A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures

Dongkeun Lee; Hoon Ryu

REVIEW 3 major objections 5 minor 52 references

A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For sparse tight-binding Hamiltonians, VQE measurement circuits reduce to one per XOR string, with O(M) grouping cost.

desk verdict Solid measurement-grouping method for sparse tight-binding Hamiltonians, but the band-gap application is under-validated because the paper never reports actual energies. read the letter →

arxiv 2505.18027 v1 pith:5WJXJNUG submitted 2025-05-23 quant-ph cond-mat.mes-hall

Dongkeun Lee , Hoon Ryu This is my paper · ORCID

classification quant-phcond-mat.mes-hall

keywords variationalquantumeigensolvertight-bindingmodelstandard-basisoperatorsGHZmeasurementfolded-spectrummethodperovskitebandgapsparseHamiltonianPauligrouping

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

Tight-binding models describe nanoscale electronic structure with sparse Hamiltonians, but using them inside a variational quantum eigensolver has been held back by measurement overhead: Pauli decomposition can produce exponentially many terms, and grouping those terms is NP-hard. This paper proposes writing the sparse Hamiltonian directly as a sum of standard-basis (SB) operators, one per nonzero matrix element, and reading out groups of them with a single extended Bell (GHZ) measurement circuit. Grouping by the XOR string between row and column indices reduces the number of circuits to the number of distinct XOR strings, with O(M) grouping cost for M nonzero elements, and at most N CNOT gates per measurement. The scheme is applied to find the band-gap energies of metal-halide perovskite supercells, where folded-spectrum VQE converges to costs matching exact diagonalization, suggesting practical TB band-structure simulation on near-term quantum devices.

What carries the argument

The load-bearing mechanism is the standard-basis (SB) operator $|z\rangle\langle z'|$, a rank-one matrix that maps one computational basis state to another. A TB Hamiltonian with $M$ nonzero elements is written as $\hat H = \sum o_{z,z'}|z\rangle\langle z'|$ with $M$ terms, each coefficient read directly from the sparse matrix. The grouping step sorts these operators by the XOR string $x = z \oplus z'$; all SB operators in a group are evaluated by one GHZ measurement circuit $U_{\mathrm{GHZ}} = H_0 \prod_{j\in X} CX_{0,j}$ applied to the state $|G\rangle|\psi\rangle$, where $G=H$ or $S\cdot H$ selects the real or imaginary part. The probability difference $p_G(0z)-p_G(1z)$ returns the expectation value, so the circuit count is the number of distinct $x$ groups, and the mapping cost is $O(M)$. The folded-spectrum cost $C(\theta)=\langle\psi(\theta)|(H-\omega I)^2|\psi(\theta)\rangle$ and a brick-wall SU(4) ansatz complete the algorithm for targeting band-edge states.

What would settle it

For a larger supercell, such as a 4x4x4 cell with 12 qubits, run the folded-spectrum VQE many times from random seeds and compare the best converged cost with exact diagonalization of $(H-\omega I)^2$; if the cost never reaches the folded ground state within tolerance, or if the number of circuits exceeds the number of distinct XOR strings $x$ plus the real/imaginary split, the paper's convergence and efficiency claims are falsified.

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Extended reading notes

Core claim

The paper's central claim is that the expectation value of a sparse tight-binding Hamiltonian can be estimated with far fewer quantum circuits than Pauli-based methods by using the standard-basis representation and GHZ measurements. Each nonzero matrix element becomes an SB operator $|z\rangle\langle z'|$, and because row and column indices are bit strings, operators sharing the same XOR string $x = z \oplus z'$ can be measured together: one circuit with ancilla gate $G \in \{H, S\cdot H\}$ and CNOTs on the qubits where $x$ has a 1 yields the real or imaginary part of all those expectation values. The number of circuits therefore drops to the number of distinct $x$ strings, with real and imaginary parts handled separately. The same grouping works when the Hamiltonian is squared for the folded-spectrum cost $C(\theta) = \langle \psi(\theta)|(H-\omega I)^2|\psi(\theta)\rangle$, and numerical experiments on CH$_3$NH$_3$PbI$_3$ supercells reproduce exact-diagonalization band-edge energies while mapping time stays linear in the number of nonzero elements.

