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

Muhammad Faizan, Muhammad Faryad

TL;DR

This paper analyzes the robustness of the Bernstein--Vazirani algorithm in the presence of Pauli noise using a density-matrix framework. It derives closed-form expressions for the success probability under bit-flip, phase-flip, and depolarizing noise and validates them against Qiskit simulations, illustrating how noise and system size jointly limit quantum advantage. The results reveal that increasing the number of qubits without improving qubit quality rapidly degrades performance, highlighting scalability challenges and the need for error mitigation. Overall, the work provides a principled understanding of noise-induced scaling effects in a foundational quantum algorithm and informs hardware-level strategies for reliable quantum computation.

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.

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

TL;DR

This paper analyzes the robustness of the Bernstein--Vazirani algorithm in the presence of Pauli noise using a density-matrix framework. It derives closed-form expressions for the success probability under bit-flip, phase-flip, and depolarizing noise and validates them against Qiskit simulations, illustrating how noise and system size jointly limit quantum advantage. The results reveal that increasing the number of qubits without improving qubit quality rapidly degrades performance, highlighting scalability challenges and the need for error mitigation. Overall, the work provides a principled understanding of noise-induced scaling effects in a foundational quantum algorithm and informs hardware-level strategies for reliable quantum computation.

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 and number of qubits . 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.

Paper Structure

This paper contains 14 sections, 1 theorem, 23 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Bernstein--Vazirani Algorithm
  3. Density Matrix Formalism
  4. Bernstein--Vazirani Algorithm in the presence of noise
  5. Bit flip noise
  6. Phase flip noise
  7. Depolarizing noise
  8. Results & Discussion
  9. Comparison with simulated results
  10. Impact of Noise on the Algorithm
  11. Performance in Low-Noise Regime
  12. Impact of System Size on the Algorithm with Fixed Noise
  13. Quantum Scalability Under Noise
  14. Conclusion and Discussion

Key Result

Proposition 1

Let $f(\mathbf x) = \mathbf s \cdot \mathbf x \bmod 2$ be a Boolean function for $\mathbf s \in \{0,1\}^n$. Then the oracle $U_f$ defined by can be written as a tensor product of single-qubit unitaries: where $Z$ is the Pauli-Z gate and $Z^{s_i}$ denotes $Z$ if $s_i = 1$, and identity $I$ otherwise.

Figures (8)

  • Figure 1: Circuit Diagram of Noiseless Bernstein--Vazirani Algorithm
  • Figure 2: Circuit representation of the oracle $U_f = \bigotimes_{i=1}^n Z^{s_i}$, where each qubit receives a $Z$ gate if the corresponding bit $s_i = 1$, and identity otherwise. This implements the phase oracle for the Boolean function $f(\mathbf{x}) = \mathbf{s} \cdot \mathbf{x} \bmod 2$.
  • Figure 3: Circuit Diagram of Bernstein--Vazirani Algorithm in the presence of noise acting on each qubit independently.
  • Figure 4: Comparison of theoretical and simulated success probabilities for the Bernstein--Vazirani algorithm in the presence of noise. The plot shows results for $n = 1$, $5$, and $9$ qubits. Theoretical results are shown as lines, while simulated results from Qiskit are marked with symbols.
  • Figure 5: Variation of the success probability with respect to the error probability $p$ and number of qubits $n$, illustrating how noise affects the reliability of the Bernstein--Vazirani algorithm.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof