Deterministic Computation as a Prerequisite for Certifiable Quantum Navigation

Recent advances in quantum sensing-including atom interferometry, optical atomic clocks, and quantum magnetometry-have enabled navigation systems capable of operating without satellite-based positioning. While these technologies dramatically improve measurement precision, they do not address a critical and largely overlooked limitation: the computational nondeterminism of modern navigation pipelines. Navigation is fundamentally a computational process, requiring the digitization, fusion, integration, and transformation of sensor data into guidance decisions. If this computation is not reproducible and replayable, navigation outputs cannot support replay-equivalent audit, certification, or post-incident forensic validation, regardless of sensor accuracy. In this paper, we show that quantum navigation systems cannot satisfy replay-equivalence certification requirements or be safely deployed at scale unless their computational semantics satisfy deterministic guarantees. We formalize this requirement using the Deterministic Computation Law (DCL), which establishes necessary and sufficient conditions for reproducible computation through canonicalization of inputs, representation-invariant state transitions, and replayable reasoning. Applying DCL to inertial, magnetic, and field-based navigation, we derive deterministic bounds on drift accumulation, establish deterministic replay reconstruction of navigation states, and show how sensor fusion becomes a replayable algebraic process under canonical ordering. We further demonstrate that deterministic navigation enables cryptographic trajectory commitments, zero-transmission verification, and complete post-incident forensic reconstruction-capabilities that are unattainable in nondeterministic navigation pipelines. Finally, we establish an impossibility result showing that nondeterministic navigation architectures cannot satisfy replay-equivalence certification requirements, independent of sensor quality or quantum precision. Together, these results show that determinism is not an optimization layer atop quantum navigation, but a prerequisite for its auditable, certifiable, and operational deployment. When paired with quantum sensing, deterministic computation enables a class of navigation systems that are simultaneously GPS-independent, replayable, and legally defensible.