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arxiv: 2603.01809 · v2 · submitted 2026-03-02 · 🪐 quant-ph · cs.IT· cs.NA· math-ph· math.IT· math.MP· math.NA

Recognition: 2 theorem links

· Lean Theorem

Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fej\'er Filtering

Chinonso Onah , Kristel Michielsen
Authors on Pith no claims yet

Pith reviewed 2026-05-15 18:28 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITcs.NAmath-phmath.ITmath.MPmath.NA
keywords CE-QAOAFejér filterconstrained quantum optimizationsuccess probability boundsfinite-depth guaranteesphase separationquantum approximate optimization
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The pith

Harmonic cost angles in CE-QAOA induce a Fejér filter that gives dimension-free lower bounds on optimal sampling probability

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

The paper studies finite-layer versions of the Constraint-Enhanced Quantum Approximate Optimization Algorithm, which operates directly on feasible states encoded in block one-hot form. By restricting the cost angles to a harmonic lattice, the cost-phase unitary behaves as a positive Fejér kernel in an auxiliary cost-dephased reference model. Under a wrapped phase-separation condition, this construction produces lower bounds on the single-shot success probability that remain valid for any finite number of layers and any finite number of measurement shots, and that do not grow worse with problem dimension. The explicit guarantee takes the ratio form q0 ≥ x/(1+x), where x depends on the layer count, a phase-gap proxy, and the mixer mass on the optimal set.

Core claim

Restricting cost angles to a harmonic lattice in CE-QAOA exposes a positive Fejér filter acting on the cost-phase unitary U_C(γ) in a cost-dephased reference model. Under a wrapped phase-separation condition, this yields dimension-free finite-depth and finite-shot lower bounds on the success probability of sampling an optimal solution, including the ratio-form guarantee q0 ≥ x/(1+x) with x = (p+1)^2 sin²(δ/2) C_β.

What carries the argument

The positive Fejér filter induced on the cost-phase unitary by harmonic-lattice restriction of the cost angles, within the cost-dephased reference model used only for analysis.

If this is right

  • The success-probability lower bound holds uniformly for all problem sizes because it is independent of dimension.
  • The bound applies after any finite number of layers p and after any finite number of shots.
  • A coherent version of the filter bound can be established without the reference model.
  • Riemann-Lebesgue averaging extends the guarantee to cost-angle choices that do not lie exactly on the harmonic lattice.

Where Pith is reading between the lines

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

  • The filtering viewpoint could be tested on small constrained problems to check whether observed success rates follow the predicted scaling with layer count p.
  • If the phase-separation condition is satisfied by typical constrained landscapes, the method offers a route to provable finite-resource performance in near-term devices.
  • Similar positive-filter constructions might supply guarantees for other variational ansatzes that currently lack dimension-free bounds.
  • Hardware-efficient approximations of positive spectral filters remain an open implementation question that could connect the analysis to actual device constraints.

Load-bearing premise

The wrapped phase-separation condition must hold and the cost-dephased reference model must correctly capture the dynamics relevant to the success-probability bound.

What would settle it

Execute the CE-QAOA circuit with harmonic cost angles on an instance that satisfies the wrapped phase-separation condition and check whether the measured single-shot success probability falls below the predicted ratio x/(1+x).

Figures

Figures reproduced from arXiv: 2603.01809 by Chinonso Onah, Kristel Michielsen.

Figure 1
Figure 1. Figure 1: Depth-p = 3 CE-QAOA for m = 3 blocks of n = 4 qubits. Each layer applies a global cost UC (γℓ) over all mn wires, followed by parallel block-local XY mixers U (j) M (βℓ). 2 Existence of constant feasibility probability Several dynamical properties of the normalized block XY mixer proposed in Def. 1 provide good materials for feasibility guarantees at finite depth. First, it affords a controllable, gapped m… view at source ↗
Figure 2
Figure 2. Figure 2: Fixed-p certification curves for the sufficient Fejér peaking bound (target 1 − ε with ε = 0.1). For each depth p, the bound certifies Pr[x ⋆ ] ≥ 1 − ε whenever Cβ ≥ Cmin(δ; ε, p), i.e. in the region above the corresponding curve. The vertical colorbar encodes p from 101 (blue) to 1010 (red). Monotonicity: increasing either the envelope mass Cβ or the phase-gap proxy δ reduce the certified depth. Conservat… view at source ↗
Figure 3
Figure 3. Figure 3: Phase diagram for the Fejér-based lower bound. Deeper green indicates higher single-shot [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

We study finite-layer alternations of the \emph{Constraint--Enhanced Quantum Approximate Optimization Algorithm} (CE--QAOA), a constraint-aware ansatz that operates natively on block one-hot manifolds. Our focus is on feasibility and optimality guarantees. We show that restricting cost angles to a harmonic lattice exposes a positive Fej\'er filter acting on the cost-phase unitary $U_C(\gamma)=e^{-i\gamma H_C}$ \emph{in a cost-dephased reference model (used only for analysis)}. Under a wrapped phase-separation condition, this yields \emph{dimension-free} finite-depth and finite-shot lower bounds on the success probability of sampling an optimal solution. In particular, we obtain a ratio-form guarantee \[ q_0 \;\ge\; \frac{x}{1+x}, \qquad x \;=\; (p{+}1)^2 \sin^2(\delta/2)\,C_\beta, \] where $q_0$ is the single-shot success probability, $C_\beta$ is the mixer-envelope mass on the optimal set, $\delta$ is a phase-gap proxy, and $p$ is the number of layers. A Coherent equivalent is proved subsequently and a Riemann--Lebesgue averaging extends the discussion beyond exact lattice normalization. We conclude by outlining coherent realizations of near-term-hardware-efficient positive spectral filters as a main open direction for this framework.

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.

Referee Report

3 major / 0 minor

Summary. The paper studies finite-layer CE-QAOA on block one-hot manifolds and derives finite-depth, finite-shot lower bounds on the single-shot success probability q0 of sampling an optimal solution. Under a wrapped phase-separation condition on the spectrum of HC, a positive Fejér filter arises in a cost-dephased reference model (analysis only), yielding the ratio guarantee q0 ≥ x/(1+x) with x = (p+1)^2 sin²(δ/2) C_β, where C_β is the mixer-envelope mass on the optimal set, δ is a phase-gap proxy, and p is the layer count. A coherent version and Riemann-Lebesgue averaging are outlined as extensions.

Significance. If the wrapped phase-separation condition holds for relevant constrained instances, the dimension-free ratio bound and explicit Fejér-filter construction would constitute a meaningful theoretical advance for QAOA variants, supplying concrete finite-p and finite-shot success-probability guarantees that are independent of Hilbert-space dimension. The use of harmonic-lattice cost angles and the explicit ratio form are strengths that could guide hardware-efficient filter realizations.

major comments (3)
  1. [Abstract] Abstract: the ratio-form guarantee q0 ≥ x/(1+x) with x = (p+1)^2 sin²(δ/2) C_β is obtained only after imposing the wrapped phase-separation condition on the spectrum of HC inside the cost-dephased reference model. The Fejér kernel is positive only under this hypothesis; without it the filter can take negative values and the lower bound on sampling probability disappears. The condition is stated as an assumption rather than derived from the problem class, and no verification is provided for any concrete constrained optimization instance.
  2. [Abstract] Abstract: the cost-dephased reference model is explicitly noted as analysis-only, leaving open whether the derived bound transfers to the physical circuit. The manuscript states that a coherent version is proved subsequently and that Riemann-Lebesgue averaging relaxes exact lattice normalization, yet neither step removes the dependence on the separation hypothesis itself.
  3. [Abstract] Abstract: the dimension-free claim is therefore conditional on an instance-dependent modeling assumption whose validity is not demonstrated, which directly affects the applicability of the finite-depth and finite-shot guarantees to typical constrained optimization problems.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, clarifying the conditional nature of the results while proposing targeted revisions for improved clarity. The wrapped phase-separation condition is central to the positivity of the Fejér filter, and we maintain that the paper's contribution lies in deriving explicit finite-depth and finite-shot guarantees under this natural hypothesis for relevant constrained instances.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the ratio-form guarantee q0 ≥ x/(1+x) with x = (p+1)^2 sin²(δ/2) C_β is obtained only after imposing the wrapped phase-separation condition on the spectrum of HC inside the cost-dephased reference model. The Fejér kernel is positive only under this hypothesis; without it the filter can take negative values and the lower bound on sampling probability disappears. The condition is stated as an assumption rather than derived from the problem class, and no verification is provided for any concrete constrained optimization instance.

    Authors: We agree that the ratio guarantee requires the wrapped phase-separation condition to ensure positivity of the Fejér kernel; without it, the filter may take negative values and the lower bound does not hold. The manuscript presents the condition explicitly as a sufficient assumption under which the dimension-free bound is derived, rather than claiming universality. This is a modeling hypothesis that captures instances with adequate phase gaps in the spectrum of HC on the feasible manifold. To address the absence of concrete verification, we will add a short paragraph explaining how the condition can be checked for specific problems (e.g., by inspecting eigenvalue spacings of HC restricted to block one-hot states). This constitutes a partial revision focused on exposition. revision: partial

  2. Referee: [Abstract] Abstract: the cost-dephased reference model is explicitly noted as analysis-only, leaving open whether the derived bound transfers to the physical circuit. The manuscript states that a coherent version is proved subsequently and that Riemann-Lebesgue averaging relaxes exact lattice normalization, yet neither step removes the dependence on the separation hypothesis itself.

    Authors: The cost-dephased reference model serves solely as an intermediate tool to identify the positive Fejér filter. The main text subsequently establishes a coherent version of the bound that applies directly to the physical CE-QAOA circuit under the same wrapped phase-separation condition. The Riemann-Lebesgue averaging extends the result beyond strict lattice normalization but, as noted, preserves the dependence on the hypothesis because positivity of the filter is essential for the probability lower bound. We will revise the relevant section to more explicitly trace the transfer from the reference model to the coherent circuit implementation. revision: partial

  3. Referee: [Abstract] Abstract: the dimension-free claim is therefore conditional on an instance-dependent modeling assumption whose validity is not demonstrated, which directly affects the applicability of the finite-depth and finite-shot guarantees to typical constrained optimization problems.

    Authors: The dimension-free character of the bound is indeed conditional on the wrapped phase-separation assumption, which is instance-dependent. This is stated clearly in the manuscript and is not presented as holding for arbitrary constrained problems. The contribution consists in supplying explicit, computable finite-p and finite-shot guarantees whenever the condition is met, which we expect to cover a meaningful subclass of block one-hot encoded optimization tasks with sufficient spectral gaps. The paper does not include explicit verification on concrete instances because its focus is the general derivation; such checks are instance-specific and are outlined as a direction for follow-up work. We therefore disagree that this conditional framing undermines the result's applicability to the relevant problem class. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under explicit assumptions

full rationale

The ratio guarantee q0 ≥ x/(1+x) with x = (p+1)^2 sin²(δ/2) C_β follows directly from positivity of the Fejér kernel applied to the cost-phase unitary inside the cost-dephased reference model, once the wrapped phase-separation condition is imposed on the spectrum of H_C. The model is explicitly labeled 'used only for analysis' and the separation condition is stated as an assumption rather than derived or fitted from the target problem class. No parameters are redefined in terms of the output bound, no self-citations supply load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The finite-depth and finite-shot claims therefore remain independent of the final success-probability expression given the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central guarantee rests on the wrapped phase-separation condition invoked to produce the Fejér filter effect; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Wrapped phase-separation condition
    Required to expose the positive Fejér filter on the cost-phase unitary in the reference model.

pith-pipeline@v0.9.0 · 5575 in / 1159 out tokens · 43583 ms · 2026-05-15T18:28:00.147193+00:00 · methodology

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Forward citations

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Reference graph

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