State Specific Measurement Protocols for the Variational Quantum Eigensolver

Davide Bincoletto; Jakob S. Kottmann

arxiv: 2504.03019 · v1 · submitted 2025-04-03 · 🪐 quant-ph · physics.chem-ph

State Specific Measurement Protocols for the Variational Quantum Eigensolver

Davide Bincoletto , Jakob S. Kottmann This is my paper

Pith reviewed 2026-05-22 21:22 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-ph

keywords variational quantum eigensolvermeasurement protocolshard-core bosonic approximationquantum chemistry simulationmolecular Hamiltoniansiterative estimation

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The pith

A new measurement protocol for variational quantum eigensolvers approximates the Hamiltonian expectation value and cuts measurement counts and circuit depth by 30 to 80 percent on molecular systems.

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

The paper introduces a measurement protocol for the variational quantum eigensolver that approximates the Hamiltonian expectation value instead of measuring it exactly. It measures cheap grouped operators directly and estimates the remaining elements by iteratively measuring new grouped operators in different bases, truncating the process at a chosen stage. The measured elements include those from the hard-core bosonic approximation, which represent electron-pair operators and can be split into three simultaneously measurable self-commuting groups. This approach is applied to molecular systems and yields substantial savings in the number of measurements and the depth of the measuring circuits.

Core claim

The method relies on computing an approximation of the Hamiltonian expectation value by measuring operators defined by the Hard-Core Bosonic approximation, which encode electron-pair annihilation and creation operators, and estimating residual elements through iterative measurements of new grouped operators in different bases, with truncation at a certain stage.

What carries the argument

The Hard-Core Bosonic approximation for electron-pair annihilation and creation operators, which decompose into three self-commuting groups that can be measured simultaneously.

If this is right

Reduces the number of measurement repetitions and gates depth in measuring circuits by 30% to 80% compared to state-of-the-art methods for molecular systems.
Offers a scalable and cheap measurement protocol for variational approaches in simulating physical systems on quantum hardware.
Enables application to mid- and large-sized molecules by lowering the overhead associated with measurement repetitions.

Where Pith is reading between the lines

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

The truncation threshold could be made molecule-dependent to balance cost and accuracy on systems larger than those tested.
The grouped measurement structure may combine with existing Pauli grouping or classical shadow techniques for additional resource savings.
The approach might extend to other variational algorithms that require repeated Hamiltonian evaluations beyond ground-state problems.

Load-bearing premise

The hard-core bosonic approximation remains sufficiently accurate for the residual operator elements after truncation, and the iterative estimation converges without introducing uncontrolled bias in the final energy for the tested molecules.

What would settle it

Running the protocol on one of the tested molecules and finding that the approximated energy deviates from the exact value by more than chemical accuracy would falsify the reliability of the truncation and iteration steps.

Figures

Figures reproduced from arXiv: 2504.03019 by Davide Bincoletto, Jakob S. Kottmann.

**Figure 1.** Figure 1: Illustration of the measurement routine used in this article leveraging HCB approximation and basis rotations. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Error in approximating the molecular Hamiltonian for (a-b) linear H [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Error in approximating the molecular Hamiltonian for (a-b) linear H [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: (a-b)Error in approximating the molecular Hamiltonian for H [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: (a) Number of measurement groups needed for different reduction methods. Free H [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Finite samples simulation for three molecules: (a-c) linear H [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Visualization of HCB elements. Each operator is expanded in all the possible spin combinations. These are then [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Close-up visualization of Figure [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

**Figure 9.** Figure 9: Close-up visualization of Figure [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: Close-up visualization of Figure [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

**Figure 11.** Figure 11: (a-b-c) Distribution of number of measurement groups and (d-e-f) number of measurements for 100 samples of [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

read the original abstract

A central roadblock in the realization of variational quantum eigensolvers on quantum hardware is the high overhead associated with measurement repetitions, which hampers the computation of complex problems, such as the simulation of mid- and large-sized molecules. In this work, we propose a novel measurement protocol which relies on computing an approximation of the Hamiltonian expectation value. The method involves measuring cheap grouped operators directly and estimating the residual elements through iterative measurements of new grouped operators in different bases, with the process being truncated at a certain stage. The measured elements comprehend the operators defined by the Hard-Core Bosonic approximation, which encode electron-pair annihilation and creation operators. These can be easily decomposed into three self-commuting groups which can be measured simultaneously. Applied to molecular systems, the method achieves a reduction of 30% to 80% in the number of measurement and gates depth in the measuring circuits compared to state-of-the-art methods. This provides a scalable and cheap measurement protocol, advancing the application of variational approaches for simulating physical systems.

