Global Readouts of Noisy Quantum Dynamics

How much of a noisy quantum many-body system can be predicted without reconstructing its full state? This paper gives a classical algorithm for global product readouts in a driven two-dimensional spin system with independent local replacement noise. At every fixed positive noise-to-interaction ratio, the algorithm predicts the expectation of any product of one-qubit Hermitian contractions from a specified product preparation. Its bit complexity is polynomial in the complete binary input, inverse additive error and confidence digits. The controls have an explicit finite description, and the time horizon is paid in binary. Full-rank biased replacement is allowed when the Hamiltonian preserves its faithful product reference. Maximally mixed replacement requires no conservation condition. The proof connects sparse observable propagation to a short-time functional that can actually be evaluated on the chosen input. Weighted hypercontractivity and moving coefficient weights control the error. Two localized strip states, with discarded reset registers retained in a positive enlarged representation, give polynomial tensor dimensions and a bounded overlap estimator. Further results give full-output sampling on architectures with a supplied bounded-cut certificate, late-time relative mixing, and an acquired local-generator table that supports predictions for new preparations and drives with explicit calibration, preparation and model-error budgets. A rare-output reduction shows that uniform relative or logarithmic probability evaluation would imply NP = RP. A noisy-code construction separately shows why accurate product means alone need not determine a distribution in total variation. The polynomial degree depends on the fixed noise ratio and faithfulness bound. The results do not cover arbitrary cooling or general interacting thermal reference states. Full-output sampling on a generic two-dimensional lattice at intermediate times remains open.