Quantum Hypercore Case Study ============================ Introduction ------------ This case study demonstrates a complete **hyper-causal DAG (Directed Acyclic Graph)** experiment that integrates counterfactual projection, finite-difference training, and post-training evaluation metrics. It simulates a dynamic system with injected anomalies to assess the **stability**, **consistency**, and **predictive robustness** of a multi-node hyper-causal architecture. Main objectives: 1. Build a synthetic dynamic environment with controlled anomalies. 2. Train a three-node hyper-causal DAG using counterfactual backends. 3. Evaluate robustness through quantitative metrics such as MAPE, Settling Time, and ROC-AUC. --- General Flow Structure ---------------------- The experiment combines deterministic simulation with causal inference and finite-difference optimization. Each backend models a dynamic node connected through directed edges forming a DAG, where causal dependencies propagate forward in time. - **SyntheticScenario**: generates a temporal dataset with sinusoidal base signals and anomaly injections. - **CFBackend**: a parametric backend with recursive :math:`\tanh` transformation and counterfactual anticipation. - **HCGraph**: connects nodes via directed edges to form a fully causal structure. - **Loss functions**: combine MSE, consistency, and coherence components. - **Finite-difference optimization**: updates parameters numerically per epoch. - **DepthScheduler**: progressively increases recursion depth through epochs. --- How to Run ---------- .. code-block:: console # From the project root python -m examples.ex_quantum_hypercore_case_study # Or directly python examples/ex_quantum_hypercore_case_study.py --- Relevant Code Snippets ---------------------- .. literalinclude:: ../../examples/ex_quantum_hypercore_case_study.py :language: python :linenos: :lines: 70-198 :caption: Definition of CFBackend (parametric backend with counterfactual anticipation) .. literalinclude:: ../../examples/ex_quantum_hypercore_case_study.py :language: python :linenos: :lines: 300-381 :caption: Main experiment function quantum_hypercore_case_study() --- Functional Explanation ---------------------- This experiment evaluates a hyper-causal DAG under perturbative dynamics and counterfactual projections. It integrates predictive, causal, and diagnostic components in a single computational framework. 1. **Synthetic Dynamic Environment** The `SyntheticScenario` generates temporal signals with injected anomalies: .. math:: x_t = \sin(\omega t + \phi) + \epsilon_t, \qquad x_{t,\text{anom}} = x_t + \delta \mathbf{1}_{t \in \mathcal{A}} where :math:`\epsilon_t` represents Gaussian noise and :math:`\mathcal{A}` marks anomaly indices. This allows controlled evaluation of the model’s ability to adapt to irregular perturbations. 2. **Counterfactual Backend** Each backend implements recursive dynamics: .. math:: S_t = \tanh(w S_{t-1} + b) and generates counterfactual future branches: .. math:: S_{t+1}^{(k)} = S_t + \mathcal{P}_k(S_t) where :math:`\mathcal{P}_k` are perturbation operators encoding “what-if” transformations. This enables simultaneous evaluation of nominal and hypothetical state trajectories. 3. **Hyper-Causal Graph Structure** The DAG is defined as: .. math:: \mathcal{G} = (\mathcal{V}, \mathcal{E}), \quad \mathcal{V} = \{A, B, C\}, \quad \mathcal{E} = \{A \to B, A \to C, B \to C\} Each node receives upstream states, applies its transformation, and emits a projected state to downstream nodes. This simulates hierarchical causal propagation within a directed structure. 4. **Loss Composition** The total loss function integrates predictive, temporal, and coherence constraints: .. math:: \mathcal{L}_{total} = \mathcal{L}_{task} + 0.5\,\mathcal{L}_{consistency} + 0.3\,\mathcal{L}_{coherence} - **Task loss**: prediction accuracy. - **Consistency loss**: stability of consecutive states. - **Coherence loss**: alignment across projected branches. Each loss term contributes to maintaining causal smoothness and coherent evolution. 5. **Training via Finite-Difference Gradients** Instead of backpropagation, gradients are computed numerically: .. math:: g_i = \frac{\mathcal{L}(\theta_i + \epsilon) - \mathcal{L}(\theta_i - \epsilon)}{2\epsilon} Parameters are updated using simple gradient descent with adaptive learning rate control. 6. **Evaluation Metrics** After training, several metrics are computed to quantify system stability and sensitivity: - **MAPE** – Mean Absolute Percentage Error. - **Overshoot** – amplitude deviation beyond target. - **Settling Time** – number of samples to reach stability within tolerance. - **Robustness** – correlation between predicted and target signals. - **Early ROC-AUC@H=3** – anomaly discrimination within a 3-step horizon. - **Recall@Lag=3** – sensitivity to delayed anomaly detection. These indicators evaluate both predictive accuracy and control stability under dynamic perturbations. --- Exact Output ------------ .. code-block:: console [Epoch 0] total_before=0.098290 total_after=0.092576 depth=[1, 1, 1] [Epoch 1] total_before=0.069731 total_after=0.062847 depth=[2, 2, 2] [Epoch 2] total_before=0.062847 total_after=0.056539 depth=[2, 2, 2] [Epoch 3] total_before=0.042616 total_after=0.041622 depth=[3, 3, 3] [Epoch 4] total_before=0.041622 total_after=0.039550 depth=[3, 3, 3] [Epoch 5] total_before=0.030317 total_after=0.029850 depth=[4, 4, 4] === Final Metrics === MAPE (%): 19094053004148.468750 Overshoot: 0.000000 Settling Time: 60 samples Robustness: 0.957343 Early ROC-AUC@H=3: 0.224138 Recall@Lag=3: 0.000000 Best epoch snapshot: {'epoch': 5, 'task': 0.019989936715902698, 'cons': 0.015169491076939604, 'coh': 0.007584946879521136, 'total': 0.02985016631822884} Telemetry JSONL → runs/quantum_hypercore_case_telemetry.jsonl Summary: { "best": { "epoch": 5, "task": 0.019989936715902698, "cons": 0.015169491076939604, "coh": 0.007584946879521136, "total": 0.02985016631822884 }, "metrics": { "mape": 19094053004148.47, "overshoot": 0.0, "settling_time": 60, "robustness": 0.9573428590759466, "early_auc_h3": 0.22413793103448276, "recall_lag3": 0.0 } }