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REVIEW 3 major objections 5 minor 119 references

Analytical Nuclear Gradients and Hessians on Quantum Hardware via Orbital-Optimized VQE with Error Mitigation

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Analytical nuclear gradients and Hessians can be computed on real quantum hardware by measuring the underlying tensor elements and correcting them with error mitigation.

desk verdict A credible proof-of-principle for analytical Hessians on real quantum hardware, with honest error accounting; the M0 noise-transfer assumption and batch selection are the real soft spots. read the letter →

arxiv 2608.08758 v1 pith:LIUK7PFY submitted 2026-08-09 physics.chem-ph quant-ph

Renato Olarte Hernandez , Karl Michael Ziems , Erik Kjellgren , Jacob Kongsted , Sonia Coriani , Stephan P. A. Sauer This is my paper · ORCID
classification physics.chem-phquant-ph
keywords nucleargradientHessianorbital-optimizedVQEtiledunitaryproductstateansatzlinearresponseerrormitigationquantumhardwarevibrationalfrequency
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 the two quantities computational chemistry most needs for geometry optimization and vibrational spectroscopy—the nuclear gradient and the nuclear Hessian—can be obtained analytically from measurements on a real quantum processor, not from finite differences or classical simulation. The strategy is to run an orbital-optimized variational quantum eigensolver with a tiled unitary product state ansatz inside an active space, then measure the reduced density matrices and the linear-response $\boldsymbol{A}$ and $\boldsymbol{B}$ matrices that the analytical derivative equations demand. A confusion-matrix error-mitigation scheme built from the same ansatz with all parameters set to zero, plus post-selection on electron number, brings the measured tensors close enough to the ideal values that the H$_2$ stretching frequency comes out at $5008 \pm 21$ cm$^{-1}$ against a $5000$ cm$^{-1}$ reference. For water the same workflow still underestimates energies by about $150$ mHa, so the paper doubles as a resource and scaling analysis of what current hardware can and cannot do.

What carries the argument

The machinery has three coupled pieces. First, the pp-tUPS ansatz—a tiled unitary product state circuit of spin-adapted single- and pair-double excitation gates, layered to approach CASSCF accuracy—produces the wavefunction on the quantum processing unit. Second, the measurement protocol directly evaluates the tensor elements the derivative equations need: the one- and two-particle reduced density matrices for the static gradient and static Hessian, and the $\boldsymbol{A}$ and $\boldsymbol{B}$ linear-response submatrices whose difference builds the electronic orbital Hessian $\mathcal{G}^{(0)}=2(\boldsymbol{A}-\boldsymbol{B})$; the response vector is then obtained by inversion of $\mathcal{G}^{(0)}$, avoiding finite differences. Third, the $\boldsymbol{M}_0$ error mitigation constructs a confusion matrix from circuits in which all ansatz parameters are set to zero, inverts it against the raw measurement statistics, and post-selects bitstrings with the correct separate $\alpha$ and $\beta$ electron counts. The analytical gradient and Hessian formulas then combine these corrected tensors with classically computed integral derivatives.

What would settle it

Build the $\boldsymbol{M}_0$ confusion matrix from circuits whose ansatz parameters are nonzero (or from the exact transpiled gates used for the property measurements), apply the same correction to the H$_2$ densities and Hessian, and compare the resulting stretching frequency with the $5008 \pm 21$ cm$^{-1}$ reported here; a shift larger than the quoted uncertainty, or corrected densities that move away from the ideal values, would show that the zero-parameter noise model does not transfer.

