Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization

Alberto Baiardi; Daniel J. Egger; Sabina Dr\u{a}goi

REVIEW 3 major objections 5 minor 1 cited by

Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that deliberately implementing an approximate, shallower version of a combinatorial problem's Hamiltonian in a QAOA Ansatz -- either a trained quadratic projection of a high-order cost function or a SWAP-truncated…

desk verdict A practical, honestly-written QAOA circuit-simplification study whose QUBO hardware results are solid but whose headline HUBO noise-robustness claim is partly an artifact of gate-count differences in the error model. read the letter →

arxiv 2505.04700 v2 pith:O2E3ZXE2 submitted 2025-05-07 quant-ph

Sabina Drăgoi , Alberto Baiardi , Daniel J. Egger This is my paper · ORCID

classification quant-ph

keywords approximatequadratizationQAOAhigh-orderunconstrainedbinaryoptimizationLABSproblemMax-Cutnoise-awareAnsatzdesignSWAPnetworksdepolarizingnoise

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

The paper tries to establish that, on noisy quantum computers, the best Ansatz for a combinatorial optimization problem is not necessarily the one that most faithfully encodes the problem. For high-order cost functions with cubic and quartic terms, it replaces the Hamiltonian used to build the Ansatz (the trial quantum circuit) by a quadratic approximation that adds no qubits; for quadratic problems, it implements only part of the Hamiltonian by limiting SWAP layers. The price is a reduction in noiseless solution quality and the loss of QAOA's asymptotic convergence guarantee. The payoff is a shallower circuit, and in simulations of 8 to 16 qubits and in hardware experiments the approximate Ansatz samples better solutions than the exact one once realistic noise is included. If this is right, a concrete design principle follows: choose Ansatz depth and problem fidelity together under the device's error budget, rather than insisting on exact problem encoding.

What carries the argument

The machinery is approximate quadratization without ancilla qubits. One variant, the hypergraph clique expansion, replaces each k-body Pauli term by a clique of pairwise ZZ edges, setting edge weights to the weighted mean that minimizes the squared deviation from the original hyperedge weights; for LABS this yields the closed form $w_{ij} = 2 - \frac{I_{ij}}{2N_{ij} + I_{ij}}$, where $I_{ij}$ marks whether the pair appears in the quadratic part and $N_{ij}$ counts hyperedges containing the pair. The variant behind the main robustness result is the variational projection $H'_2(\theta) = \sum_{i>j} \theta_{ij} Z_i Z_j + \sum_i \theta_i Z_i$, whose weights are optimized together with the QAOA angles to minimize the energy of the original quartic Hamiltonian $H_C$. For quadratic problems, the analogous mechanism is SWAP-layer truncation: only $k$ of the $n-2$ SWAP layers needed for full connectivity are applied, so the circuit evolves a sub-Hamiltonian $H_C(k)$ that is a subgraph of the full cost Hamiltonian, and $k$ becomes a tunable noise-versus-fidelity knob.

What would settle it

Run the same depth-two quadratized-versus-standard QAOA comparison on LABS instances with 16, 20, and 24 variables at depolarizing strengths from 0.001 to 0.01; if for the best 5% of samples the standard Ansatz matches or beats the quadratized Ansatz at the noise strengths where the paper reports a threshold advantage, the central robustness claim is falsified. A second check would execute Max-Cut on more than three 40-node three-regular graphs over all SWAP-layer counts and depths; if the optimal number of SWAP layers is always zero or always maximal, the noise-aware design claim collapses.

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Extended reading notes

Core claim

The central discovery is that approximate quadratization -- projecting a quartic Hamiltonian onto a fully connected quadratic Hamiltonian whose ZZ coefficients are optimized as variational parameters -- yields a QAOA Ansatz that is markedly less sensitive to depolarizing noise than the standard depth-two QAOA Ansatz for the 12-variable LABS problem. In the paper's noisy simulations, the best one percent of samples from the quadratized Ansatz barely degrade as noise grows, while standard QAOA energies worsen noticeably; at the ten-percent best-samples level the quadratized Ansatz is 25% and 36% less noise-sensitive than the line-transpiled and all-to-all QAOA Ansatze. The paper also shows, on 40-node Max-Cut circuits, that the measured approximation ratio rises with the number of SWAP layers only up to an optimal point, after which extra layers or extra QAOA depth reduce solution quality. Both findings support the paper's thesis that a noisy device can deliver better solutions from an approximate implementation of the full problem structure than from the exact one.

Load-bearing premise

The load-bearing assumption is that a trained quadratic Hamiltonian captures enough of the low-energy structure of the original high-order problem that sampling from it yields good solutions once noise is added, and this assumption is currently demonstrated on only a single 12-variable LABS instance, with the paper explicitly noting that the usual QAOA convergence guarantee is lost.

Editorial extensions

If this is right

For dense quartic problems such as a fully-connected four-local Hamiltonian, the quadratized Ansatz reduces the two-qubit gate count from $O(n^4)$ to $O(n^2)$; for LABS the reduction is $O(n)$.
For the 12-variable LABS instance studied, the quadratized Ansatz samples sub-optimal but high-quality solutions, for example a 97.1% approximation ratio with 95.7% probability at depth two, while using fewer two-qubit gates than the standard Ansatz.
Because the quadratized Ansatz is less noise-sensitive, there is a threshold noise strength above which it outperforms standard QAOA; the paper identifies such thresholds for the best 1% and 5% of samples.
On hardware, for 40-node three-regular Max-Cut instances, the measured approximation ratio has an interior optimum in the number of SWAP layers and QAOA depth, so adding problem structure beyond that point lowers quality.
The sampling overhead needed to recover noiseless approximation ratios through CVaR post-selection follows the estimate $1/\sqrt{\gamma}$ based on two-qubit gate fidelities, so it can be predicted before execution.

Reading between the lines

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

A natural extension the paper leaves implicit is to use the trained quadratic Hamiltonian as a warm start for QAOA parameters of the exact problem, potentially improving convergence of the full algorithm.
The clique expansion's failure on LABS suggests a testable design rule for approximate quadratization: the quadratic projection should preserve the ground-state Hamming-weight structure of the original Hamiltonian, since converting weight-1 or weight-3 minima into weight-2 minima destroyed the solution quality.
The SWAP-truncation idea generalizes to any connectivity-limited compilation: instead of a fixed $k$, one could order Hamiltonian terms by their SWAP cost or estimated error and include them in decreasing fidelity order; the paper's $k$-layer scheme is the special case where all included terms share one cost class.
Because the HUBO robustness result rests on a single 12-variable instance, the practical scope remains open; a noisy simulation across 16 to 24 variables would show whether the quadratization advantage grows, shrinks, or disappears with system size.

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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 paper does two useful things. First, it trains a quadratic Hamiltonian H'_2(theta) as the QAOA generator for quartic HUBOs, which is genuinely different from ma-QAOA or time-block ansatze and avoids adding ancillas. Second, it gives a clean hardware study of SWAP-layer truncation for Max-Cut on 40-node graphs, showing a real optimum in (k,p) under noise. That second part is the strongest contribution: the ibm_fez data, the MPS-based training, and the simple 1/sqrt(gamma) model for the fitted CVaR quantile all hang together, and the comparison against the earlier Eagle result is informative. The paper is also transparent about losing the QAOA convergence guarantee and about the heuristic nature of the method.

The soft spots are concentrated in the HUBO robustness claim, and one of them is more serious than the reader flagged. Figure 5(a) compares a 374-CZ quadratized circuit against 2094-CZ and 600-CZ standard QAOA circuits under depolarizing noise after each two-qubit gate. For the quadratized circuit, the noiseless survival fraction is (1-lambda)^374, which stays above the plotted alpha=0.01 for all lambda up to 0.01. So its flat best-1% curve can be populated entirely by noiseless shots, and the 'statistically insensitive to noise' statement at alpha=0.01 is largely a gate-count artifact, not evidence that the quadratization preserves useful structure. The alpha=0.05 and 0.1 panels are less vacuous, but the abstract's unqualified robustness claim leans on the alpha=0.01 panel. The HUBO evidence is also a single 12-variable LABS instance, with no code or data release, and the MPS bond dimension of 20 is not convergence-tested. None of this invalidates the QUBO part, which is well-supported.

Who should read this? Anyone building QAOA-style heuristics for current hardware, especially those interested in SWAP routing or CVaR-based post-selection. The quadratization idea is worth knowing about even if the evidence for it is thin. The paper deserves peer review, but a referee should push for more HUBO instances, a noise model that accounts for gate-count differences properly, and a bond-dimension check.

Referee Report

3 major / 5 minor

Summary. The paper addresses the excessive circuit depth and gate count of QAOA for high-order (cubic/quartic) cost Hamiltonians and for dense QUBOs. It proposes two approximate quadratizations of HUBOs—a hypergraph clique expansion and a variational fully connected quadratic Hamiltonian H'_2(θ)—that avoid ancilla overhead, and a SWAP-layer truncation method for QUBOs. Noiseless numerics on 12-qubit LABS show that the variational quadratization samples good suboptimal states with much shallower circuits; noisy simulations purport to show improved noise robustness over standard QAOA. Hardware experiments on ibm_fez with three 40-node 3-regular Max-Cut instances show an optimal number of SWAP layers and QAOA depth, and a CVaR post-selection model reproduces noiseless approximation ratios.

Significance. If established, the core message—that approximating the problem structure in the Ansatz can outperform implementing the full QAOA cost operator under realistic noise—would be practically useful, especially for HUBOs where exact implementation is hopeless on current hardware. The resource-scaling analysis in Table I and Fig. 1 is a useful quantitative contribution, and the hardware study is a genuine benchmark with three graphs and 50,000 samples per point, including a simple analytical noise model for the CVaR post-selection. The paper is also transparent that the variational quadratization is heuristic and that the QAOA convergence guarantee is lost. However, the HUBO noise-robustness evidence is not yet convincing for the reasons given in the major comments; the single-instance, same-cost-function training makes the result a proof of principle rather than a demonstrated method.

major comments (3)

[§III.C, Fig. 5(a)] The flatness of the purple curve at α=0.01 is quantitatively explained by the gate count alone. With 374 CZ gates, the noiseless survival probability under the depolarizing model is (1−λ)^374 ≥ 0.99^374 ≈ 0.023 for λ∈[0.001,0.01], which is larger than α=0.01. Thus the best 100 of 10,000 samples can be entirely noiseless at every plotted noise strength, so the 'statistically insensitive' behavior does not demonstrate that the quadratization preserves useful problem structure under noise. For the standard line Ansatz (2094 CZ gates), the noiseless fraction at λ=0.01 is 0.99^2094 ≈ 8×10^−10, far below α, so its best 1% necessarily includes corrupted samples; the comparison is therefore biased by circuit size. The authors should repeat the analysis for α above the noiseless fraction, explicitly account for the finite-shot noiseless contribution, or otherwise show that the advantage persists after controlling for gate count. The α=0.05 and α=0.1 panels are less vacuous, but the abstract's unqualified 'more robust to noise' relies on the α=0.01 statement.
[Abstract vs §III.C] The abstract states that the noise robustness is demonstrated 'through simulations of systems of 8 to 16 qubits with variable noise strengths,' but the noisy simulations in §III.C cover a single 12-variable LABS instance only. The 8–16 qubit data in Fig. 1 and Table I are circuit-resource counts, not solution-quality simulations under noise. This discrepancy overstates the evidence; the claims in the abstract should match the actual experimental scope, or additional noisy-simulation results at other sizes should be provided.
[Appendix D and §IV, Fig. 6] The noiseless MPS reference used for Fig. 6(a) and for the CVaR fit in Fig. 6(c) is computed at bond dimension 20 with no convergence test. For the deepest circuits (40 qubits, k=9, p=3) the MPS truncation error could be non-negligible, and without a χ-dependence study the claims that the noiseless approximation ratio increases monotonically with k and that the fitted α recovers the noiseless value with RMSE=10^−6 are not fully supported. The authors should report at least one convergence check (e.g., χ=10, 20, 40 for one graph) and specify how the RMSE was evaluated. The hardware finding of an optimal (k,p) in Fig. 6(b) is independent of this issue, but the noiseless interpretation is not.

minor comments (5)

[§III.C] The set 'α∈{0.01,0.05.0.1}' contains a typographical error; it should read 'α∈{0.01,0.05,0.1}'.
[§III.B] There is a missing space in 'Thequadratizationyieldsaninteresting tradeoff'.
[§II, Eq. (1)] The product symbol in Eq. (1) appears as 'pY' in the text; the typesetting should be corrected to a proper product notation.
[Appendix D] The authors cite Ref. [74] for the MPS simulator, but the main text does not state which library or implementation was used; a short sentence identifying the software would aid reproducibility.
[Reproducibility] The HUBO simulation code and data are not released, while the QUBO code is linked in Ref. [71]; making the HUBO code available would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the quadratized-ansatz performance is empirically compared with a standard QAOA baseline optimized on the same objective, and the fitted CVaR quantile is explicitly labeled as a fit.

full rationale

The paper's load-bearing derivation chain is not circular. The quadratized ansatz in Eq. (11) is defined by the variational minimization in Eq. (10) of ⟨ψ'_2|H_C|ψ'_2⟩, and the subsequent figures evaluate exactly this H_C energy; however, the standard QAOA baseline is optimized on the same target Hamiltonian, so the comparison is between two trained ansatz families under an identical noise model rather than a fit being relabeled as a prediction. The noise-robustness claim is empirical and driven by circuit size: the quadratized circuit has 374 two-qubit gates versus 600 and 2094 for the two standard-Ansatz variants, and Fig. 5 reports how the best-α sample energies change with depolarizing strength; that dependence is not imposed by the training objective. The clique expansion uses an algebraic l2 fit (Eq. (7)) and is then scored on H_C, but the paper does not claim the fit equals the score. In Section IV the CVaR quantile α is explicitly fitted ('Here, we fit α such that the approximation ratio measured with CVaRα matches the noiseless one obtained with the MPS simulator'), and the theoretical α_th in Eqs. (18)-(19) is a separate parameter-free estimate; the fitted value is not renamed as a prediction. Self-citations such as Refs. [41], [53], [58], and [59] are used for standard SWAP-network compilation and hardware baselines, not as authority to forbid alternatives, and no uniqueness theorem is imported. No step reduces an equation to its own input by construction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper does not introduce new physical entities. The central claims rest on variational parameters fitted to the optimization objective, a heuristic ansatz structure, a depolarizing noise model, and an MPS simulation parameter, all of which are acknowledged or under-tested in the text.

