Neural-network warm-start for QAOA-MaxCut initialization
Domain: quantum (ai-for-quantum, qaoa, meta-learning)
Topic: Neural-network warm-start for QAOA-MaxCut: comparing MLP-predicted initial parameters against random and fixed initialization across in-distribution and out-of-distribution graph families
Venue: ARC-Bench 2026
Headline metric: iterations_to_target_ratio (minimize)

Research briefing:
The Quantum Approximate Optimization Algorithm (QAOA) prepares a
parameterized quantum state to approximate the maximum cut of a graph.
At depth p=1 the state depends on two real parameters (beta, gamma),
and the algorithm proceeds by classical optimization of those two
parameters against the average cut size sampled from measurements of
the prepared state. The bottleneck for practical QAOA is the initial
parameter guess. Random initialization frequently falls into the
barren-plateau region or onto sub-optimal local minima, requiring many
classical iterations to escape. Brandao et al. 2018 showed
theoretically that the optimal (beta, gamma) values concentrate
around a problem-independent point on certain graph ensembles, so a
fixed initialization like (pi/4, pi/4) is a strong default.

Beyond concentration, a richer strategy is to train a classical
neural network offline to predict the optimal (beta, gamma) given
features of the input graph. Verdon et al. 2019 introduced the idea
as quantum-aware meta-learning. Jain et al. (Quantum 2022, "Graph
neural network initialisation of QAOA") showed that a graph neural
network trained on a few thousand small graphs can predict initial
parameters that reduce the number of subsequent classical
optimization iterations by 5-10x compared to random initialization,
with measurable transfer to out-of-distribution graph families.

The empirical question this topic addresses is whether the MLP-warm-start
strategy actually generalizes off-distribution: a model trained on
Erdős-Rényi graphs of one density should ideally retain its advantage
on regular graphs of a different size. We also want to verify whether
the simpler "fixed init at (pi/4, pi/4)" baseline (from Brandao 2018)
is competitive with the MLP-predicted init at p=1, since the
concentration result suggests that simple defaults may be hard to
beat in the low-depth regime.

Implementation guidance is provided by the quantum-qiskit skill at
stage 10. Implement QAOA on the qiskit AerSimulator statevector
backend; the small graph sizes (n=6 to n=8) make exact statevector
simulation cheap. The MLP is sklearn.neural_network.MLPRegressor
trained offline on a precomputed set of (graph_features, optimal_beta,
optimal_gamma) tuples obtained by grid search on 200 training
graphs. Optimization at test time uses COBYLA with a hard cap of 20
iterations.

Experimental protocol. Two graph families serve as the test bed.
In-distribution: Erdős-Rényi G(n=6, p=0.5) graphs (matched to the MLP
training distribution). Out-of-distribution: 3-regular graphs at n=8
(different size and edge density). Four conditions cover the
initialization strategies: random init, fixed init (beta=gamma=pi/4),
MLP-init evaluated in-distribution, and MLP-init evaluated
out-of-distribution. (MLP-init has 2 conditions because the model
trained on Erdős-Rényi G(n=6, p=0.5) is evaluated on both graph
families to test transfer.) Each (condition, graph family) cell is
averaged over 3 different random graph instances drawn within that
family. Total cells: 4 conditions x 2 graph families x 3 seeds = 24
cells.

Research question: *Does a small MLP trained to predict QAOA-MaxCut
initial parameters from graph features (degree distribution, edge
density, spectral gap) provide a faster warm-start than random
initialization or the Brandao 2018 fixed (pi/4, pi/4) initialization,
and does the advantage transfer to graphs from a different
distribution than the MLP was trained on?*

Hypotheses:
statement: On in-distribution Erdős-Rényi graphs at n=6, the MLP-predicted initialization reaches approximation ratio 0.9 in at most 10 COBYLA iterations on average (3-graph mean), while random initialization requires at least 18 iterations to reach the same target., statement: On out-of-distribution 3-regular graphs at n=8, MLP-predicted initialization still beats random initialization by at least 20 percent in iteration count (3-graph median), demonstrating partial transfer of the learned initialization heuristic., statement: At p=1 the fixed initialization (beta=gamma=pi/4) is competitive with MLP-predicted initialization: the gap in 3-graph-median iteration count between fixed init and MLP-init is at most 30 percent on in-distribution graphs, consistent with the Brandao 2018 concentration prediction that QAOA-1 parameters concentrate around a problem-independent point.