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Extended reading notes

Core claim

The central claim is that the novelty lies in the explicit and corrected measurement, on a quantum processing unit, of the tensor elements required by the analytical equations for nuclear derivatives: the one- and two-electron reduced density matrices for the static terms, and the $\boldsymbol{A}$ and $\boldsymbol{B}$ linear-response matrices whose combination $\mathcal{G}^{(0)}=2(\boldsymbol{A}-\boldsymbol{B})$ forms the electronic orbital Hessian needed for the relaxed (response) term. The gradient follows the standard analytical-derivative expression with Pulay connection terms, and the Hessian adds the static second-derivative terms plus the response term $f^{(1)}\lambda^{(1)}$ solved by full-space measurement and inversion of $\mathcal{G}^{(0)}$. On the hardware side, each required expectation value is obtained from the pp-tUPS ansatz circuit and corrected with the $\boldsymbol{M}_0$ confusion matrix built from the same ansatz with all parameters set to zero, followed by post-selection that keeps only bitstrings with the correct separate $\alpha$ and $\beta$ electron counts. Demonstrated on H$_2$, the mitigated energies sit within $9$ mHa of full configuration interaction, the gradient modulus within $6$ mHa/Bohr, and the retrieved H$_2$ stretching frequency is $5008 \pm 21$ cm$^{-1}$ versus the $5000$ cm$^{-1}$ finite-difference reference; on water the same protocol still underestimates the energy by roughly $150$ mHa and demands substantially more QPU time.

Load-bearing premise

The whole correction scheme assumes that the noise of the real, deep, transpiled circuits is the same as the noise seen in the shallow calibration circuits with all ansatz parameters set to zero, and no experiment in the paper checks that transfer; the reported statistics also come from only the 35 of 46 runs that were not discarded for high qubit errors.

Editorial extensions

If this is right

  • Geometry optimizations and harmonic vibrational frequencies of small molecules can be run on current noisy quantum hardware using analytical derivatives instead of numerical finite differences, with per-geometry QPU times on the order of minutes for four-qubit systems.
  • The same measured $\boldsymbol{A}$ and $\boldsymbol{B}$ matrices serve both the electronic Hessian and the static property gradients, so for small active spaces no extra circuits are needed beyond those already used for the energy.
  • The errors in individual tensors (especially the 2-RDM and the $\boldsymbol{A}$ matrix) are larger than the final nuclear Hessian error, indicating beneficial error cancellation among the contributing terms.
  • The water experiment sets a concrete scaling boundary: an 8-qubit, 414-depth transpiled circuit with 290 Pauli groups costs hours of QPU time and still underestimates energies by about 150 mHa, so resource and noise characterization are the current bottleneck.
  • For larger systems, the authors propose avoiding explicit full electronic Hessian measurement via Hessian-vector products and a Davidson-style iterative solver, which would replace the exponential confusion-matrix cost with a more scalable procedure.

Reading between the lines

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

  • If the zero-parameter confusion matrix faithfully models the noise of the optimized circuits, the same measurement pipeline should transfer to other response properties—polarizabilities, NMR shieldings, hyperfine couplings—since those share the same $\boldsymbol{A}$ and $\boldsymbol{B}$ building blocks; this is a testable extension rather than a claim the paper makes.
  • A direct experimental check of the transfer assumption would be to build the $\boldsymbol{M}_0$ matrix from circuits with nonzero ansatz parameters and compare the corrected 1- and 2-RDMs; systematic differences would quantify the hidden bias in all mitigated Hessians.
  • The observed error cancellation between 2-RDM and $\boldsymbol{A}$ errors suggests that allocating measurement shots preferentially to the tensor elements with the largest variance could improve accuracy at fixed total shot count, something the paper does not explore.
  • Because only 35 of 46 hardware batches survived the qubit-error selection, the reported statistics describe a filtered subset; repeating with error-mitigated selection criteria that do not discard data would test whether the selection itself inflates the apparent accuracy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents an analytical implementation of nuclear gradients and Hessians within an orbital-optimized VQE framework using the pp-tUPS ansatz, with the required 1-RDM, 2-RDM, and linear-response A/B tensor elements measured on IBM Pittsburgh hardware and corrected by the M0 confusion-matrix error-mitigation scheme with post-selection. Ideal simulations for H2/STO-3G (AS(2,2)) and H2O/STO-3G (AS(4,4)) are compared with PySCF FCI/CASSCF and Dalton references, and QPU results for H2 give mitigated energy errors below 9 mHa, gradient errors below 6 mHa/Bohr, and a stretching frequency of 5008±21 cm−1 versus a 5000 cm−1 finite-difference reference. The water QPU results are single evaluations with mitigated energies 123–201 mHa below the reference, and the Hessian experiment retains only 35 of 46 batches after discarding runs on noisy qubits.