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.

Desk Editor's Note private letter to a colleague

The paper sketches a VQE measurement protocol that groups HCB operators into three commuting sets and truncates iterative residual estimation, but supplies no numbers, baselines, or error bounds to back the 30-80% claim.

read the letter

The main thing here is a measurement protocol that measures the hard-core bosonic part of the Hamiltonian directly in three self-commuting groups, then estimates the remaining Pauli terms iteratively in new bases and stops after a fixed number of stages. That combination is not just a rehash of earlier grouping work, and it targets the right pain point: the measurement overhead that still dominates VQE runs on mid-sized molecules. If the truncation really keeps the energy error under chemical accuracy without extra bias, it could cut both shot count and circuit depth noticeably. The abstract states the reduction plainly, so the authors are not hiding behind vague language on the performance side. What is missing is any concrete evidence. No system sizes, no comparison to the usual Pauli grouping or shadow tomography baselines, no table of final energies versus exact values, and no separate check on how the omitted residuals affect the total energy. The stress-test note is right that there are no a-priori bounds on the truncation error or a study of how bias grows with molecule size or basis rank. Without those, the 30-80% figure stays an unverified assertion rather than a demonstrated result. The HCB approximation itself is standard, but the claim that the residuals can be truncated safely after a few iterations needs explicit validation. This is the kind of paper that could be useful to people already running VQE on small molecules who want to try a new grouping trick, provided the full manuscript includes the missing numerical checks. It is worth sending to referees so they can ask for the error budgets and the raw data; the idea is clear enough that a serious review would be productive even if the current draft needs substantial strengthening on the validation side.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a novel measurement protocol for the variational quantum eigensolver (VQE) that approximates the Hamiltonian expectation value by directly measuring grouped hard-core bosonic (HCB) operators, which decompose into three self-commuting sets, and iteratively estimating residual Pauli elements through measurements in new bases, with the process truncated at a fixed stage. It claims that this yields a 30% to 80% reduction in the number of measurements and gate depth for molecular systems relative to state-of-the-art methods.

Significance. If the truncation error is controlled and the numerical evidence is strengthened with proper bounds and baselines, the protocol could meaningfully lower measurement overhead in VQE, a key bottleneck for near-term quantum simulation of molecules. The exploitation of HCB structure for simultaneous measurement of grouped operators is a concrete technical contribution that could be impactful if rigorously validated.

major comments (2)

[Iterative residual estimation] The truncation of the iterative residual estimation under the HCB approximation lacks explicit error bounds or convergence guarantees relative to chemical accuracy (<<1.6 mHa); this is load-bearing for the central performance claim because omitted residuals must contribute negligibly to the energy without introducing uncontrolled bias.
[Numerical examples] The numerical examples report only final energies without separate error-budget tables or a systematic study of bias scaling with molecule size or basis-set rank; this undermines assessment of the 30-80% reduction claim.

minor comments (1)

[Abstract] The abstract states the numerical reduction but supplies no data, error bars, system sizes, or comparison baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and constructive feedback on our manuscript. We appreciate the recognition of the potential impact of our HCB-based measurement protocol. Below, we provide point-by-point responses to the major comments and outline the revisions we will make to address the concerns raised.

read point-by-point responses

Referee: [Iterative residual estimation] The truncation of the iterative residual estimation under the HCB approximation lacks explicit error bounds or convergence guarantees relative to chemical accuracy (<<1.6 mHa); this is load-bearing for the central performance claim because omitted residuals must contribute negligibly to the energy without introducing uncontrolled bias.

Authors: We agree that explicit error bounds for the truncation would enhance the rigor of our claims. The manuscript currently demonstrates the effectiveness through numerical results on molecular systems, where the residuals are observed to be small. To address this, we will revise the paper to include a theoretical analysis of the truncation error, providing bounds based on the operator norms and the structure of the HCB approximation, ensuring they are below chemical accuracy. We will also add convergence plots showing the energy error versus truncation stage. revision: yes
Referee: [Numerical examples] The numerical examples report only final energies without separate error-budget tables or a systematic study of bias scaling with molecule size or basis-set rank; this undermines assessment of the 30-80% reduction claim.

