A Mathematical Theory of Correlated Error Accumulation in Quantum Circuits

<title>Abstract</title> <p> We develop a rigorous multi-level mathematical framework for correlated error accumulation in quantum circuits, inspired by the layered theoretical development of randomised benchmarking. Three explicit axioms (A1–A3) with measurable violation conditions replace ad hoc assumptions. The Correlated Error Accumulation Theorem (CEAT) proves that for C(k)∼k ^−β , the accumulation operator T ⁿ (C)=ΣC(|i − j|) satisfies T ⁿ (C)∼n ^α with α = max(1,2 − β), proved via Euler–Maclaurin expansion. A Stability Theorem with three cases (additive, multiplicative, and truncated perturbations) establishes robustness; a Universality Theorem defines the result precisely through the theory of regular variation. The perturbation bound |C ^GHZ  − φ²/4|≤φ⁴/48 (proved via alternating series) and five explicit validity conditions B1–B5 connect the theorem to circuit infidelity. All five NLS regularity conditions are proved directly for the K–R model; consistency (Jennrich), asymptotic normality (White), and finite-n upward bias O(n ^β−1 ) are established and Monte Carlo–verified. A Sample Complexity Theorem gives the minimum qubit count N(β,ε). Bayes-optimal regime boundaries are derived from simulation-calibrated distributions, yielding α*=1.57 at n = 2–10. Appendix D characterises four failure modes for Axiom A2. Four figures verify all major results. </p>