[Paper Review] Effects of noise on performance of Bernstein-Vazirani algorithm

Archi Gupta, Priya Ghosh+2 more|arXiv (Cornell University)|May 31, 2023

Optical Network Technologies4 citations

TL;DR

This paper investigates the robustness of the Bernstein-Vazirani quantum algorithm under various forms of glassy (quenched) disorder affecting Hadamard gates, modeling realistic noise in NISQ-era quantum devices. It shows that success probability decreases with increasing disorder strength across all distributions, but for large secret strings, performance depends only on the disorder's mean and spread—demonstrating quantum advantage persists even under noise.

ABSTRACT

The Bernstein-Vazirani (BV) algorithm offers exceptional accuracy in finding the hidden bit string of a function. We explore how the algorithm performs in real-world situations where noise can potentially interfere with its performance. In order to assess the impact of imperfect equipments, we introduce various forms of glassy disorders into the effect of the Hadamard gates used in the Bernstein-Vazirani circuit. We incorporated disorders of five different forms, viz., Haar-uniform with finite cutoff, spherical Gaussian, discrete circular, spherical Cauchy-Lorentz, and squeezed. We find that the effectiveness of the algorithm decreases with increasing disorder strength in all cases. Additionally, we demonstrate that as the number of bits in the secret string increases, the success probability of correctly guessing the string becomes increasingly insensitive to the type of disorder and instead depends only on the mean and spread of the disorder. We compare our results with the performance of the analogous classical algorithm in the presence of similar noise. When the length of the secret string is small or moderate, the quantum BV algorithm is found to be more efficient compared to the classical algorithm for almost all types of disorders under consideration, unless the strength of the disorder is very high and the disorder follows a discrete circular distribution. However, if we move to extremely large secret strings, the success probability of the disordered BV algorithm merges with the success probability of the disordered classical algorithm for all considered disorders having arbitrary strengths. The limit on the length of the string after which the efficiency of the quantum algorithm becomes equivalent to the classical algorithm depends on the amount of disorder and not on the type of disorder.

Motivation & Objective

• To evaluate the performance of the Bernstein-Vazirani algorithm under realistic noise conditions affecting quantum hardware.
• To model noise as glassy (quenched) disorder in Hadamard gates, representing imperfections in quantum gate operations.
• To compare quantum and classical algorithm performance under identical noise conditions, particularly for long secret strings.
• To determine whether the algorithm's success probability depends on the specific form of disorder or only on its statistical properties (mean and spread).

Proposed method

• Introduces five types of glassy disorder into Hadamard gates: Haar-uniform with finite cutoff, spherical Gaussian, discrete circular, spherical Cauchy-Lorentz, and squeezed distributions.
• Models each noisy Hadamard gate as a rotation with random angles θk and φk drawn from specified probability distributions.
• Computes the final state of the quantum circuit using density matrix formalism and evaluates the probability of measuring the correct secret string.
• Averages the success probability over many realizations of the disorder to obtain disorder-averaged performance metrics.
• Applies the central limit theorem to analytically show that for large n, all disorders with well-defined mean and standard deviation have equivalent impact on performance.
• Compares quantum results with a classical counterpart where the same disorder is introduced into classical logic gates.

Experimental results

Research questions

• RQ1 How does the success probability of the Bernstein-Vazirani algorithm degrade under different types of glassy disorder in Hadamard gates?
• RQ2 Does the form of the disorder (e.g., uniform, Gaussian, Cauchy) significantly affect algorithm performance, or is it only the mean and spread that matter?
• RQ3 For large secret strings, does the algorithm's performance become independent of the specific disorder type and depend only on its statistical parameters?
• RQ4 How does the quantum algorithm’s performance compare to the classical algorithm under identical noise conditions, especially for long secret strings?
• RQ5 Under what conditions does the quantum advantage of the Bernstein-Vazirani algorithm break down in the presence of noise?

Key findings

• The success probability of the Bernstein-Vazirani algorithm decreases monotonically with increasing disorder strength across all five disorder types.
• For large secret strings (n=10), the success probability becomes insensitive to the specific form of disorder and depends only on the disorder’s mean and standard deviation, as predicted by the central limit theorem.
• The classical algorithm becomes exponentially inefficient with increasing string length, even in the noiseless case, highlighting the quantum advantage.
• The quantum algorithm outperforms the classical one for all disorder types and strengths, except in the case of strong discrete circular disorder with moderate n (n=2), where classical performance briefly surpasses quantum.
• Fitting curves for success probability show exponential decay with disorder strength, with parameters depending on n; for n=10, the decay is nearly complete with dU=0 and dG=dC=0.
• For n=10, the fitting parameter b for all distributions (uniform, Gaussian, Cauchy, discrete) increases significantly (e.g., bU=10, bG=10, bC=9.3, bD=35.4), indicating a sharp drop in success probability at high disorder strength.