Load-bearing premise

The scheme assumes the chosen brick-wall variational circuit can express the band-edge states and that random-start gradient optimization finds the true minimum of the folded cost; neither is guaranteed for larger supercells.

Editorial extensions

If this is right

For a sparse TB Hamiltonian with $M$ nonzero elements, Hamiltonian mapping and grouping cost $O(M)$, so circuit-preparation overhead does not grow exponentially with system size.
The number of measurement circuits scales with the number of distinct XOR strings $x$, which for the tested perovskites is far smaller than raw Pauli decomposition or qubit-wise-commuting Pauli grouping.
The GHZ measurement uses at most $N$ CNOT gates, avoiding the deeper measurement circuits associated with simultaneous measurement of many commuting Pauli strings.
Applying folded-spectrum VQE twice, with $\omega = 0$ for the valence band maximum and then $\omega = -E_{\mathrm{VBM}}$ for the conduction band minimum, yields band-gap estimates consistent with exact diagonalization for $1\times1\times2$, $1\times2\times2$, and $2\times2\times2$ CH$_3$NH$_3$PbI$_3$ supercells.
The shot-count bound derived in the paper gives a practical rule for allocating samples to achieve a target error, and the early-optimization simulations stay within the predicted confidence intervals.

Reading between the lines

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

The same $O(M)$ SB grouping applies to any sparse Hermitian matrix whose nonzero pattern is known, so the method could be reused for other single-particle effective Hamiltonians, such as $k\cdot p$ or Wannier-interpolated models, not just tight-binding.
Because the folded-spectrum cost squares the Hamiltonian, the effective operator is $(H-\omega I)^2$, whose sparsity pattern grows; the paper's circuit-count comparison implicitly assumes this squared matrix remains sparse enough, which may not hold for denser couplings.
On real hardware, gate errors and decoherence add noise beyond the finite-sampling error analyzed here, so a hardware demonstration is needed to test whether the circuit-count advantage survives in practice.
The empirical ansatz depths of 4, 7, and 8 layers suggest that expressivity demands grow with supercell size; a systematic depth-scaling study would test whether the scheme remains trainable as the qubit count grows.

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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

The core contribution is a clean specialization of Kondo et al.'s extended Bell measurement to sparse tight-binding Hamiltonians. Grouping SB operators by the XOR string x, handling Hermitian pairs with z <= z', and splitting real/imaginary parts are sensible and efficient: the O(M) grouping cost is a real advantage over Pauli grouping, which needs NP-hard optimization. The derivations in Appendices A and B are correct, and the shot-count bound in Eq. (10) is a reasonable rule of thumb. The circuit-count comparison in Fig. 7 is convincing: SB grouping needs far fewer circuits than QWC and is competitive with GC, without the overhead of solving clique covers. For anyone doing VQE on sparse empirical Hamiltonians, this is genuinely useful.

The soft spot is the application claim. The abstract says the scheme gives solutions that follow the accurate ones, but the paper never reports EVBM, ECBM, or the band-gap energy. The only quantitative evidence is convergence of the folded-spectrum cost to a loose 1.5e-2 threshold. A low cost means the state has small variance around omega, but it does not pin down the expectation value of H unless the state is exactly an eigenstate. Since the second optimization uses omega = -EVBM from the first step, errors compound. Without the actual energies and their errors against exact diagonalization, the central applied claim is unsupported. This is a load-bearing gap for the perovskite demonstration, though not for the measurement method itself.

The empirically chosen ansatz depth (4, 7, 8 layers) and the absence of trainability analysis are secondary; that is common in VQE and does not invalidate the method. The noisy-sampling test covers only the smallest cell and only early iterations, so it is a limited sanity check rather than a robustness study.

Overall, this is a solid methods paper with an under-validated application section. The measurement grouping deserves peer review, but the referee should require the authors to report final energies with error bars, and to discuss how the cost threshold translates to energy accuracy. I would engage with the paper in review, but I would not trust the band-gap numbers until that evidence appears.