Significance. If the error-mitigation pipeline is valid, the paper demonstrates a complete QPU workflow for analytical first and second nuclear derivatives in an active-space framework, including an explicit tensor-measurement protocol for the response equations. Strengths include the use of standard Helgaker–Jørgensen equations, independent classical references from PySCF and Dalton, explicit circuit resource tables, and a quantitative flagship frequency prediction that is directly falsifiable on the same hardware. The central load-bearing assumption, however, is that the M0 confusion matrix measured with all ansatz parameters equal to zero transfers to the much deeper transpiled measurement circuits; this is not validated in the manuscript. In addition, the reported hardware statistics exclude 11 of 46 Hessian runs, so the headline frequency uncertainty is conditional on a selected subset. These issues are fixable in a revision with additional control experiments and complete data reporting.

major comments (3)
  1. [§2.5.1, Eqs. (38)–(42), and Table 2] The M0 calibration circuits are structurally much shallower than the circuits actually used to measure the RDMs and the A/B matrices. Because the tUPS tile in Eq. (9) reduces to the identity when all parameters are zero, the calibration states in Eq. (39) contain essentially no entangling gates, whereas the transpiled H2 measurement circuit has depth 98 and 29 entanglers and the H2O circuit has depth 414 and 257 entanglers (Table 2). The inversion in Eq. (41) therefore assumes that gate and crosstalk noise is independent of ansatz parameters and of transpilation. No parameter-matched calibration, noisy-simulator cross-check, or condition-number/bias analysis is provided. Since every corrected density, A/B element, and hence the 5008±21 cm−1 frequency depends on this assumption, the central claim requires validation: calibrate M0 at the actual optimized parameters for at least one geometry, compare corrected expectation values against exact values on a noisy simulator with gate-dependent noise, and report the spectrum or condition number of M0.
  2. [§4.2.2, Fig. 8, and Table 5] The Hessian statistics are computed after discarding 11 of 46 batches because they “were run on unreliable qubits with high error rates,” yet no pre-defined, result-independent exclusion rule is given. The reported 5008±21 cm−1 and the tensor-error statistics in Table 6 are therefore conditional on a selected subset, and the quoted uncertainty does not characterize full run-to-run variability. Please report all 46 batches, state the exclusion criterion (for example, a calibration threshold fixed before data taking), and show the sensitivity of the stretching frequency to inclusion/exclusion. The negative average translational eigenvalues (−407±171 cm−1) also indicate that the measured Hessian is not positive semidefinite; a discussion of this instability is needed to qualify the reliability of the retrieved force constants.
  3. [§4.1.4, Table 4] The water QPU results, although explicitly labeled as single evaluations, constitute an in-manuscript test of the M0 noise-transfer assumption and they are not consistent with it: the mitigated energies lie 123–201 mHa below the CASSCF reference, meaning the correction overshoots by an amount comparable to the raw error. Because the same M0 pipeline is used for the H2 Hessian, this overcorrection raises the possibility that the H2 agreement is partly coincidental. The manuscript should either identify a mechanism for the water overcorrection (for example, parameter-dependent gate errors, crosstalk, or state-preparation errors) or, at minimum, substantially soften the claim that the approach demonstrates good performance beyond H2.
minor comments (5)
  1. [§3.1] A step-size convergence test for the finite-difference CASSCF Hessian reference should be reported; 0.001 Å is reasonable, but the sensitivity of the 5000 cm−1 reference to this step is not documented.
  2. [§4.2.2, Table 6] Units are missing or inconsistent across rows (for example, “∆1-RDM Max15.571±6.543” has no unit, while later rows quote mHa); add units to every row and state explicitly that both “Max” and “Distance” quantities are in the property’s own units.
  3. [§4.1.3 and Table 6] The text states that the mitigated gradient precision is “∼ 1mHa/Bohr,” while Table 6 reports a mean gradient-modulus error of 7.261±5.894 mHa/Bohr; these numbers need to be reconciled.
  4. [§2.5.1, Eq. (39)] Since all tUPS parameters are zero, |x0⟩ is just |x⟩ for the defined ansatz; the notation should be explained or simplified, and the phrase “while considering the gate noise drifting” should be made mathematically precise.
  5. [§4.2.2, Table 5] The first two eigenvalues are labeled x/y translations; for a finite system these should be near zero, and the mean −407 cm−1 suggests broken translational symmetry. A brief explanation of why these modes are not projected out would clarify the Hessian quality assessment.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the analytical gradients and Hessians follow standard Helgaker-Jørgensen response equations, the ideal ansatz is benchmarked against independent PySCF/Dalton references, and QPU error-mitigation concerns are validity limitations rather than circular reductions.