free parameters (4)

Quadratic Hamiltonian weights theta=(theta_ij, theta_i) in H'_2(theta) = not reported
Optimized with COBYLA to minimize the energy of H_C; the HUBO quadratization's success depends on these values. Location: Eqs. (10)-(12).
QAOA angles beta, gamma = not reported
Standard variational parameters optimized for each problem; they are free parameters for the central benchmarks. Location: Eqs. (1), (9), (11).
CVaR quantile alpha (Fig. 6c) = fitted per graph-depth-k, not tabulated
Fitted with COBYLA so that hardware approximation ratios match MPS noiseless values; used to estimate sampling overhead. Location: Section IV, Fig. 6(c).
Clique expansion edge weights w_ij = weighted mean of hyperedge weights (App. C)
Chosen by minimizing Eq. (7), but the clique method is shown to perform poorly; included for completeness.

assumptions (4)

domain assumption Finite-depth QAOA can find good approximate solutions for combinatorial optimization problems, but performance must be benchmarked per problem.
The paper adopts the standard heuristic status of finite-depth QAOA, citing Refs. [17,22].
ad hoc to paper A variational quadratic Hamiltonian H'_2(theta) can approximate the low-energy structure of the quartic Hamiltonian H_C after optimization.
No proof or guarantee is given; the paper states the approach is inherently heuristic and only demonstrates it on one 12-variable LABS instance.
domain assumption Depolarizing noise applied after each two-qubit gate is a faithful model for hardware noise in the HUBO robustness study.
Used in Section III.C for all noisy HUBO simulations; no hardware validation of HUBO circuits is presented.
domain assumption MPS with bond dimension 20 provides accurate energies for the 40-qubit Max-Cut circuits in the noiseless benchmark.
Used in Appendix D and Fig. 6(a); no convergence test with respect to bond dimension is reported.

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Cite this review

Pith. "Pith review of Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization." pith.science (2026). https://pith.science/paper/O2E3ZXE2

@misc{pith2026250504700,
  author       = {Pith},
  title        = {Pith review of: Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O2E3ZXE2}},
  note         = {Machine review of arXiv:2505.04700}
}

read the original abstract

Combinatorial optimization problems have wide-ranging applications in industry and academia. Quantum computers may help solve them by sampling from carefully prepared Ansatz quantum circuits. However, current quantum computers are limited by their qubit count, connectivity, and noise. This is particularly restrictive when considering optimization problems beyond the quadratic order. Here, we introduce Ansatze based on an approximate quadratization of high-order Hamiltonians which do not incur a qubit overhead. The price paid is a loss in the quality of the noiseless solution. Crucially, this approximation yields shallower Ansatze which are more robust to noise than the standard QAOA one. We show this through simulations of systems of 8 to 16 qubits with variable noise strengths. Furthermore, we also propose a noise-aware Ansatz design method for quadratic optimization problems. This method implements only part of the corresponding Hamiltonian by limiting the number of layers of SWAP gates in the Ansatz. We find that for both problem types, under noise, our approximate implementation of the full problem structure can significantly enhance the solution quality. Our work opens a path to enhance the solution quality that approximate quantum optimization achieves on noisy hardware.

Figures

Figures reproduced from arXiv: 2505.04700 by the authors.

**Figure 1.** Circuit complexity of e −iγHC for Hfull 4 (teal) and LABS (blue) compared to a fully-connected QUBO (purple). Panel (a) shows the number of CZ gates as a function of problem size. Panel (b) shows the CZ gate depth of the transpiled circuits. The continuous lines connect raw data points. The horizontal dashed line corresponds to the number of CZ gates that fit within the median qubit T1 time on ibm_fez. Furthermore,… view at source ↗

**Figure 2.** Construction of shallow Ansatze. (a) and (b) show [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Convergence of the quadratizations as function [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Cumulative distribution function of the LABS [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Evolution of expected performance of depth-two QAOA for LABS versus strength of depolarizing noise for a system [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Performance of QAOA as a function of depth and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Decomposition of an RZZ gate followed by a SWAP gate into a basis gate set made of single-qubit rotations and a CX. Here, two consecutive CX gates cancel. This construction is typical in SWAP networks on linear connectivity, see [PITH_FULL_IMAGE:figures/full_fig_p009…

Discussion (0). Continue with ORCID to comment.

Forward citations

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