Experiment design:
research_question: Does an MLP trained on small Erdős-Rényi graph features predict QAOA-MaxCut initial parameters that reduce COBYLA iteration count compared to random and fixed init, and does the advantage transfer to graphs of a different distribution than the MLP was trained on?
conditions: name: mlp_init_in_distribution
description: Proposed method evaluated in-distribution. sklearn.neural_network.MLPRegressor(hidden_layer_sizes=(32, 16), activation='relu', max_iter=2000, random_state=seed) trained offline on 200 Erdős-Rényi G(n=6, p=0.5) graphs with (degree_mean, degree_var, edge_density, n_edges) as features and grid-searched (beta_optimal, gamma_optimal) as targets. At test time, predict initial (beta, gamma) for a fresh in-distribution graph, then run COBYLA QAOA optimization starting from that initialization., name: mlp_init_out_of_distribution
description: Proposed method evaluated out-of-distribution. Same MLP as mlp_init_in_distribution (trained on Erdős-Rényi G(n=6, p=0.5)). At test time, evaluated on 3-regular graphs at n=8: predict initial (beta, gamma) using the same feature extractor, then run COBYLA QAOA optimization.
baselines: random_init: classical baseline. Initial (beta, gamma) drawn from Uniform[0, pi/2] x Uniform[0, pi/2] using a per-graph seed independent of optimization seed. Then COBYLA QAOA from that point., fixed_init_pi_over_4: classical baseline from Brandao 2018. Initial (beta, gamma) = (pi/4, pi/4). Then COBYLA QAOA. Tests whether the concentration prediction holds in practice at p=1.
metrics: name: iterations_to_target_ratio
direction: minimize
description: Number of COBYLA iterations required to reach approximation ratio (cut_size / optimal_cut_size) >= 0.9 on each graph, with a hard cap at 20 iterations (cells that don't reach 0.9 within 20 iterations report 20 and success=False). Primary metric. 3-graph median per condition x graph_family cell., name: final_approximation_ratio
direction: maximize
description: approximation ratio (cut_size / optimal_cut_size) achieved after exactly 20 COBYLA iterations, regardless of whether the target threshold was met. 3-graph mean., name: success_rate
direction: maximize
description: Fraction of the 3 random graph instances in a cell that reached approximation ratio 0.9 within the 20-iteration budget.
datasets: name: erdos_renyi_n6_p05
source: networkx.erdos_renyi_graph(n=6, p=0.5, seed=graph_instance_seed)
description: In-distribution test set. 3 fresh random Erdős-Rényi graphs G(n=6, p=0.5), distinct from the 200 graphs used for MLP training. Optimal MaxCut value computed by brute-force enumeration over the 2^6 cuts. QAOA p=1 on AerSimulator statevector backend., name: regular_3_n8
source: networkx.random_regular_graph(d=3, n=8, seed=graph_instance_seed)
description: Out-of-distribution test set. 3 random 3-regular graphs at n=8 (different size and edge density than the n=6 ER training distribution). Optimal MaxCut value by brute-force enumeration over 2^8 cuts. QAOA p=1 on AerSimulator statevector backend.
compute_requirements: gpu_required: no
estimated_wall_clock_sec: 1200

Requirements:

