Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut

1 Introduction

• Earlier parameter-transfer results addressed unweighted MaxCut and the Sherrington–Kirkpatrick model, while weighted MaxCut includes practically relevant problems such as network community detection.The paper identifies this as a gap in prior work on transferring parameters across problem instances.Source: §1 · p. 1
• A single BFGS run initialized with transferred parameters recovers directly optimized approximation ratios to high precision in 96.35% of cases.The reported match is up to 10^-7.Source: §1 · p. 2
• The proposed scheme rescales typical median optimized parameters from unweighted MaxCut instances for weighted instances with varying sizes and weight distributions.The approach is presented as a lower-cost alternative to direct parameter optimization.Source: §1 · p. 2
• The authors report that the transfer scheme can reduce QAOA parameter-optimization cost for weighted problems of practical interest.They provide the data in QAOAKit and code examples for applying their findings.Source: §1 · p. 2

2.1 Setup

• QAOA parameters are chosen to maximize the expected objective, and performance on an instance is measured by its approximation ratio.The depth is p, and the parameter landscape varies β and γ for a fixed objective and depth.Source: §2.1 · p. 3

2.2.1 Triangle-free graph

• For triangle-free graphs at p = 1, weighted objective terms oscillate at frequencies set by edge weights, so their phases can align at small γ1.In the four-node example, this alignment gives a high approximation ratio near γ1 ≈ 0.23π, although a larger-γ1 optimum can be better.Source: §2.2.1 · pp. 3–4, Figure 2 · p. 4

2.2 Objective landscapes with weights

• Small γ_l can align objective-term phases in generic graphs, while larger values may randomize them; numerically, a high-quality maximum appears near zero across layers.The analysis extends the landscape intuition from triangle-free p = 1 graphs to more general cases and motivates numerical evaluation at p ∈ {1, 2, 3}.Source: §2.2.2 · pp. 4, 6

2.2.2 Generic graph

• For generic weighted graphs, γ* scales approximately as O(1/|w|), and rescaling by average absolute edge weight can align weighted and unweighted landscapes near γ1 = 0.The Cauchy-weighted example remains rugged with many local maxima, but retains a prominent maximum near zero; transferring unweighted parameters can seed local optimization.Source: §2.2.2 · p. 5
• For objectives differing only by a positive scale factor w, replacing γ with γ/w preserves the approximation ratio at any QAOA depth.For MaxCut, the objective scales linearly with edge weights, making average absolute edge weight the relevant scaling measure.Source: §2.1 · p. 3, §2.2.2 · p. 5

2.3 Scaling rule for parameter transfer

• The transfer procedure rescales median preoptimized parameters using average degree and average absolute edge weight before generating angles for a new instance.The γ degree factor arctan(1/√(d−1)) is a heuristic for the non-triangle-free and p > 1 transfer setting, chosen for empirical performance at the low depths studied.Source: §2.3 · p. 6

3 Results of Numerical experiments

• A 2.0 p.p. median approximation-ratio decrease with transferred parameters falls to 1.2 p.p. with KDE across the full dataset.Figure 4 compares direct optimization and transfer for p = 1, 2, 3 and varying weight distributions.Source: Figure 4 · p. 7, §3 · p. 9
• A fixed set of median parameters, derived from optimized 9-node unweighted graphs, is rescaled and transferred across graph sizes and weight distributions.The evaluation dataset contains 34,701 weighted graphs with up to 20 nodes.Source: §3 · pp. 6–8
• The median transfer gap grows from 1.2 p.p. at p = 1 to 3.0 p.p. at p = 3, and from 1.0 p.p. for uniform (0, 1) weights to 3.6 p.p. for exponential weights.The reported gap also increases with weight variance; even the p = 3 exponential case has a 4.5 p.p. decrease.Source: §3 · p. 7, Figure 5 · p. 8
• 96.35% of instances recover the directly optimized approximation ratio to 10^-7 precision after one BFGS run from transferred median parameters.In 96.34% of cases, the recovered parameters also match the directly optimized parameters up to 10^-4 after accounting for symmetries.Source: §3 · p. 8
• Transferred and optimized β values remain close, while γ distances increase on distributions where transfer performs worse and remain below 0.1 at the median.Some growth in parameter distance with p reflects the increasing dimensionality of parameter space.Source: §3 · p. 8, Figure 6 · p. 9
• For truncated Cauchy weights, direct transfer is less effective, but a BFGS run from transferred parameters recovers equivalent ratios in 37.44% of cases.These weights have high variance and greater variation between instances; 99.9% of large β-distance cases have optimized γ components of both signs.Source: §3 · pp. 8–9, Figure 6 · p. 9
• Ten additional KDE samples reduce the median transfer gap by 40%, from 2.0 p.p. to 1.2 p.p.; KDE improves on median-only transfer in 89.5% of cases.The method evaluates the median parameters plus 10 samples and reports the best approximation ratio.Source: §3 · p. 9