full rationale

The derivation chain is self-contained. The nuclear gradient (Eq. 22) and Hessian (Eqs. 26-37) are taken from the standard Helgaker-Jørgensen formulation, with the QPU contribution limited to measured 1-RDM, 2-RDM, and A/B response matrices. The ideal tUPS results are not assumed; they are numerically compared against independent FCI (H2) and CASSCF (H2O) references from PySCF and Dalton, and the ansatz itself is attributed to Burton's prior work rather than to a self-citation. The M0 error-mitigation scheme is fully specified in Sec. 2.5.1, and although it is cited to the authors' earlier work, it is described in the paper and has prior external applications; this is a normal method citation, not a load-bearing self-citation. The skeptical concern that the zero-parameter M0 calibration (Eq. 39) may not transfer to the deep transpiled measurement circuits is a correctness and sensitivity question about an unvalidated noise-transfer assumption, not an instance of a prediction reducing to its inputs by construction. No fitted parameter is relabeled as a prediction, and no uniqueness theorem or ansatz is imported solely via a self-citation. The claim that pp-tUPS approaches CASSCF in the infinite-layer limit reflects the ansatz design, but the reported agreement is verified numerically against independent classical codes, so it is not circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper adds no fundamental constants and no invented physical entities. It relies on established MCSCF derivative theory and on three domain assumptions: ansatz fidelity, M0 noise transfer, and operator-space completeness. The shot budget is an empirical protocol choice, and the finite-difference step is a numerical reference choice.

free parameters (2)
  • Measurement shot budget per Pauli string = 12000
    Hand-selected after a shot-count scan (Fig. 6, Table 3); larger budgets sometimes gave worse results, so the chosen value is data-dependent.
  • Finite-difference step for CASSCF Hessian reference = 0.001 Angstrom and 0.001 radian
    Chosen for the central-difference reference Hessian; affects validation of the method but not the QPU measurement itself.
assumptions (5)
  • standard math Helgaker-Jorgensen analytical derivative and CP-MCSCF response equations are valid for the oo-VQE wavefunction.
    The entire gradient/Hessian formalism is taken from Ref. [103] and assumed valid for the tUPS wavefunction; invoked in Secs. 2.4.1 and 2.4.2.
  • domain assumption One-layer tUPS is equivalent to FCI for H2 and two-layer tUPS is equivalent to CASSCF for H2O.
    Relied on for the claim that measured densities are the true correlated densities; supported by ideal simulations in Secs. 4.1.1 and 4.1.2.
  • domain assumption The M0 confusion matrix built from zero-parameter ansatz circuits characterizes noise on the fully parameterized circuits.
    Eq. 39 uses U(0) to build the confusion matrix, and Eq. 41 applies its inverse to raw measurements of the actual circuit; the transfer is assumed, not tested.
  • domain assumption The naive spin-adapted single and double excitation operators span the response space.
    The authors state in Sec. 2.3 that naive operators are exact for H2 but an approximation for the H2O active-space calculation.
  • domain assumption The finite-difference reference Hessian at 0.001 Angstrom or 0.001 radian is converged.
    PySCF lacks CASSCF Hessians, so the reference is numerical; the step choice is a standard but unverified convergence choice in Sec. 3.1.

