Task 06: use TensorCircuit native jaxode (1.50x)

optimized_solutions/challenge-06/README.md

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+# Task 06 — use TensorCircuit's native `jaxode` backend
+
+> **Take-home insight:** the acceleration comes from selecting TensorCircuit's
+> native `jaxode` ODE backend for the unchanged adaptive evolution. Gate fusion
+> is secondary; automatic `dt0` and sparse BCOO Hamiltonians do not help.
+
+## Factor speedups
+
+| Factor | Measured speedup | Decision |
+|---|---:|---|
+| TensorCircuit native `jaxode` | **1.529x** single-screen | Keep — dominant |
+| Exact `RZ → RY → RZ` Euler fusion | 1.152x cold; 1.004x steady | Keep — secondary |
+| Diffrax automatic `dt0` | 1.001x | Discard — neutral |
+| BCOO Hamiltonian actions | 0.29x–0.30x | Discard — regression |
+
+![Task 06 factor ablation](factor-ablation.svg)
+
+*Figure — Each panel normalizes the recommended implementation to `1.0x`.
+Removing native `jaxode` produces the dominant slowdown. Euler fusion helps
+mainly in the cold kernel, while sparse BCOO actions are substantially slower.
+The panels retain their stated canonical-screen or isolated-profile context
+and are not multiplied.*
+
+## What the factors mean
+
+- **Native `jaxode`:** run the same `tc.timeevol.ode_evol_global` problem
+  through TensorCircuit's faster native backend without changing the vector
+  field, tolerances, endpoints, or step bound.
+- **Euler fusion:** replace each exact three-gate Euler sequence with one
+  phase-corrected TensorCircuit `U` gate while retaining all parameters.
+- **Automatic `dt0`:** let the solver choose its initial step instead of using
+  the expert's explicit value.
+- **BCOO actions:** replace the small termwise Hamiltonian products with sparse
+  matrix multiplication; this stack makes them substantially slower.
+
+## End-to-end result
+
+All five matched pairs passed. Mean runtime fell from `41.4259 s` to
+`27.5366 s`, for a mean paired speedup of **1.50446x** (5/5 candidate wins).
+These are same-container local-engine measurements.