Per-shot observable expectations and Pauli-target OBSERVABLE_INCLUDE (like Clifft's EXP_VAL) #249

Motivation

Clifft's end-to-end magic state cultivation results (arXiv:2604.27058) rely on its EXP_VAL P instruction. At its position in the circuit, it returns the exact value ⟨ψ|P|ψ⟩ for each shot as a float, without noise and without collapsing the state. The probe is placed just before the final MPP that measures Y_L (convert_s_to_t.py, run_end2end_bruteforce.py).

|T⟩ is not a Pauli eigenstate, so a sampled MPP Y_L comes out wrong in (1−1/√2)/2 ≈ 14.6% of shots even without errors. Estimating an infidelity of 1e-6 to within 10% then takes about 2.5e13 shots, compared with about 1e8 using the exact per-shot value. Tsim can't reproduce these runs today.

Proposal

1. Accept OBSERVABLE_INCLUDE(k) P with Pauli targets. Tsim currently rejects them in src/tsim/core/parse.py. Stim already defines this as a noiseless Pauli term of observable k that doesn't collapse the state, so no new instruction is needed. With S[T] tags, Stim reads the circuit as the S-proxy, where the term is deterministic, so the DEM and the decoder can still be built from the same circuit.
2. Return the per-shot expectation ⟨(−1)^obs⟩ as a float. CUDA backend #219 already does this with observable_expectations, but only on CUDA. Add it to the CPU lane executor (in float64) and to sample().

Caveats

• Conditioning: the expectation is only exact if it is conditioned on every detector used downstream, both for postselection and for decoding. When a postselected detector is settled after the observable's measurement, CUDA backend #219 falls back to the sampled ±1. Detectors that the decoder reads are not covered. It may be better to schedule the observable's measurement after all detectors of its component.
• Non-Clifford states: a sampled bit for a term that doesn't collapse the state is only well-defined if the term comes at the end of the circuit or commutes with everything after it. Anticommuting terms at the same point, such as X_L, Y_L and Z_L, have expectations but no joint distribution of bits, so they must not be lowered to sequential MPPs. A first version could accept only terms at the end of the circuit and evaluate each one without collapsing the state.

Related: #21 (DEM for non-deterministic observables).