Analysis of the Bernstein--Vazirani Algorithm in the presence of Pauli Noise

TL;DR

This paper analytically examines how the Bernstein--Vazirani quantum algorithm's success rate is affected by different types of Pauli noise, highlighting the importance of qubit quality for maintaining quantum advantage.

Contribution

It provides exact success probability formulas under various Pauli noise models, revealing the impact of noise on quantum speedup as system size grows.

Findings

Success probability decreases with noise and system size.

Performance degradation varies among noise types.

Scaling without improving qubit quality reduces quantum advantage.

Abstract

We analytically investigate the robustness of the Bernstein--Vazirani algorithm in the presence of bit flip, phase flip, and depolarizing noise using the density matrix formalism. We derive the exact expressions for the algorithm's success probability as a function of the error probability $\boldsymbol{p}$ and number of qubits $\boldsymbol{n}$. The analysis compares the three noise models and reveals how performance degrades with increasing system size under standard Pauli noise models. Most importantly, we show that scaling up quantum systems without simultaneously improving qubit quality leads to a sharp decline in ideal quantum speedup.