Referee Report

3 major / 5 minor

Summary. The paper proposes a VQE protocol for sparse tight-binding Hamiltonians in first quantization. The Hamiltonian is decomposed into standard-basis (SB) operators, and expectation values are estimated with a GHZ measurement that simultaneously collects all strings z for a fixed XOR string x; Algorithm 1 groups the SB operators in O(M) time. A shot-count bound is derived in Appendix B. The method is applied to folded-spectrum VQE for the VBM and CBM of CH3NH3PbI3 supercells using an SU(4) brick-wall ansatz, and the circuit count and mapping time are compared with Pauli grouping methods.

Significance. The measurement-grouping construction is a genuine contribution: it avoids the NP-hard Pauli grouping step, uses at most N CNOT gates per circuit, and the derivations in Appendices A and B are mathematically sound. If the band-gap application were properly validated, the paper would serve as a useful practical guideline for NISQ-era tight-binding simulations. As it stands, the numerical evidence for the central application claim is incomplete: no final band-gap energies are reported, and the expressivity and trainability of the ansatz are assumed rather than demonstrated.

major comments (3)

[Sec. III, Fig. 5, Eq. (11)] The central claim that the VQE solutions 'follow well the accurate ones' (abstract and Sec. III) is not supported by the reported data. The only quantitative result is that C(theta) converges near the exact-diagonalization value of C, with a convergence threshold of 1.5e-2. A small value of C means small variance of H around omega, but it does not identify which eigenvalue is selected, and the threshold is not an energy tolerance. Moreover, the CBM calculation uses omega = -E_VBM obtained from the first minimization, so any error in E_VBM shifts the CBM target. The paper should report E_VBM, E_CBM, and the band gap for each supercell with exact-diagonalization comparisons, and ideally state fidelities or energy errors, rather than only the cost function.
[Sec. III, ansatz-depth paragraph and Fig. 5] The expressivity of the SU(4) brick-wall ansatz and the success of ADAM optimization are assumed rather than demonstrated. Depths of 4, 7, and 8 layers are chosen empirically for the three supercells, and optimization starts from randomly determined variational parameters. Because VQE optimization is NP-hard in general and barren-plateau phenomena are a known risk for layered ansatze, the scalability claim for larger supercells rests on an unverified premise. Please provide a multi-seed analysis, the distribution of final costs and energies, and evidence that the relevant folded-spectrum minimum is reached reliably.
[Sec. III, Fig. 6 and Eq. (10)] The finite-sampling experiment does not validate Eq. (10) for the actual convergence target. The chosen error rates epsilon1 = 0.5 and epsilon2 = 0.1 are more than an order of magnitude larger than the 1.5e-2 convergence threshold used in Fig. 5, and Fig. 6 shows only the early stage of optimization. Showing that early cost values lie inside the 2-epsilon confidence intervals is consistent with Eq. (10) as a bound but says nothing about whether the final converged band-gap energies are accurate under finite sampling. Please run the full optimization with finite shots and report final energy errors, or explicitly restrict the claim to early iterations.

minor comments (5)

[Sec. II.B.3, Eq. (3)] The string z' is used before the bit string x is introduced; please define z' = z XOR x explicitly at the first occurrence.
[Sec. II.B.4, last paragraph] The text contains a typo ('original appraoch') and the condition z <= z' for Hermitian pairs should be stated in terms of the binary-index ordering used in Algorithm 1.
[Fig. 5 caption] The caption contains typos ('unictells', 'unitells') and should state that the CBM row targets the eigenvalue closest to omega = -E_VBM, not a global minimum of the conduction band.
[Fig. 7 and Sec. III] The efficiency comparison reports only the number of circuits and the mapping time; because the SB and GC measurement circuits differ in CNOT count, a comparison of total CNOT count or total shot count would strengthen the claimed superiority.
[Reproducibility] No data or code availability statement is given; releasing the tight-binding parameters, supercell geometries, and PennyLane scripts would aid reproducibility of Figs. 5-7.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GHZ measurement grouping, shot-count bound, and folded-spectrum runs are derived or benchmarked independently, and the only author-overlap citation is corroborative.

full rationale

Score 0. The central efficiency claim — that expectation values of a sparse TB Hamiltonian require only as many circuits as there are distinct XOR strings x — is derived, not assumed. Appendix A derives the GHZ-measurement identity (Eqs. A1–A5) from which the simultaneous readout of all operators |z⟩⟨z⊕x| for fixed x follows, and Algorithm 1 then groups by that derived property; the circuit count is a mathematical consequence of the measurement scheme, not a fitted or target-dependent quantity. The shot estimate (Eq. 10) is obtained in Appendix B from the sample-variance identity (Eq. 9) by Lagrange multipliers with a state-independent upper bound (Eq. B5); no target or fitted value enters it. The band-gap protocol uses the folded-spectrum cost C(θ)=⟨ψ|(H−ωI)^2|ψ⟩ with ω=0 and then ω=−EVBM, where EVBM is the output of the first optimization; the second step's reference value is not chosen to force the reported gap. The ansatz (Anselmetti et al., Shende et al.), the GHZ/Bell measurement idea (Kondo et al.), and the folded-spectrum method (Peruzzo et al.) are all external citations; the paper explicitly attributes Algorithm 1 to Kondo et al. and claims only the Hermitian-pair and real/imaginary reductions as its own modification, a legitimate incremental claim. The only overlapping-author citation, Ref. [21] (Hoon Ryu), supports the empirical sp3 TB model of halide perovskites alongside the external Ref. [19] (Boyer-Richard et al.), whose externally fitted parameters define the Hamiltonian; this self-citation is corroborative, not load-bearing, and the numerically validated target (the exact-diagonalization value of the cost function) is computed from the same Hamiltonian independently of any fitted parameter. Numerical validation compares the VQE cost against the ED value of the same cost function, which is a legitimate self-consistency benchmark of the solver against classical exact diagonalization of the same model. The empirically chosen ansatz depth (4, 7, 8 layers) is a tuning parameter, not an input derived from the target result. Per the completeness rule, one non-circularity gap is flagged: the paper never reports absolute EVBM, ECBM, or band-gap energies, so the claim that solutions "follow well the accurate ones" rests on cost-function convergence (Fig. 5) rather than on reported energy values; this is an evidence gap for the application claim, not a circular reduction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The scheme introduces no new physical entities. Its quantitative claims rely on published TB parameters (inputs from prior literature), an empirically tuned ansatz depth, and a loose convergence criterion; these are the main hidden inputs.

free parameters (2)

SU(4) ansatz layer count = 4, 7, 8 (for 1x1x2, 1x2x2, 2x2x2 supercells)
Number of ansatz layers empirically determined for each supercell (Sec. III); affects whether the target eigenstates are reachable and hence whether the folded-spectrum minimization converges.
Cost convergence threshold = 1.5e-2
Optimization stopped when the cost function fluctuates within 1.5e-2 (Fig. 5 caption); this loose tolerance may allow the final state to deviate from the exact eigenstate, and the resulting energy error is not reported.

assumptions (6)

domain assumption Tight-binding model with sp3 basis and parameters from [19,21] accurately describes CH3NH3PbI3 electronic structure
The paper uses a published empirical sp3 TB parametrization; if these parameters are wrong, the 'accurate' band-gap values from exact diagonalization are also wrong, so the benchmark is only as good as the model.
domain assumption VBM lies closer to zero than CBM, so setting omega=0 first finds VBM
Stated in Sec. III: 'the VBM is usually closer to zero than the CBM as the hole effective mass in solid systems is generally much larger than the electron effective mass.' The two-step procedure depends on this.
ad hoc to paper The SU(4) brick-wall ansatz with the chosen layer count can represent VBM/CBM eigenstates and ADAM finds the global minimum of the folded-spectrum cost
Layer count is empirically tuned per supercell; no trainability or barren-plateau analysis is provided. VQE optimization is NP-hard in general [11], so convergence to the exact eigenstate is an assumption.
standard math The GHZ measurement probability-difference formula (Eq. 6) computes the desired expectation values
Derived in Appendix A from the action of CNOTs on computational basis states; relies on standard linear algebra.
standard math The sample-variance upper bound in Eq. (10) is valid and guides shot allocation
Derived in Appendix B using Var(A)=E[A^2]-E[A]^2 and the inequality Var(Sum_z c_z |z><z xor x|) <= Sum_z |c_z|^2 p(z) <= max_z |c_z|^2; standard probability bounds.
standard math The folded-spectrum cost function minimum corresponds to the eigenvalue closest to omega
Minimizing <(H - omega I)^2> yields the eigenstate with eigenvalue closest to omega, by the variational principle applied to (H - omega I)^2.

