When Is a Probabilistic Qubit Model Valid? Subspace Leakage as a Quantitative Diagnostic for Quantum Architectures

Probabilistic qubit models are indispensable tools in quantum engineering, yet their domain of validity is often assumed rather than explicitly examined. Standard reduced descriptions, including Lindblad master equations, Pauli error channels, and fidelity-based benchmarks, implicitly rely on the assumption that non-computational dynamics remain negligible. In realistic devices, however, physical qubits are embedded in larger Hilbert spaces, and leakage into non-computational degrees of freedom can invalidate these approximations. In this work, I present a minimal, matrix-based framework that treats subspace leakage as a quantitative diagnostic for the validity of probabilistic qubit models. Qubits are defined as projected subspaces of a global state space, and leakage is quantified by the population residing outside the computational subspace. Rather than addressing the origin of quantum probability, the framework focuses on identifying when reduced probabilistic descriptions remain controlled approximations and when higher-dimensional models become necessary. To make this notion concrete, a phenomenological leakage model is applied to a distance-3 rotated surface code. A code-dependent leakage tolerance, ε_QEC, is derived using a two-fold logical error degradation criterion. The resulting tolerance is found to lie on the order of 10⁻⁴ per cycle and depends strongly on whether leakage predominantly affects measurement processes or manifests as data errors, spanning nearly a factor of four for fixed code distance. This sensitivity is not captured by fidelity-based metrics alone. By providing an explicit validity condition for reduced probabilistic models, this work complements existing benchmarks and introduces isolation as an independent diagnostic axis. The framework is broadly applicable across qubit platforms and error-correcting codes, clarifying when probabilistic modeling assumptions in quantum hardware design remain justified.