Scoring rubric:
requirements: An empirical study of neural-network warm-start for QAOA-MaxCut at p=1. The agent must (a) precompute 200 Erdős-Rényi G(n=6, p=0.5) training graphs with their optimal (beta, gamma) via grid search to label the MLP training set, (b) train sklearn.MLPRegressor to predict (beta, gamma) from graph features, (c) at test time, predict (beta, gamma) and run COBYLA QAOA optimization starting from that init for up to 20 iterations, (d) compare against random and fixed (pi/4, pi/4) initialization baselines on both in-distribution (ER G(n=6, p=0.5)) and out-of-distribution (3-regular n=8) test graphs, and (e) score H1/H2/H3 with numerical evidence.
judging_note: QAOA warm-start studies are scored on (i) whether the MLP is actually trained offline (cells reporting identical MLP predictions across different graph instances indicate the model is broken or untrained), (ii) whether the 4 initialization strategies actually produce different starting (beta, gamma) points (must be verified by sanity check), (iii) whether COBYLA is run from each init for exactly 20 iterations and the approximation_ratio trajectory is recorded, and (iv) whether H1/H2/H3 are supported by 3-graph-median numerical evidence keyed by (condition, graph_family). Quantitative result leaves use a graded scale.
sub_tasks: requirements: Code-development bucket: QAOA p=1 circuit built via qiskit.circuit.library or hand-coded RX/RZ + CNOT correctly, MLP trained offline on graph features, and 4 initialization conditions actually produce different starting points.
sub_tasks: requirements: QAOA p=1 circuit for MaxCut implemented correctly: H^{otimes n} initial state, then U_C(gamma) = product over edges of exp(-i * gamma * Z_i Z_j), then U_B(beta) = product over qubits of exp(-i * beta * X_i). Run on qiskit AerSimulator statevector backend; observable is the cut Hamiltonian sum_{(i,j) in E} (1 - Z_i Z_j) / 2. Optimization via qiskit_algorithms.optimizers.COBYLA(maxiter=20).
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: Offline phase: generate 200 random ER G(n=6, p=0.5) graphs with a different RNG than the test set. For each, compute the optimal (beta_optimal, gamma_optimal) via a 20x20 grid search on (beta, gamma) in [0, pi/2]^2 maximizing the QAOA p=1 expected cut size. Extract 4 graph features (degree_mean, degree_var, edge_density, n_edges) and fit sklearn.neural_network.MLPRegressor(hidden_layer_sizes=(32, 16), max_iter=2000) to map features -> (beta_optimal, gamma_optimal). The trained MLP is reused for both in-distribution and out-of-distribution test conditions.
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: The 4 initialization strategies produce different starting (beta, gamma) per test graph. Sanity check: for 3 test graphs, log the initial (beta, gamma) for each of the 4 conditions and verify they are not all equal. Each condition's COBYLA optimization is run for exactly 20 iterations, and the cut size at each iteration is recorded so the iterations_to_target_ratio can be computed by replaying the trajectory.
task_category: Code Development
finegrained_task_category: Experimental Setup, requirements: Execution-validity bucket: all cells ran and produced numerically valid outputs.
sub_tasks: requirements: At least 22 cells out of 24 expected (4 conditions x 2 graph families x 3 seeds) completed and produced an iterations_to_target_ratio value (clamped to 20 if not reached). Missing more than 2 cells (10 percent) without documented justification fails this leaf.
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Numerical validity: iterations_to_target_ratio is in [1, 20] (integer-valued, since COBYLA budget is 20); final_approximation_ratio is in [0, 1]; success_rate is in [0, 1]. At least one cell across the entire 24-cell sweep reaches AR >= 0.9 within 20 iterations (otherwise threshold is uncalibrated).
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Results bucket: quantitative tests of H1/H2/H3 plus a per-hypothesis writeup.
sub_tasks: requirements: Quantitative test of H1. On in-distribution ER G(n=6, p=0.5) graphs, does mlp_init_in_distribution reach AR>=0.9 in <= 10 COBYLA iterations on 3-graph mean, while random_init requires >= 18? 100% if MLP <= 10 AND random >= 18, 67% if MLP at least 30 percent better than random, 33% if MLP marginally better than random, 0% if random equals or beats MLP.
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Quantitative test of H2. On out-of-distribution 3-regular n=8 graphs, does mlp_init_out_of_distribution beat random_init by at least 20 percent in 3-graph-median iterations_to_target_ratio? 100% if mlp gain >= 20 percent, 67% if mlp gain 10-20 percent, 33% if mlp marginally faster, 0% if no transfer (random matches or beats MLP).
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Quantitative test of H3. On in-distribution graphs, is the 3-graph-median iteration-count gap between fixed_init_pi_over_4 and mlp_init_in_distribution <= 30 percent (i.e. fixed init is competitive with MLP at p=1, consistent with Brandao 2018 concentration)? 100% if gap <= 30 percent, 67% if 30-60 percent, 33% if 60-100 percent, 0% if MLP is more than 2x better than fixed init (would refute the concentration interpretation).
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Writeup of at least 200 words (submission/README.md ## Agent-produced writeup section) with explicit supported / refuted / inconclusive verdict for each of H1/H2/H3 backed by per-(condition, graph_family) median iteration counts and approximation ratios. References Brandao et al. 2018 concentration result and Jain et al. Quantum 2022 GNN-init paper. Discusses whether the observed out-of-distribution transfer rate is high enough that MLP-init is worth the offline training cost compared to fixed init.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation