Breaking the Million-Shot Tax: Reverse Bayesian Retrodictive Smoothing (RBRS) via 3-Forward / 2-Backward Fixed-Lag Quantum Trajectory Control

While recent milestones in fault-tolerant quantum computing—most notably surface-code scaling and magic state cultivation—have reduced logical gate error rates by over an order of magnitude, the information extraction layer remains bottlenecked by the Standard Quantum Limit (SQL). Extracting expectation values, eigenvalues, or variational gradients to precision ε = 10⁻³ via conventional projective sampling requires S ≈ O(1/ε²) ≈ 10⁶ shots per circuit evaluation, compounded by non-Markovian 1/f hardware drift that degrades coherence across sequential runs. In this paper, we introduce Reverse Bayesian Retrodictive Smoothing (RBRS), a real-time closed-loop estimation and control framework operating on an interleaved 3-Step Forward Filtering / 2-Step Backward Retrodictive Smoothing (3F/2B) cadence. Rather than treating quantum measurements as isolated classical coin flips or collapsing the computational register mid-flight, RBRS couples weak dispersive ancilla readouts with operator-algebraic smoothing via the Past Quantum State (PQS) formalism and the Quantum Petz Recovery Map. By anchoring the backward retrodictive boundary to known physical symmetry invariants (such as particle-number conservation, gauge invariance, and stabilizer parity subspaces) rather than an unknown final answer, the 2-step retrodictive pass separates algorithmic phase evolution from correlated hardware drift in sub-microsecond FPGA control loops. We prove via the Quantum Cramér-Rao Bound (QCRB) that active feedforward basis alignment driven by the 3F/2B smoother maximizes Quantum Fisher Information (QFI) across sequential macro-cycles, transitioning effective convergence from SQL scaling O(1/√S) to Heisenberg-like adaptive scaling O(1/S) and compressing required hardware execution runs from 10⁶ to ≤ 10³. Version 2.0 Revision Note: This updated release expands Section 2.2 to explicitly formalize the orthogonal dual-channel observability architecture—separating out-of-subspace relaxation/leakage scrubbing (via the stabilizer/symmetry projector Π_S) from in-subspace non-Markovian 1/f phase velocity tracking (via SLD-aligned weak dispersive readout  in the conjugate transverse basis). Additionally, Section 3.2 has been expanded (Eqs. 14–15) to show the explicit two-parameter (S, M) Quantum Cramér-Rao Bound substitution and concrete hardware verification at M = 64 coherent macro-cycles (24.32 µs at 380 ns/cycle) and Petz recovery efficiency η_Petz = 0.5, yielding S ≈ 977 ≤ 10³ shots.