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Pith. "Pith review of Analytical Nuclear Gradients and Hessians on Quantum Hardware via Orbital-Optimized VQE with Error Mitigation." pith.science (2026). https://pith.science/paper/LIUK7PFY

@misc{pith2026260808758,
  author       = {Pith},
  title        = {Pith review of: Analytical Nuclear Gradients and Hessians on Quantum Hardware via Orbital-Optimized VQE with Error Mitigation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LIUK7PFY}},
  note         = {Machine review of arXiv:2608.08758}
}
read the original abstract

Nuclear gradients and Hessians are fundamental quantities in computational chemistry, essential for a wide range of applications including geometry optimization, vibrational spectroscopy, and molecular property calculations. In this work, we present their analytical implementation on quantum hardware. The methodology is formulated within an active-space framework combining orbital optimization and linear-response theory. On the quantum-computing side, the approach employs the tiled unitary product state (tUPS) ansatz to directly evaluate the tensor elements required for solving the response equations. Moreover, the expectation values are corrected using an adapted confusion-matrix error-mitigation scheme in combination with post-selection criteria. The resulting workflow is assessed on molecular hydrogen and on water through the calculation of potential energy surfaces, nuclear gradients, Hessians, and vibrational frequencies, enabling the evaluation of both its capabilities and current limitations. The results demonstrate good performance for the hydrogen molecule, whereas the water molecule provides a more demanding test of quantum-hardware resources and highlights the trade-offs associated with error-mitigation strategies. The quantified analysis of the results identify the main sources of errors, suggesting improvement directions for more accurate quantum computer applications.

Figures

Figures reproduced from arXiv: 2608.08758 by the authors.

Figure 1
Figure 1. The tiled-UPS ansatz with the perfect pairing ordering as proposed by Burton [96]. The first two operator columns correspond to the first layer (blue), and the combination of the first and second columns (blue and green) corresponds to a two-layer ansatz, and so on. The pp-tUPS ansatz is embedded within the oo-VQE framework, where the ground-state energy is obtained by variational minimization over both sets of para… view at source ↗
Figure 2
Figure 2. H2 classical FCI and ideal one-layer tUPS energies and gradients. (Bottom) Hydrogen’s stretching PES and its associated gradient modulus. (Top) The energy and gradient errors with respect to the classical FCI reference. At first glance, the results from the classical FCI reference and the UPS wavefunction are in excellent agreement between each other for the energy and also for the gradient results. The gradient (fo… view at source ↗
Figure 3
Figure 3. H2 /AS(2,2)/STO-3G ground state shot noise simulation. The average and standard deviation of 100 simulations of the electronic energy are shown versus the number of shots. The ’Inf’ label stands for an infinite limit of shots. Under the light of these results, the shot noise magnitude is estimated to 1-2 mHa. Hence, shot noise is important only when the results are close to the desired chemical accuracy. 4.1.2. H2O … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: H2O/AS(4,4)/STO-3G classical CASSCF and ideal a) one-layer and b) two-layer tUPS energies and gradients. (Bottom) Water stretching PESs and their associated gradient modulus. (Top) The energy and gradient errors with respect to the classical CASSCF reference. The one-l…
Figure 5
Figure 5. Figure 5: H2O/AS(4,4)/STO-3G ground state shot noise simulation. The average and standard deviation of 100 simulations of the electronic energy are shown versus the number of shots. The ’Inf’ label stands for an infinite limit of shots. 4.1.3. H2 quantum experiment Using the sam…
Figure 6
Figure 6. Figure 6: H2 /AS(2,2)/STO-3G stretching energies: simulated reference and real measurements on IBM’s Pittsburgh QPU using different number of shots. (Bottom) Simulated FCI reference energy (black). The colored lines represent the QPU measurements using different number of shots.…
Figure 7
Figure 7. Figure 7: H2 /AS(2,2)/STO-3G stretching energies and gradients: simulated reference versus real measurement on IBM’s Pittsburgh QPU using 12 000 shots. (Bottom) Simulated FCI reference energy (black) and gradient (gray). Average raw QPU-measured energy (red). Average mitigated e…
Figure 8
Figure 8. Figure 8: The 35 measurement batches of the ground state geometry properties performed on IBM’s Pittsburgh QPU backend. The energies, gradient, and frequencies are compared to the ideal fermionic wavefunction results. Similar to the gradient results, the raw energies are on aver…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.