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Cite this review

Pith. "Pith review of A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures." pith.science (2026). https://pith.science/paper/5WJXJNUG

@misc{pith2026250518027,
  author       = {Pith},
  title        = {Pith review of: A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5WJXJNUG}},
  note         = {Machine review of arXiv:2505.18027}
}

read the original abstract

We propose a cost-efficient measurement scheme of the variational quantum eigensolver (VQE) for atomistic simulations of electronic structures based on a tight-binding (TB) theory. Leveraging the lattice geometry of a material domain, the sparse TB Hamiltonian is constructed in a bottom-up manner and is represented as a linear combination of the standard-basis (SB) operators. The cost function is evaluated with an extended version of the Bell measurement circuit that can simultaneously measure multiple SB operators and therefore reduces the number of circuits required bythe evaluation process. The proposed VQE scheme is applied to find band-gap energies of metal-halide-perovskite supercells that have finite dimensions with closed boundaries and are described with a sp3 TB model. Experimental results confirm that the proposed scheme gives solutions that follow well the accurate ones, but, more importantly, has the computing efficiency that is obviously superior to the commutativity-based Pauli grouping methods. Extending the application scope of VQE to three-dimensional confined atomic structures, this work can serve as a practical guideline for handling TB simulations in the noise-intermediate-scale quantum devices.

Figures

Figures reproduced from arXiv: 2505.18027 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Reference graph

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Representation of TB Hamiltonian with SB Operators The SB in linear algebra, a set ofL-dimensional vectors having a single nonzero element of 1, can span the entire vector space RL or CL. The computational basis {|z⟩} for N qubits in quantum computing is nothing but the SB of 3 the pure quantum state, where z ≡ z1 · · ·zN for each zj ∈ {0, 1}. Adopting th...

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The computational basis {|z⟩} for N qubits in quantum computing is nothing but the SB of 3 the pure quantum state, where z ≡ z1 · · ·zN for each zj ∈ {0, 1}

Representation of TB Hamiltonian with SB Operators The SB in linear algebra, a set ofL-dimensional vectors having a single nonzero element of 1, can span the entire vector space RL or CL. The computational basis {|z⟩} for N qubits in quantum computing is nothing but the SB of 3 the pure quantum state, where z ≡ z1 · · ·zN for each zj ∈ {0, 1}. Adopting th...

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In a quantum circuit, the Bell measurement can be implemented using a sequence of a CNOT gate, a Hadamard (H) gate, and Z-basis measure- ments, as shown in Fig

GHZ Measurement Circuit The Bell measurement, being also known as the Bell state measurement, is a joint measurement scheme that projects two qubits onto one of the four Bell states: {|00⟩ ± |11⟩, |01⟩ ± |10⟩}. In a quantum circuit, the Bell measurement can be implemented using a sequence of a CNOT gate, a Hadamard (H) gate, and Z-basis measure- ments, as...

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According to Eq

Expectation Value of SB Operator The GHZ measurement can be used to evaluate the ex- pectation value of a SB operator with an N -qubit ansatz V (θ) and a state |ψ⟩ ≡V (θ)|0⟩. According to Eq. (3), it is easy to get the expectation value of SB operators that consist solely of projective measurements |zj⟩⟨zj| on individual qubits, as depicted in Fig. 1(c) t...

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(4) and (6) clearly indicate the expectation value of a TB Hamiltonian having M nonzero elements can be estimated with O(2M ) circuits

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