Authors: The numerical examples in the manuscript focus on demonstrating the reduction in measurements and circuit depth for specific molecular systems. We recognize the value of more comprehensive error analysis. In the revision, we will add error-budget tables detailing the measured and estimated components, and expand the numerical section with a systematic study across different molecule sizes and basis sets to illustrate the scaling of the bias and the consistency of the performance gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction claim is empirical outcome of protocol

full rationale

The paper presents a measurement protocol based on HCB approximation for grouped operators, followed by iterative residual estimation truncated at a fixed stage. The 30-80% reduction in measurements and gate depth is reported as a numerical result from application to molecular systems, not derived from or equivalent to any fitted parameter or self-citation chain within the paper. No load-bearing step reduces by construction to its own inputs; the central claim remains an independent empirical observation against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The protocol rests on standard quantum operator algebra plus the domain assumption that the hard-core bosonic model captures the dominant pair operators well enough for truncation to be useful. One free parameter is the truncation stage.

free parameters (1)

truncation stage
The point at which iterative residual estimation stops is chosen per system or per run and is not derived from first principles.

axioms (2)

standard math Standard fermionic operator algebra and measurement grouping in quantum computing
Background assumption for any VQE measurement protocol.
domain assumption Hard-core bosonic approximation accurately encodes the relevant electron-pair operators
Invoked to justify decomposing operators into three self-commuting groups.

pith-pipeline@v0.9.0 · 5705 in / 1344 out tokens · 34218 ms · 2026-05-22T21:22:50.840717+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking (D=3) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The measured elements comprehend the operators defined by the Hard-Core Bosonic approximation... These can be easily decomposed into three self-commuting groups which can be measured simultaneously.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the method achieves a reduction of 30% to 80% in the number of measurement and gates depth... by truncating at a certain stage

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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L. Ding, S. Knecht, and C. Schilling, Quantum Information-Assisted Complete Active Space Optimiza- tion (QICAS), The Journal of Physical Chemistry Let- ters 14, 11022 (2023)

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Reascos, G

L. Reascos, G. F. Diotallevi, and M. Benito, Uni- versal solution to the Schrieffer-Wolff Transforma- tion Generator, quant-ph 10.48550/arXiv.2411.11535 (2024), arXiv:2411.11535

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G. F. Diotallevi, L. Reascos, and M. Benito, SymPT: A comprehensive tool for automating effective Hamilto- nian derivations, quant-ph 10.48550/arXiv.2412.10240 (2024), arXiv:2412.10240. 16 APPENDIX

work page doi:10.48550/arxiv.2412.10240 2024
[51]

Visualization of HCB elements The operators αk, βkl, γkl and δkl presented in Sec- tion IIA account for multiple creation and destruction fermionic operators with all possible spin combinations. In Figure 7 we display a visual representation of all the terms that one needs to take into consideration and why we can interpret them as paired-electrons or qua...

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[52]

F ull results tables Tables I and II show all the numerical results dis- played in Figures 5(a) and 5(c)

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[53]

Close-up visualization Figures 8, 9 and 10 display close-up visualizations of Figures 2 and 3 and 10 molecules from Figure 4, which represent errors in approximating the molecular Hamiltonian operators of the selected examples

work page
[54]

17 (a) Akk (b) Bkl (c) Ckl (d) Dkl Figure 7

Distribution of number of measurements groups and number of measurements for free H 6 samples Figure 11 presents the full distribution of the number of measurement groups and number of measurements for the 100 samples of free geometry H6 that achieves an error below 2 mEh, as shown in Figure 5(a) and Figure 5(c). 17 (a) Akk (b) Bkl (c) Ckl (d) Dkl Figure ...

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[1] [1]

or even to O(N) [14]. The trade-off however is the need for entangling gates in the diagonalizing circuits, which can be a concern considering the fidelity of multi-qubit operators with the hardware being used. Recent results have shown however that using FC grouping together with a circuit optimization procedure to reduce CNOT gate counts can still resul...

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This corresponds to the number operator

Two electrons are destroyed in orbitalk and cre- ated in orbitalk. This corresponds to the number operator. αk = X σ hkka† kσakσ = X σ hkknkσ (8)

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βkl = X σ,σ′ gkklla† kσa† kσ′alσ′alσ (9)

Two electrons are destroyed in orbitall and cre- ated in orbitalk. βkl = X σ,σ′ gkklla† kσa† kσ′alσ′alσ (9)

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This corresponds to the number operators ofk and l

One electron is destroyed in orbitall and created in the same orbitall and another electron is de- stroyedinorbital k andcreatedinthesameorbital k. This corresponds to the number operators ofk and l. γkl = X σ,σ′ gkllka† kσa† lσ′alσ′akσ = X σ,σ′ gkllknkσnlσ′ (10)

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δkl = X σ,σ′ gklkla† kσa† lσ′akσ′alσ (11) In Appendix 1 we present a simple visualization of how the operators act on electron pairs

One electron is destroyed in orbitall and created in orbital k and another electron is destroyed in orbital k and created in orbitall. δkl = X σ,σ′ gklkla† kσa† lσ′akσ′alσ (11) In Appendix 1 we present a simple visualization of how the operators act on electron pairs. Finally, the HCB Hamiltonian has the following ex- pression: HHCB = X k αk + X kl (βkl +...

