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arxiv: 2504.01694 · v2 · submitted 2025-04-02 · 🪐 quant-ph · cs.ET

Iterative Interpolation Schedules for Quantum Approximate Optimization Algorithm

Anuj Apte , Shree Hari Sureshbabu , Ruslan Shaydulin , Sami Boulebnane , Zichang He , Dylan Herman , James Sud , Marco Pistoia This is my paper

Pith reviewed 2026-05-22 21:52 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET
keywords QAOAparameter schedulesinterpolationorthogonal basisquantum optimizationSherrington-KirkpatrickLABS problem
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The pith

Iterative interpolation of orthogonal basis coefficients constructs effective QAOA schedules at depths exceeding 1000 layers.

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

The paper develops an iterative approach to optimize QAOA by representing parameter schedules as expansions in orthogonal functions and refining the expansion while increasing the number of layers. This reduces the optimization burden from 2p independent parameters to a small fixed number of basis coefficients at each stage. The method is supported by bounds from approximation theory that limit how many coefficients are needed when the schedules are smooth. It demonstrates improved results on the Sherrington-Kirkpatrick model, portfolio optimization, and LABS problems, including near-optimal performance at unprecedented depths for LABS and exact solutions for SK with modest depth increases. Readers interested in variational quantum algorithms would care because this sidesteps the parameter explosion that otherwise limits practical QAOA depths.

Core claim

Optimal QAOA parameter schedules can be expressed in an orthogonal function basis and constructed iteratively by optimizing a small number of coefficients while increasing circuit depth. Jackson's theorem provides bounds on the number of coefficients needed under Lipschitz continuity. This produces better results than prior methods on three benchmark problems, including near-optimal LABS merit factors at depths exceeding 1000 layers and exact SK solutions at modest depths.

What carries the argument

Iterative interpolation of QAOA parameter schedules expressed in an orthogonal function basis, with coefficient count increased alongside circuit depth.

If this is right

  • Better performance is achieved with fewer optimization steps than existing methods on SK, portfolio optimization, and LABS.
  • Near-optimal merit factors are reached for the largest LABS instances using schedules with more than 1000 layers.
  • Mild growth in QAOA depth suffices to solve the SK model exactly.
  • High-quality schedules become feasible at circuit depths an order of magnitude beyond previous reach.

Where Pith is reading between the lines

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

  • If the smoothness of optimal schedules holds across other variational algorithms, the same interpolation approach could reduce optimization cost in those settings as well.
  • Numerical exploration of QAOA scaling at depths of thousands of layers becomes practical, allowing direct checks of performance trends that were previously out of reach.
  • The observation of exact SK solutions at modest depths suggests a possible relationship between problem structure and the depth at which QAOA saturates that could be tested on related spin-glass models.
  • Alternative choices of orthogonal basis or adaptive rules for adding coefficients might yield further reductions in the number of optimization steps required.

Load-bearing premise

Optimal QAOA parameter schedules are smooth enough functions of layer index to be approximated well by a small number of orthogonal basis coefficients.

What would settle it

A direct test showing that the number of coefficients required grows rapidly with depth or that full independent optimization of all 2p parameters outperforms the interpolated schedules at large p on LABS would falsify the efficiency claim.

Figures

Figures reproduced from arXiv: 2504.01694 by Anuj Apte, Dylan Herman, James Sud, Marco Pistoia, Ruslan Shaydulin, Sami Boulebnane, Shree Hari Sureshbabu, Zichang He.

Figure 1
Figure 1. Figure 1: FIG. 1. As [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: the first panel shows a smoothly varying sched￾ule obtained for p = 100, the second panel demonstrates the rapid decay of mode coefficients in the Chebyshev basis, and the third panel confirms that reconstruction using only the dominant modes preserves the schedule’s performance. Consider a family of orthonormal functions on the unit interval fn, satisfying the orthonormality condition: Z 1 0 fn(t)fm(t)dt … view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the performance of II for the LABS problem across different N values. As demon- [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the scaling of QAOA depth with system size for the SK model and LABS with parameters opti￾mized using II. For the SK model, we randomly generate 600 instances of the problem for system sizes in the range [10, 28] and run II with a maximum depth of p = 150. We report the depth required to reach a 25% overlap with the optimal state and present the median depth required, as well as a 10% confidence inte… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The scaling of QAOA circuit depth as a function [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The fraction of failing instances with varying over [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA uses a parameterized circuit with $p$ layers, where higher $p$ yields better solutions, but requires optimizing $2p$ independent parameters, which is challenging at large $p$. We present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing the work of Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and coefficient count until convergence, our method constructs high-quality schedules for large $p$. We provide theoretical justification using Jackson's theorem and Lipschitz continuity to bound the required number of basis coefficients for a given accuracy. Our approach achieves better performance with fewer optimization steps than existing methods across three benchmark problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. Additionally, we observe that a mild growth in QAOA depth suffices to solve the SK model exactly, a result of independent theoretical interest.

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

2 major / 1 minor

Summary. The paper proposes an iterative interpolation method for QAOA that parameterizes optimal schedules in an orthogonal function basis, optimizes a reduced set of coefficients, and iteratively increases depth p and coefficient count. It invokes Jackson's theorem plus Lipschitz continuity to bound the required coefficients, and reports empirical gains (better performance, fewer optimization steps) on SK, portfolio optimization, and LABS instances, including near-optimal LABS merit factors at p>1000 and exact SK solutions at modest depth.

Significance. If the central claims hold, the method would provide a practical route to high-depth QAOA with reduced classical optimization cost, extending the reachable regime for variational quantum optimization heuristics. The reported SK-model observation would also be of independent theoretical value.

major comments (2)
  1. [Theoretical justification section] Theoretical justification (Jackson's theorem and Lipschitz continuity): the a-priori bound on basis-coefficient count is stated to follow from Jackson's theorem once a Lipschitz constant L is known, yet no numerical estimate of L, no realized truncation error versus p, and no comparison of the recovered schedule against direct 2p-parameter optimization are supplied for the SK, portfolio, or LABS Hamiltonians. This verification is load-bearing for the p>1000 claims.
  2. [Results on LABS and SK model] Experimental claims (LABS p>1000 and SK exact solution): the statements that the iterative procedure yields near-optimal merit factors and solves SK exactly rest on the basis truncation being sufficiently accurate; without reported approximation-error curves or direct-optimization baselines at the same p, the performance advantage cannot be isolated from possible under-optimization of the reference methods.
minor comments (1)
  1. [Abstract and Method] The abstract and method description should explicitly state the number of orthogonal coefficients retained at each iteration and the convergence criterion used.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful review and constructive feedback. We agree that additional numerical verification of the theoretical bounds and truncation accuracy would strengthen the manuscript and will revise accordingly.

read point-by-point responses

  1. Referee: Theoretical justification (Jackson's theorem and Lipschitz continuity): the a-priori bound on basis-coefficient count is stated to follow from Jackson's theorem once a Lipschitz constant L is known, yet no numerical estimate of L, no realized truncation error versus p, and no comparison of the recovered schedule against direct 2p-parameter optimization are supplied for the SK, portfolio, or LABS Hamiltonians. This verification is load-bearing for the p>1000 claims.

    Authors: We agree that the manuscript would benefit from explicit numerical validation. In the revised version we will add estimates of L obtained from the observed smoothness of the optimized schedules for each Hamiltonian, truncation-error curves versus p and coefficient count, and direct 2p-parameter optimization comparisons at moderate p where computation remains feasible. These additions will directly support the high-p claims. revision: yes

  2. Referee: Experimental claims (LABS p>1000 and SK exact solution): the statements that the iterative procedure yields near-optimal merit factors and solves SK exactly rest on the basis truncation being sufficiently accurate; without reported approximation-error curves or direct-optimization baselines at the same p, the performance advantage cannot be isolated from possible under-optimization of the reference methods.

    Authors: We concur that isolating the performance gain requires explicit demonstration of truncation accuracy. The revision will include approximation-error curves versus number of coefficients and p. For the SK model we will add details on verification of exact solutions. Direct-optimization baselines will be supplied for moderate p; at p>1000 such baselines are intractable, so we will rely on the added error curves and iterative convergence to substantiate the claims. revision: partial

standing simulated objections not resolved
  • Direct 2p-parameter optimization baselines cannot be supplied at p>1000 because the classical optimization cost grows linearly with p and becomes prohibitive.

Circularity Check

0 steps flagged

No significant circularity; constructive optimization procedure is self-contained.

full rationale

The paper describes an iterative algorithm that expresses QAOA schedules in an orthogonal basis, optimizes a small number of coefficients, and increases depth until convergence. Performance claims (better results with fewer steps, near-optimal LABS merit factors at p>1000, exact SK solutions) are empirical outputs of running this procedure on external benchmark instances, not quantities defined in terms of the fitted coefficients or recovered by construction. Jackson's theorem is invoked as an external a-priori bound rather than a self-citation or ansatz smuggled from prior work by the same authors. No step reduces a claimed result to a renaming, a fitted input relabeled as prediction, or a uniqueness theorem imported from overlapping authorship. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger is therefore minimal and provisional.

axioms (1)
  • domain assumption Optimal QAOA parameter schedules are Lipschitz continuous functions of layer index.
    Invoked to apply Jackson's theorem for bounding the number of basis coefficients needed.

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Forward citations

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