4 Discussion

• Rescaled parameters from a single typical vector at each depth transfer across tens of thousands of weighted MaxCut instances with performance near direct optimization.The approach reduces the time-consuming and difficult task of finding high-quality QAOA parameters.Source: §4 · pp. 9–10
• The evidence covers only depths p ≤ 3; whether concentration extends to larger depths and graph sizes remains a prospect rather than a demonstrated result.The discussion notes larger p may be needed for future applications.Source: §4 · p. 10
• Success on large typical-case ensembles does not establish success on scientifically or industrially relevant instances; whether those problems suit transfer remains open.The paper identifies truncated Cauchy weights as an extreme example of transfer limitations.Source: §4 · p. 10

Data availability

• The dataset generated for this study is available in QAOAKit, with direct datafile URLs listed in Refs..Source: Data availability · p. 10

A Details of the instances, parameter optimization, and analysis

• The experiments use positive uniform, signed uniform, positive exponential, and signed truncated Cauchy weight distributions to probe variance and negative-weight effects.Source: A Details of the instances, parameter… · p. 12
• The truncated Cauchy weights are bounded by −10^3 and 10^3, producing high-variance samples distinct from the full Cauchy distribution.The exponential distribution is generated with rejection sampling and a cutoff, though the supplied passage does not give its complete value.Source: A Details of the instances, parameter… · pp. 12–13
• Parameters are optimized from random β and γ initializations using BFGS; the chosen result is the seed with the highest approximation ratio, not a guaranteed global optimum.The number of random seeds increases with depth and is larger for truncated Cauchy instances.Source: A Details of the instances, parameter… · p. 13
• The KDE metadistribution uses Gaussian-kernel density estimation, with bandwidth selected by five-fold cross-validation and samples generated by adding Gaussian noise to randomly selected training points.The model is trained using scaled optimized parameters for nonisomorphic 9-node unweighted MaxCut graphs.Source: A Details of the instances, parameter… · p. 13

• For Cw = wC with w > 0, parameters (β, γ/w) on Cw achieve the same QAOA approximation ratio as (β, γ) on C.Source: Proof γ scales with objective value · p. 13
• The expected objective for Cw at (β, γ/w) equals w times the expected objective for C at (β, γ).Source: Proof γ scales with objective value · p. 13
• The maximum and minimum objective values for Cw are each w times the corresponding values for C, supporting the equality of normalized approximation ratios.Source: Proof γ scales with objective value · pp. 13–14

A.1 Parameter symmetries and the parameter distance

• Time reversal maps (β, γ) to (−β, −γ), while β periodicity maps (β, γ) to (β ± π/2, γ).Source: §A.1 · p. 14
• Accounting for symmetries avoids distances driven by arbitrary choices of symmetry sector rather than parameter differences.Source: §A.1 · p. 14
• The parameter distance is made unique by minimizing separation over symmetry-related optimized parameters.Source: §A.1 · p. 14

Disclaimer

• The document is for information purposes and disclaims representations, warranties, and liability regarding the information’s completeness, accuracy, reliability, or effects.Source: Disclaimer · p. 14
• It is not investment research or advice, nor a recommendation or solicitation concerning financial products or transactions.Source: Disclaimer · p. 14
• The U.S. Government retains a license to reproduce and distribute the manuscript, and the Department of Energy will provide public access under its plan.Source: Disclaimer · p. 14