Access Resources and Stability of Error-Correction Capabilities in Continuous Strategic Codes

Abstract

Continuous quantum error correction depends on which parts of a measurement record remain available to the controller at recovery. We study this temporal access using continuous strategic codes, formulated as open continuous process tensors contracted with causal instruments and conditional recovery. For monitored stabilizer codes, a common-optimal-correction criterion identifies when records can be merged without loss and yields a factorization for separated CSS inference. Terminal decision labels and causal persistent memory carry different costs: five persistent classical states reach the complete-record optimum in a finite-strength counting model, whereas its two-window diffusive counterpart has nine terminal decision labels and requires unbounded memory for exact causal recovery. Uniform lower and constructive upper bounds establish an optimal fidelity deficit of order $m^{-2}$ for $m$ persistent states in the diffusive model. Counting and diffusive stability bounds translate open-process approximation into guarantees for deployed-controller performance and retained access advantages, quantified by explicit quantizers and model-mismatch curves. The same process--instrument framework connects storage with persistent environments and irreversible noise to spacetime interrogation and record-selected paths for a logical $T$ gate. These results specify which record distinctions recovery needs, the memory cost of approximating them, and the logical protection retained under model error.