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REVIEW 3 major objections 3 minor 91 references

Systematic improvement of the quantum approximate optimisation ansatz for combinatorial optimisation using quantum subspace expansion

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

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Pith's one-line read QAOA plus a quantum subspace expansion step systematically improves maximum-independent-set solutions and overtakes plain QAOA beyond about 75 nodes.

desk verdict QAOA + QSE is a credible small-N improvement for MIS, but the N=75 crossover rests on an inverted ratio in Eq. (14) and an uncontrolled Fermi-Dirac extrapolation. read the letter →

arxiv 2506.18594 v1 pith:SP6ETXJO submitted 2025-06-23 quant-ph nucl-th

Yann Beaujeault-Taudière This is my paper · ORCID
classification quant-phnucl-th
keywords QAOAquantumsubspaceexpansiongeneratorcoordinatemethodmaximumindependentsetErdős–Rényigraphsapproximationratiofidelitygatecount
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 sets out to show that appending a quantum subspace expansion (QSE) step to a QAOA-prepared state systematically improves the approximation ratio and fidelity for the maximum independent set problem on Erdős–Rényi graphs. In simulations on graphs with 2 to 10 nodes, adding eight trial states roughly doubles the approximation ratio at $N=10$ and raises the fidelity from about 0.15 to above 0.96, while also correcting the Hamming-weight and parity errors left by QAOA. Combining a fitted fidelity curve with logical CNOT and T gate counts for three kernel-estimation strategies, the paper estimates that QAOA-plus-QSE surpasses plain QAOA in cost-to-solution for graphs above about 75 nodes. This matters because QSE is a post-processing layer that can be added on top of an existing optimiser rather than a replacement for it.

What carries the argument

The central object is the quantum subspace expansion (QSE), a generator-coordinate method in which the QAOA output $|\Phi_0\rangle$ is used to build $K$ trial states $|\chi_k\rangle = e^{-i\hat{H}_C t_k}|\Phi_0\rangle$ at equally spaced times $t_k$, and the cost Hamiltonian is diagonalised in the span of these states through the generalised eigenvalue problem $H f = E S f$. Uniform time spacing gives the Hamiltonian and overlap kernels an index-difference (Toeplitz) structure, so the number of matrix elements to estimate grows only linearly in $K$. The overlap kernel is regularised by deleting eigenvectors with eigenvalues below $\varepsilon_\text{cut}=10^{-3}$. The cost comparison is carried by the inequality $\frac{F_\mathrm{GCM}}{F_\mathrm{QAOA}} \gtrsim 4(K-1)(1+1/L') f$, with $f$ set by the kernel-estimation circuit, and by leading-order CNOT/T counts for three estimation strategies: direct Pauli grouping, real-time-evolution finite differences, and linear combination of unitaries.

What would settle it

Run the QAOA-plus-QSE protocol on Erdős–Rényi graphs of density $1/2$ with $L=20$ and $K=8$ at $N=20,30,50,75$; if the measured fidelity ratio $F_\mathrm{QSE}/F_\mathrm{QAOA}$ stops increasing along the fitted curve, or if the logical CNOT/T counts at a fixed approximation ratio do not favour QSE, the predicted crossover at $N>75$ is refuted.

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Extended reading notes

Core claim

The central claim is that a generator-coordinate method in its quantum subspace expansion form, applied to the state produced by a QAOA circuit, yields systematically better solutions to the maximum independent set problem, with the improvement growing as the number $K$ of trial states increases. For Erdős–Rényi graphs of average density $1/2$ and a 20-layer QAOA optimised one layer at a time, the QSE step improves the approximation ratio and fidelity at every tested size ($N=2$ to $10$) and drives the Hamming-weight and parity errors essentially to zero. A Fermi-Dirac fit to the fidelity decay, combined with explicit leading-order CNOT and T gate counts for three ways of computing the Hamiltonian and overlap kernels, gives the estimate that $K=8$ trial states already make the combined method cheaper per solution than plain QAOA for graphs with more than about 75 nodes. The same construction, on a regular time grid, can implement a Gaussian energy filter or imaginary-time evolution, which is what makes the subspace expansion improve with $K$.

Load-bearing premise

The load-bearing premise is that the fidelity decay of QAOA and of QAOA-plus-QSE, measured only for graphs of 2 to 10 nodes, keeps following the fitted S-shaped Fermi-Dirac curves introduced in Sec. V and used in Sec. VIII.C all the way to 75 nodes and beyond; the paper itself says the precise extrapolated value 'bears little meaning,' but the claimed advantage beyond 75 nodes is exactly that extrapolation.

