Noise Sensitivity of Deutsch's Quantum Algorithm

Abstract

Noise accumulation remains a central limitation on the reliable execution of quantum algorithms. This study analyzes the sensitivity of Deutsch’s algorithm to global depolarizing noise using the density-matrix formalism. First, a general analytical expression is derived for the success probability of a quantum circuit subjected to repeated depolarizing channels. The expression relates algorithmic performance to the ideal success probability, the rank of the success projector, and the cumulative contraction factor across circuit layers. The framework is then applied to the two-qubit Deutsch circuit, with a global depolarizing channel introduced after each of its three effective unitary layers. This yields the closed-form success probability PD(λ)=(1+λ3)/2, where λ denotes the survival parameter of the channel. The analytical result is numerically verified in MATLAB for all four Boolean oracles over 404 parameter–oracle combinations. The maximum discrepancy between the analytical and numerical probabilities is 6.661×10-16, while the maximum trace error is 7.772×10-16, indicating agreement within double-precision floating-point accuracy. The results demonstrate how repeated global depolarization progressively reduces the ideal contribution to the output state and drives the classification probability toward the random-guessing limit. Although the model does not represent hardware-specific noise, it provides an exactly solvable benchmark for studying cumulative noise effects in elementary quantum circuits.