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reference orbitals

Choose orbital bases B = {Bk}. The orbital bases are given as unitaryN × N ma- trices which operates on the initial set of orbitals, which we will call “reference orbitals”. Note how- ever, that they do not need to be “Hartree-Fock” orbitals, they merely define the reference point for the given matrices. Each matrix inB is compiled into a orbital rotation...

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This will provide the wavefunction of interest |Ψ⟩ = U |0⟩ (18) where U is the quantum circuit, and|0⟩ the quan- tum register

Choose a quantum circuit to prepare the quantum state of interest. This will provide the wavefunction of interest |Ψ⟩ = U |0⟩ (18) where U is the quantum circuit, and|0⟩ the quan- tum register. This is the state of which we aim to compute the expectation value

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Therefore, we can isolate it and elaborate on the latter term, which we will callH ′ for convenience

Iteratively approximate the expectation value of H First, H is transformed into ˜H R1 HCB = (R1HR † 1)HCB (19) and H ′ = (R1HR † 1)res (20) To reconstructH we need to consider both oper- ators, but the former term can be evaluated eas- ily since all the terms can be collected into three commuting groups, as shown before. Therefore, we can isolate it and e...

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⟨Ψ| H |Ψ⟩ ≈ X Rk∈R ⟨Ψ| R† k ˜H Rk HCBRk |Ψ⟩ (21) The crucial point of the method is that an accurate approximation is bound to a correct choice of the or- bital rotations

Accumulate all contributions At the end of the procedure we have collected a series of operators that, if not truncated, accumu- late to the original HamiltonianH. ⟨Ψ| H |Ψ⟩ ≈ X Rk∈R ⟨Ψ| R† k ˜H Rk HCBRk |Ψ⟩ (21) The crucial point of the method is that an accurate approximation is bound to a correct choice of the or- bital rotations. In the following, we ...

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reference orbitals

By arranging the values in rows {0, 1}, {2, 3}, corresponding to the graph nodes, the operation can be represented by the matrix: RG1 π 2 = R{{0,1},{2,3}} π 2 = = R{0,1} π 2 R{2,3} π 2 = = 1√ 2   1 1 0 0 −1 1 0 0 0 0 1 1 0 0 −1 1   (22) The coefficients show that the orbitals are now in an equal superposition, thus, we can interpret the first row as...

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We will represent such circuits graphically as URG1 π 2 ≡ , URG2 π 2 ≡ (24) where the lines represent spatial orbitals (and therefore 2-qubitsinmostencodings). Thecorrespondingunitary operators are: URG1 = UR{{0,1},{2,3}} π 2 = = UR{0,1} π 2 UR{2,3} π 2 = = e π 4 (a† 0↑a0↑+a† 1↓a1↓−h.c.) e π 4 (a† 2↑a2↑+a† 3↓a3↓−h.c.) (25) and URG2 = UR{{0,3},{1,2}} π 2 =...

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Values of Scenario I and II are expressed in the Reordered Jordan-Wigner encoding

Here we considered only linear system examples since results are compatible. Values of Scenario I and II are expressed in the Reordered Jordan-Wigner encoding. (c) Number of measurements required to compute all the contributions given by the general procedure, estimated as in Section IIIB. This represents the cost for each shot of a VQE algorithm. For acr...

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Visualization of HCB elements The operators αk, βkl, γkl and δkl presented in Sec- tion IIA account for multiple creation and destruction fermionic operators with all possible spin combinations. In Figure 7 we display a visual representation of all the terms that one needs to take into consideration and why we can interpret them as paired-electrons or qua...

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F ull results tables Tables I and II show all the numerical results dis- played in Figures 5(a) and 5(c)

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Close-up visualization Figures 8, 9 and 10 display close-up visualizations of Figures 2 and 3 and 10 molecules from Figure 4, which represent errors in approximating the molecular Hamiltonian operators of the selected examples

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17 (a) Akk (b) Bkl (c) Ckl (d) Dkl Figure 7

Distribution of number of measurements groups and number of measurements for free H 6 samples Figure 11 presents the full distribution of the number of measurement groups and number of measurements for the 100 samples of free geometry H6 that achieves an error below 2 mEh, as shown in Figure 5(a) and Figure 5(c). 17 (a) Akk (b) Bkl (c) Ckl (d) Dkl Figure ...

work page