Editorial extensions

If this is right

  • For Erdős–Rényi graphs of density $1/2$ and a fixed 20-layer QAOA, the estimated crossover graph size beyond which QAOA-plus-QSE beats plain QAOA in logical-gate cost is about 75 at $K=8$; larger $K$ is expected to lower that threshold.
  • At every tested size and on both the cube graph and the $K^+_{3,3}$ graph, the QSE step improves approximation ratio and fidelity monotonically with $K$; $K=8$ gives fidelity above 0.98 on the two test graphs and above 0.96 at $N=10$ on random graphs.
  • Because the QSE layer is independent of how the initial state was prepared, the same post-processing can be stacked on QAOA variants, warm-started circuits, or adiabatic protocols, as the conclusion argues.
  • Under the optimistic assumption that QAOA finds near-optimal solutions, the real-time-evolution kernel estimator has the best asymptotic leading-order gate count among the three methods considered, scaling as $N^2 \log^p N$ rather than $N^5$ or $N^6/\log^2 N$.

Reading between the lines

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

  • Editorial: the specific crossover $N\approx75$ is an extrapolation from a fit on $N=2$ to $10$; if the true fidelity decay deviates from the fitted Fermi-Dirac curves, the crossover could move substantially or disappear, even though the small-$N$ improvements are direct numerical results.
  • Editorial: the gate-count comparison fixes the QAOA depth at $L=20$ and sequential layer-by-layer optimisation; allowing the depth to grow with $N$ may erode the advantage, while using QSE to reduce the depth needed for a target accuracy may strengthen it.
  • Editorial: since the QSE construction acts as a Gaussian energy filter, the same eight-state recipe might serve as a verification or error-mitigation layer: a large gap between the filtered energy and the bare variational energy could flag a failed QAOA optimisation.
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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 / 3 minor

Summary. The paper proposes appending a quantum subspace expansion (QSE) step, formulated as a generator coordinate method, on top of a QAOA-prepared state for the maximum independent set problem on Erdős–Rényi graphs. The QSE kernel matrix elements are estimated with Hadamard tests, and the resulting generalized eigenvalue problem is solved classically. Numerical statevector experiments for graph sizes N=2 to 10 (14 random graphs per size) show systematic improvements in approximation ratio, fidelity, Hamming-weight error, and parity error when QSE with up to K=8 trial states is added. A cost model in logical CNOT and T gates is developed for three kernel-evaluation strategies, and a Fermi–Dirac fit to the fidelity data is used to extrapolate a graph-size crossover N*≈75 beyond which QSE is claimed to surpass plain QAOA. The paper also discusses an LCU-based re-encoding of the QSE state and compares the three kernel-estimation methods.

Significance. If the numerical results are taken at face value, the paper contributes a useful and simple post-processing layer for QAOA: the QSE step demonstrably raises fidelity and approximation ratio for small MIS instances, and the detailed logical-gate accounting for the Hadamard/Pauli, real-time-evolution, and LCU approaches is a valuable reference. The main quantitative claim, however, is the extrapolated crossover at N*≈75, and that claim currently rests on an equation-level inversion error and on an unvalidated empirical scaling law fitted to only nine data points. The core small-N result is likely sound, but the headline crossover is not supported as written and needs either substantial correction and validation or removal/qualification.

major comments (3)
  1. [Sec. VIII.C, Eq. (14)] The fidelity ratio in Eq. (14) is inverted relative to the cost condition in Eq. (12). Eq. (12) requires F(GCM)/F(QAOA) to exceed a polynomial factor, but the left-hand side of Eq. (14) is β_QAOA/(1+e^{Nα_QAOA}) × (1+e^{Nα_QSE})/β_QSE = F_QAOA/F_QSE. Since the stated fits have α_QSE < α_QAOA, this ratio decays exponentially with N and cannot cross the growing polynomial right-hand side; therefore the reported value N*=75 cannot be obtained from Eq. (14) as printed. The corrected condition should place F_QSE/F_QAOA on the left, and the factor 2K should be reconciled with the 4(K−1)(1+1/L′) factor appearing in Eq. (12).
  2. [Secs. V and VIII.C] The Fermi–Dirac form F(N)=β(1+e^{αN})^{-1} is introduced as a simple empirical fit and is fitted only to N=2..10 with 14 graphs per size, but the manuscript reports no fitted values of α and β, no goodness-of-fit measures, and no uncertainty estimates. The crossover location is exponentially sensitive to the difference α_QAOA−α_QSE, so extrapolating this fit to N≈75 is not quantitatively justified. The statement in the abstract and conclusion that the approach surpasses QAOA for graphs of size greater than 75 is therefore unsupported by the evidence shown. The authors should either supply validation of the scaling law (for example, tests at larger N or on held-out graph densities) with explicit fit uncertainties, or rewrite the abstract and conclusion to present N* as only an illustrative extrapolation, consistent with the caveat already stated in Sec. VIII.C.
  3. [Sec. VIII.A] The statement that 'the QSE is guaranteed to deliver improved solution with respect to the QAOA' is too strong if applied to the fidelity rather than the variational energy. The trial subspace includes the QAOA state when the evolution-time grid contains t=0, so the generalized-eigenvalue solution cannot have higher energy than the QAOA state; however, the K=3 row in Fig. 7 shows that the fidelity can deteriorate relative to QAOA. The sentence should specify that the guarantee applies to the cost value, not to the fidelity.
minor comments (3)
  1. [Throughout] There are several typographical errors: 'Addtionally' (Sec. I), 'Erdös' (abstract and elsewhere) should be 'Erdős', 'degrade' should be 'degrades' (Sec. VIII.C), 'correpond' in the Fig. 8 caption, 'yieding' and 'smaler' (App. B), 'assummption' (App. A.2), and a missing closing parenthesis after 'see App. B' in Sec. IV.
  2. [Fig. 8 caption] The caption states that the shaded areas correspond to one standard deviation, but it does not specify whether the spread is over the 14 random graph instances only or also includes the QAOA angle optimization runs; please clarify.
  3. [Sec. VIII.A] The parameters L′ and ε_cut are introduced in the text, but the relation between L′ and the number of layers L used in the cost estimate of Eq. (12) and Eq. (14) is not made explicit near their first use; please define all symbols consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the small-N QAOA-plus-QSE improvements are directly simulated, and the N*>75 crossover is a caveated extrapolation rather than a fitted parameter masquerading as an independent prediction.

full rationale

The paper's core small-N claim of systematic fidelity and approximation-ratio improvement is obtained from explicit statevector simulations on Erdős-Rényi graphs of size N=2 to 10 (Fig. 8). These results are observed, not derived from the fitted Fermi-Dirac scaling, so the main qualitative conclusion does not reduce to its inputs. The N*>75 statement is an extrapolation: Section V introduces the fidelity scaling as a 'simple empirical fit', and Section VIII.C inserts that fit into the cost condition to 'extract' a critical graph size. This is a model-based extrapolation, and the paper itself warns that 'the precise value obtained here bears little meaning'. That makes the crossover estimate fragile, but not circular: the predicted crossover is not an input to the fit, and the fitted parameters are not defined in terms of the crossover. The only self-citation, the author's earlier GCM work [30], is used as methodological background and for the generic symmetry-restoration property; it is not load-bearing for the present numerical claims. The apparent inversion of the fidelity ratio in Eq. (14) and the lack of reported fit parameters are correctness and reproducibility concerns, not instances of circular reasoning under the rubric used here.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central N* estimate rests on several hand-chosen parameters and an assumed fidelity scaling law. The small-N improvement is empirical, but the crossover claim is a fit-based extrapolation.

