Wave-based dispatch strategies can be optimized using machine learning to improve the efficiency of quantum algorithms for Hamiltonian subspace diagonalization.

A machine-learned dispatch policy — mapping a target Hamiltonian's structural features (locality, term count, symmetry sector, subspace dimension) to a choice among ≥3 wave-based circuit construction strategies (e.g., Trotterized time-evolution, qubitized block-encoding + QSP/QSVT, variational quantum subspace expansion) — reduces expected two-qubit gate count by ≥20% and/or reduces number of measurement shots to reach chemical accuracy (1.6 mHa) by ≥20%, relative to the best single fixed (non-adaptive) dispatch strategy applied uniformly across a benchmark suite of ≥15 molecular/spin Hamiltonians (4–20 qubits), evaluated on both noiseless simulators and one real NISQ device, at statistical significance p<0.05 (paired test across benchmark instances, n≥15, Wilcoxon signed-rank).

1. Assemble benchmark suite: 15–25 Hamiltonians spanning molecular (H2, LiH, BeH2, H2O, N2 in minimal/STO-3G to 6-31G bases, 4–20 qubits after tapering) and spin-model (TFIM, Heisenberg chains, 4–16 sites) systems.
2. Define ≥3 candidate dispatch strategies with fully specified, tunable implementations (Trotter product formulas with variable order/step count; qubitization/QSVT block-encoding; ADAPT-VQE style adaptive ansatz).
3. For each Hamiltonian × strategy pair, run exhaustive/grid hyperparameter search to obtain best-achievable gate count and shot count at fixed target accuracy (1.6 mHa) — this produces ground-truth labels and per-instance oracle performance.
4. Featurize each Hamiltonian (Pauli-term count, locality histogram, commutation graph sparsity, spectral gap estimate, symmetry sector dimension).
5. Train ML dispatch model (gradient-boosted trees as primary; small MLP as secondary) via 5-fold cross-validation to predict which strategy (and coarse hyperparameters) minimizes resource cost.
6. Hold out 30% of instances (stratified by system type) as final test set, untouched during training/model selection.
7. Evaluate: (a) noiseless statevector/tensor-network simulation for gate count and shot count; (b) execution on ≥1 real NISQ device (or high-fidelity noise-model simulator, e.g., Qiskit FakeBackend calibrated to real device) for circuit fidelity and effective accuracy.
8. Compare ML-dispatch vs. each fixed single-strategy baseline vs. simple-heuristic baseline using paired statistics.
9. Ablation: remove ML model, replace with random dispatch and with oracle (upper bound) to bound achievable gain.

# Phase A: Ground truth generation
for H in benchmark_hamiltonians:
    features[H] = extract_features(H)  # term count, locality, symmetry, spectral gap
    for strategy in [Trotter, Qubitization_QSVT, ADAPT_VQE]:
        best_cost[H][strategy] = grid_search_min_resources(
            H, strategy, target_accuracy=1.6e-3_Ha)
    oracle_label[H] = argmin(best_cost[H])

# Phase B: Train dispatch model
X_train, y_train = build_table(features, oracle_label, cost)
model = XGBoostClassifier(objective='multi:softprob').fit(
    X_train, y_train, cv=5, nested=True)
model_baseline_heuristic = DecisionTree(max_depth=3).fit(X_train, y_train)

# Phase C: Evaluate on held-out set
for H in test_hamiltonians:
    strategy_pred = model.predict(extract_features(H))
    circuit = build_circuit(H, strategy_pred, tuned_hparams)
    result_noiseless = simulate_statevector(circuit)
    result_noisy = execute_on_backend(circuit, backend=real_or_fake_device)
    log_resources(circuit), log_accuracy(result_noisy)

# Phase D: Statistics
compare(ml_dispatch_results, fixed_baseline_results, oracle_results,
        heuristic_results, test='wilcoxon', alpha=0.05)