration of established scaling.

What carries the argument

The carrying object is the sampled determinant subspace $X = \{x_1,\dots,x_D\}$ of bitstrings measured from the LUCJ circuit, together with the projected Hamiltonian $P_X \hat{H} P_X$, whose lowest eigenvalue is the SQD energy. The LUCJ ansatz, a shallow composition of orbital rotations and spin-balanced diagonal Coulomb interactions, is initialized from CCSD amplitudes by double factorization with a single repetition, so the circuit is a sampler rather than a variational ansatz. A self-consistent configuration-recovery procedure repairs bitstrings with wrong spin-resolved particle numbers, subsamples into five batches, and iterates until both the minimum batch energy and the spin-orbital occupations stabilize; selected-CI diagonalization of the sparse projected Hamiltonian then yields the ground-state estimate.

What would settle it

Take the H2O/cc-pVDZ system at a stretched O-H bond length, say $R_{\mathrm{OH}}=1.5$ Å, and compare the SQD energy against a full diagonalization of the same active-space Hamiltonian; if the deviation there is much larger than the 6.34 mHa found at equilibrium, the sampled subspace is missing determinants in the multireference region.

Watch

Extended reading notes

Core claim

The central claim is that a shallow, one-repetition local unitary cluster Jastrow (LUCJ) circuit, initialized from CCSD t1 and t2 amplitudes, can act as a structured sampler whose measured bitstrings define a determinant subspace in which classical diagonalization recovers the ground-state energy of the active-space Hamiltonian. For the cc-pVDZ basis, the SQD potential-energy curves reproduce the equilibrium-region trends of the CCSD curves for all three systems. The deviations from the same-active-space CASCI references are 0.00 mHa for HeH+, 2.51 mHa for ArH+, and 6.34 mHa for H2O at equilibrium, and the corresponding deviations from CCSD are 0.00, 0.04, and 2.67 mHa. The H2O result demonstrates execution of the full workflow on a 46-qubit polyatomic active space.

Load-bearing premise

The load-bearing premise is that the one-shot sampling circuit built from classical coupled-cluster results has enough overlap with the true ground state at every geometry—if it misses the important electron configurations, the final energy will be biased.

Editorial extensions

If this is right

If the claim holds, SQD offers a near-term route to molecular ground-state energies that avoids repeated on-hardware expectation-value estimation, using only shallow circuit sampling plus classical selected-CI diagonalization.
The workflow is executable on a 46-qubit polyatomic active space on current superconducting processors, so the practical reach of hardware-assisted quantum chemistry extends beyond two-center ions.
Because the LUCJ parameters come from CCSD, the SQD-CCSD agreement is an internal-consistency check, and independent benchmarks such as same-active-space CASCI or experiment are needed to validate absolute accuracy.
The growing deviation from CASCI, from 0.00 to 2.51 to 6.34 mHa, identifies subspace completeness rather than sampling noise as the next limiting factor, since the reported intra-pool batch spread is much smaller.
The method can be viewed as a quantum-assisted selected configuration interaction: the quantum device proposes determinants, and classical diagonalization decides the weights.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable consequence is that enlarging the subspace, for example with two LUCJ repetitions or a re-optimized circuit, should reduce the H2O deviation below 6.34 mHa while keeping the shot count fixed; if it does not, the remaining error is not subspace truncation.
The stretched O-H region is the natural stress test: if a same-active-space CASCI comparison at $R_{\mathrm{OH}}=1.5$ Å shows a sharply growing deviation, then the single-repetition CCSD-initialized sampler misses the determinants that dominate as the bond breaks.
Because all recovery iterations reuse the same finite shot pool, the reported convergence may underestimate run-to-run hardware variability; repeated independent executions would give a truer picture of the method's stability.
If SQD is extended to larger active spaces, the sparse structure of the projected Hamiltonian means the classical diagonalization step will stay manageable as long as the sampled subspace remains compact; the bottleneck would then be whether the shallow sampler keeps finding the important determinants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a useful, honestly reported feasibility data point, not a methodological breakthrough. The new content is hardware-executed SQD potential curves for HeH+, ArH+, and a 46-qubit H2O active space, with transparent same-active-space CASCI deviations of 0.00, 2.51, and 6.34 mHa. It deserves a serious referee, but the reported deviations should be read as single-run demonstrations, not robust error estimates.

What is actually new: the specific hardware results and the 46-qubit benchmark. SQD itself, the LUCJ ansatz, and CCSD-initialized circuits are all prior work, and the paper does not pretend otherwise. What it does well is transparency. The authors explicitly label the SQD–CCSD agreement as an internal-consistency benchmark because the LUCJ parameters come from CCSD amplitudes. They give a more independent check against same-active-space CASCI and report the gaps without spin. They also provide circuit depths, shot counts, and make clear that no variational optimization was performed on the device. That candor is genuinely useful.

The soft spots are real but not hidden. The 6.34 mHa H2O residual is above chemical accuracy, and the paper does not identify its source. The convergence plots reuse the same 20,000-shot pool, so the plateaus only show that the recovery procedure stabilizes on one finite dataset; they do not demonstrate subspace completeness. Given the LUCJ ansatz uses one repetition and only nearest-neighbor/on-site diagonal-Coulomb terms, it is entirely plausible that the remaining gap is a systematic subspace limitation of the sampler rather than a finite-shot or noise artifact. On that point the stress-test note is right: more hardware shots will not reduce the deviation if the missing determinants are never generated. The min-batch estimator, taking the minimum over five subsamples, also makes the results look better than the central tendency would, and there are no repeated independent QPU runs to provide error bars.

The central feasibility claim holds up: SQD on current IBM hardware can track CCSD-quality curves for these benchmarks and handle a 46-qubit active space. That is worth knowing. What would strengthen the paper for publication at a higher level: a few repeated runs, an uncertainty estimate on the min-batch estimator, and a same-basis CCSD/6-31G reference to separate basis-set error from method error. Code and data should be released.

This paper is for people tracking practical quantum chemistry on NISQ hardware, especially those interested in SQD's strengths and limits. It is not a methods paper; it is an application with honest caveats. I would send it to peer review and ask the authors to address the repetition and error-bar issues in revision.

Referee Report

3 major / 5 minor

Summary. The paper reports hardware-assisted Sample-Based Quantum Diagonalization (SQD) calculations for the ground states of HeH+, ArH+, and H2O, using shallow LUCJ circuits initialized from RHF-CCSD t1/t2 amplitudes on IBM quantum hardware. The authors compute potential-energy curves with 6-31G and cc-pVDZ basis sets, compare equilibrium-region energies against same-active-space CASCI and CCSD references, and report SQD-CASCI deviations of 0.00, 2.51, and 6.34 mHa for HeH+, ArH+, and H2O, respectively. The central claim is that the SQD workflow is feasible for these benchmarks, including a 46-qubit H2O active space, while explicitly acknowledging that the SQD-CCSD comparison is an internal-consistency check and that repeated independent executions are needed before drawing scalability conclusions.

Significance. If the reported deviations are representative, the paper provides useful evidence that SQD can produce chemically meaningful equilibrium energies for small molecules and a polyatomic active space on near-term hardware. The manuscript is transparent in several important ways: the CASCI references diagonalize the same frozen-core active-space Hamiltonian used by SQD, the circuit resource counts and sampling parameters are reported in detail, and the text explicitly cautions that the SQD-CCSD agreement is partly internal consistency because the LUCJ parameters come from CCSD amplitudes. These strengths make the feasibility claim defensible as a demonstration. However, the independent validation is concentrated at three equilibrium geometries, and the origin of the largest SQD-CASCI gap, the 6.34 mHa H2O deviation, is not resolved; whether it is a hardware-noise effect or a systematic limitation of the one-repetition CCSD-initialized LUCJ sampler is load-bearing for the paper's broader applicability claim.

major comments (3)

[Sec. 2.2 / Sec. 2.6 / Table 2] The 6.34 mHa H2O SQD-CASCI gap may be dominated by a systematic subspace limitation of the sampling circuit rather than by hardware noise. The LUCJ ansatz uses L=1, a restricted nearest-neighbor/on-site diagonal-Coulomb pattern, and parameters fixed from CCSD amplitudes with no hardware-side optimization, so the sampled determinant manifold is anchored to a single-reference CCSD-like state. Because the convergence analysis in Fig. 2 reuses the same 20,000-shot empirical pool across recovery iterations, the plateaus show stability of the recovery procedure on that finite pool, not completeness of the sampled subspace. To support the feasibility claim, the authors should perform a noiseless classical simulation of the same LUCJ circuit for H2O (or an alternative state-preparation scheme) and show whether the 6.34 mHa gap persists; if it does, the deviation is an ansatz-expressibility error and additional hardware shots or error mitigation will not remove it.
[Sec. 2.3 / Sec. 2.6 / Table 1(b)] The paper's single-run measurement pool and min-over-batches estimator do not separate hardware noise from other error sources. The final SQD energy is defined as the minimum over B=5 subsamples of one empirical pool, and the reported 0.0423 mHa subsampling spread is explicitly not an independent QPU uncertainty. The conclusion that the H2O deviation 'cannot be attributed to intra-pool batch fluctuations alone' is reasonable, but the feasibility claim would be considerably stronger with at least a few independent hardware executions at one or two geometries. Without such repetitions, the statement that the deviation reflects 'state preparation, finite-shot hardware sampling, configuration recovery, and subspace construction' remains a list of possibilities rather than a diagnosis.
[Sec. 3 / Fig. 1] The potential-energy-curve comparisons to CCSD are only an internal-consistency benchmark, because the LUCJ parameters are initialized from CCSD amplitudes and the paper itself calls the CCSD comparison an internal-consistency check. The independent same-active-space CASCI references are reported only at the three equilibrium geometries in Table 2, not along the scanned curves. As a result, the claim that the SQD curves 'closely follow' CCSD and reproduce the equilibrium-region trends is not independently validated for the stretched regions, where the RHF-CCSD reference itself becomes unreliable. The authors should provide CASCI reference points at selected stretched geometries (at least for the smaller HeH+ and ArH+ systems, where FCI in the active space is cheap) to verify that the SQD subspace remains accurate where the CCSD benchmark is not a valid reference.

minor comments (5)

[Sec. 3 / Fig. 2] The manuscript contains duplicated text blocks in Section 3, including the passage beginning 'ROH = 0.95782Å.Theupperpanelsshow...' and repeated wording in the Fig. 2 caption; these should be removed.
[Sec. 2.6] The sentence 'Under this configuration, four α and four β electrons were explicitly treated' is ambiguous because it applies only to ArH+ and H2O, not to HeH+; specify the electron counts separately for each molecule.
[Abstract / full text] There are run-together words resulting from the text extraction (for example, 'Forthepresentcalculations,butstringsweregenerated'), which should be corrected in the final typeset version.
[Table 1(b)] The entry mentioning 'ffsim.qiskit.PRE_INIT' introduces the ffsim package without defining it; either name the package and version or provide a reference.
[Sec. 2.3 / Eq. (11)] Using the minimum over B subsamples as the final estimator is not a standard unbiased estimator; the paper should clarify the rationale or report both the minimum and the mean over batches.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: independent same-active-space CASCI references anchor the accuracy claims, while the CCSD-seeded SQD–CCSD comparison is explicitly disclosed as an internal-consistency check.

full rationale

The accuracy claims rest on SQD–CASCI deviations (0.00, 2.51, 6.34 mHa for HeH+, ArH+, H2O) in Table 2, where the CASCI reference is obtained by FCI diagonalization of the same frozen-core active-space Hamiltonian and is independent of the CCSD t1/t2 amplitudes used to build the LUCJ sampler. Although the SQD–CCSD agreement in Fig. 1 is expected to be partially self-consistent because Sec. 2.2 initializes the LUCJ parameters from RHF-CCSD amplitudes, the paper explicitly says this comparison “should be regarded as an internal-consistency benchmark rather than as a fully independent validation” (Sec. 3). The convergence analysis (Fig. 2) also reports the deviation to CASCI, not CCSD. The H2O 6.34-mHa CASCI deviation, the single measurement pool, the truncated LUCJ, and the size-dependent error are all acknowledged as limitations rather than hidden circular steps. No load-bearing self-citation or imported uniqueness theorem appears. The derivation chain (LUCJ sampling → subspace projection → deterministic diagonalization → CASCI comparison) is self-contained for the feasibility claim made.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central result rests on the active-space Hamiltonians (Table 1a), the CCSD-initialized LUCJ sampling circuits, the configuration-recovery settings, and the min-batch estimator. The CASCI reference is exact within the same active space, so it is a computed benchmark rather than an axiom. The main hand-chosen elements are listed as free parameters; the CCSD-initialized LUCJ heuristic and the recovery estimator are the key ad hoc assumptions.

free parameters (6)