free parameters (7)
  • Fermi-Dirac fit parameter alpha_QAOA = not reported
    Decay rate of the QAOA fidelity fit, Eq. (14), fitted to N=2 to 10 fidelity data; determines how fast F(QAOA) drops, critical for the N* crossover.
  • Fermi-Dirac fit parameter beta_QAOA = not reported
    Plateau scale for the QAOA fidelity fit, from the same fit used in Eq. (14).
  • Fermi-Dirac fit parameter alpha_QSE(K) = not reported
    Decay rate of QSE fidelity for K trial states, fitted to N=2 to 10 data; the inequality alpha_QSE less than alpha_QAOA drives the exponential growth of the fidelity ratio.
  • Fermi-Dirac fit parameter beta_QSE(K) = not reported
    Plateau scale for the QSE fidelity fit, used together with alpha_QSE to compute N*.
  • QAOA layer count L = 20
    Chosen by hand; affects the gate cost in Eq. (A4) and the retained depth L'.
  • QSE evolution time range = [-pi(1-1/K), pi(1-1/K)]
    Chosen by hand to give Toeplitz kernels; the grid may or may not contain t=0 depending on the parity of K.
  • Overlap truncation threshold epsilon_cut = 10^{-3}
    Chosen by hand to regularize the Hill-Wheeler equation; the paper states the parameters are chosen somewhat arbitrarily.
assumptions (6)
  • standard math The Rayleigh-Ritz variational principle applied to the GCM subspace yields the optimal approximation to the ground state within that subspace.
    Used to justify solving the generalized eigenvalue problem (8).
  • domain assumption The MIS cost Hamiltonian (1) correctly encodes the maximum independent set problem as a ground state search.
    Standard mapping from MIS to an Ising-type Hamiltonian, used throughout.
  • ad hoc to paper Fidelity scaling of QAOA and QSE follows F(N) = beta (1 + e^{alpha N})^{-1} for all graph sizes.
    Assumed in Sec. V and used in Sec. VIII.C to extrapolate N*; no derivation and no validation beyond N=10.
  • ad hoc to paper The QSE is guaranteed to deliver an improved solution compared with QAOA.
    Stated in Sec. VIII.A; contradicted by the K=3, K+_{3,3} result in Sec. VIII.B where odd-parity states are incorrectly cancelled.
  • domain assumption The logical gate counts for Clifford+T decompositions (Tables I and II) give a fair cost measure for early fault-tolerant implementations.
    Counts assume specific decompositions of RZZ, CRZZ, and CnZZ gates; other decompositions or error-correction overheads could change the crossover.
  • domain assumption The time-evolved states e^{-i H_C t_k} |Phi_0> form a sufficiently expressive subspace for MIS ground states at the chosen grid.
    The grid is chosen for Toeplitz structure, not guaranteed to contain the QAOA state; for even K, t=0 is not in the grid.

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Pith. "Pith review of Systematic improvement of the quantum approximate optimisation ansatz for combinatorial optimisation using quantum subspace expansion." pith.science (2026). https://pith.science/paper/SP6ETXJO

@misc{pith2026250618594,
  author       = {Pith},
  title        = {Pith review of: Systematic improvement of the quantum approximate optimisation ansatz for combinatorial optimisation using quantum subspace expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SP6ETXJO}},
  note         = {Machine review of arXiv:2506.18594}
}
abstract

The quantum approximate optimisation ansatz (QAOA) is one of the flagship algorithms used to tackle combinatorial optimisation on graphs problems using a quantum computer, and is considered a strong candidate for early fault-tolerant advantage. In this work, I study the enhancement of the QAOA with a generator coordinate method (GCM), and achieve systematic performances improvements in the approximation ratio and fidelity for the maximal independent set on Erd\"os-R\'enyi graphs. The cost-to-solution of the present method and the QAOA are compared by analysing the number of logical CNOT and $T$ gates required for either algorithm. Extrapolating on the numerical results obtained, it is estimated that for this specific problem and setup, the approach surpasses QAOA for graphs of size greater than 75 using as little as eight trial states. The potential of the method for other combinatorial optimisation problems is briefly discussed.

Figures

Figures reproduced from arXiv: 2506.18594 by the authors.

Figure 2
Figure 2. FIG. 2. Circuit implementing the Hadamard test for esti [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. |0⟩ H Rz(ϕ) • H |Ψ0⟩ / N e −iHˆC (tj+tk′ −tk) FIG. 3. Circuit implementing the Hadamard test plus RTE method for estimating ⟨e −iHˆC tj ⟩kk′ ≡ ⟨Ψk|e −iHˆC tj |Ψk′ ⟩. The circuit is run for all values of tj , yielding the ex￾pectation values {⟨e −iHˆC tj ⟩kk′}j=1,··· ,p. These are even￾tually combined in a post-processing step, giving Hkk′ = Pp j=1 aj ⟨e −iHˆC tj ⟩kk′ + O(t p ). The same matrix elements can be ued to… view at source ↗
Figure 1
Figure 1. FIG. 1. Circuit for the evaluation of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Circuit implementing the Hadamard test plus LCU [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. LCU circuit encoding the state [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Probabilities of measuring different bit-strings with [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Probabilities of measuring different bit-strings with [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Approximation ratio (top left), fidelity (top right), error in the Hamming weight/particle number (bottom left), and [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 10
Figure 10. Figure 10: The C nX gates can then be implemented in mul￾tiple manners [84–87]. Which decomposition is optimal will, among others, be dictated by hardware-related con￾siderations, lying outside the scope of the present work. The final gate count given in table II reflects a spec…
Figure 10
Figure 10. Figure 10: FIG. 10. Top: decomposition of a [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

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