LUCJ circuit repetitions L (nreps) = 1
One LUCJ repetition retained with no optimization of the factorization (optimize=False). This truncation determines the sampled determinant distribution. Sec. 2.2 and Table 1(b).
Diagonal-Coulomb interaction pattern = same-spin nearest-neighbor and opposite-spin on-site pairs
Restriction of the diagonal-Coulomb interactions to these pairs is a chosen truncation; it limits the expressibility of the LUCJ state. Sec. 2.2.
Number of batches B and samples per batch = B=5, 350 samples per batch
Subsampling parameters chosen for SQD post-processing; the final energy is the minimum over batches, which affects the reported error. Sec. 2.3, 2.6 and Table 1(b).
Convergence tolerances for recovery iterations = epsilon_E=2e-4 Eh, epsilon_n=2e-4
Stopping criteria for the configuration-recovery iterations; affect the converged energy. Sec. 2.4 and Table 1(b).
Random seed for subsampling = 24
Seed for subsampling; influences the batch composition and the final min-batch energy. Table 1(b).
Frozen-core orbital choices = Ar: 5 core orbitals frozen; H2O: 1 oxygen core orbital frozen; HeH+: none
These hand-chosen active-space definitions define the reference Hamiltonian for all comparisons. Sec. 2.6 and Table 1(a).

assumptions (5)

domain assumption The Born-Oppenheimer approximation separates nuclear and electronic motion, and the clamped-nuclei electronic Hamiltonian (Eq. 1) is the target.
Invoked at the start of Sec. 2.1; standard in quantum chemistry but a modeling assumption.
domain assumption The active-space Hamiltonian obtained after freezing core orbitals and choosing a basis set faithfully represents the molecular ground-state chemistry of interest.
Active-space definitions in Table 1(a) restrict the Hilbert space; the SQD, CASCI, and CCSD values all refer to this same reduced Hamiltonian.
ad hoc to paper The LUCJ ansatz with one repetition, initialized from CCSD t1/t2 amplitudes, produces a sampled determinant distribution with substantial overlap with the true active-space ground state.
This heuristic is specific to this work; Sec. 2.2 states the parameters are initialized from CCSD amplitudes and not varied. If the overlap is insufficient, the SQD subspace may miss key determinants.
standard math The Jordan-Wigner mapping and the second-quantized fermionic Hamiltonian (Eq. 1) are valid.
Standard fermion-to-qubit mapping; Sec. 2.1.
ad hoc to paper The configuration-recovery procedure converges to a physically meaningful determinant set, and the min-batch energy is a reliable estimator of the active-space ground-state energy.
The recovery algorithm and the min-over-batches estimator (Sec. 2.3, 2.4) are specific to the SQD workflow; the paper reports convergence criteria but no independent QPU repetition.

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Cite this review

Pith. "Pith review of Ground-State Energy Estimation of HeH$^{+}$, ArH$^{+}$, and H$_2$O via Sample-Based Quantum Diagonalization." pith.science (2026). https://pith.science/paper/YP2WBCSA

@misc{pith2026260806415,
  author       = {Pith},
  title        = {Pith review of: Ground-State Energy Estimation of HeH$^+$, ArH$^+$, and H$_2$O via Sample-Based Quantum Diagonalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YP2WBCSA}},
  note         = {Machine review of arXiv:2608.06415}
}

abstract

Accurate ground-state energies are essential for understanding molecular structure, chemical bonding, and reaction energetics in quantum chemistry. In this work, we investigate the ground-state properties of the molecular systems HeH$^+$, ArH$^+$, and H$_2$O using Sample-Based Quantum Diagonalization (SQD), a hybrid quantum-classical framework designed for near-term quantum devices. Unlike variational approaches such as VQE, which require deep parameterized circuits and repeated expectation-value measurements, SQD reconstructs a low-energy determinant subspace directly from measured bitstrings. For the present calculations, bitstrings were generated on IBM quantum hardware using shallow local unitary cluster Jastrow (LUCJ) circuits whose parameters were constructed from the $t_1$ and $t_2$ amplitudes of coupled-cluster singles and doubles (CCSD) calculations based on restricted Hartree--Fock (RHF) references. From these samples, we compute ground-state potential-energy curves of HeH$^+$, ArH$^+$, and H$_2$O with the 6-31G and cc-pVDZ basis sets. For all three systems, the SQD results obtained with the cc-pVDZ basis closely follow the corresponding same-basis CCSD energies and reproduce the equilibrium-region trends of the potential-energy curves. HeH$^+$ and ArH$^+$ were chosen as simple yet astrophysically important molecular-ion benchmarks, while H$_2$O was included as a representative polyatomic molecule to assess the applicability of SQD beyond diatomic ionic systems. At the adopted equilibrium geometries, the deviations from the same-active-space CASCI references are 0.00, 2.51, and 6.34 mHa for HeH$^+$, ArH$^+$, and H$_2$O, respectively. These results demonstrate the feasibility of hardware-assisted SQD for the present benchmark systems and motivate further studies of its accuracy and computational scaling for larger molecular active spaces.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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