Quadratic Advantage with Quantum Randomized Smoothing Applied to Time-Series Analysis

TL;DR

The paper tackles certifiable robustness in quantum machine learning by analyzing how data encoding and perturbation types interact under randomized smoothing. It introduces QuAdRo, a Grover-based method that achieves a quadratic sampling advantage over classical randomized smoothing while relying on basis-state encoding. A constrained $k$-Hamming perturbation model is proposed, with efficient quantum-state preparation methods that preserve meaningful perturbations for basis encodings. The methodology is validated on time-series classification with Bag-of-Words preprocessing, showing improved certified robustness with fewer samples. The results suggest a viable path for scaling quantum robustness certificates to more complex tasks beyond classical capabilities.

Abstract

As quantum machine learning continues to develop at a rapid pace, the importance of ensuring the robustness and efficiency of quantum algorithms cannot be overstated. Our research presents an analysis of quantum randomized smoothing, how data encoding and perturbation modeling approaches can be matched to achieve meaningful robustness certificates. By utilizing an innovative approach integrating Grover's algorithm, a quadratic sampling advantage over classical randomized smoothing is achieved. This strategy necessitates a basis state encoding, thus restricting the space of meaningful perturbations. We show how constrained $k$-distant Hamming weight perturbations are a suitable noise distribution here, and elucidate how they can be constructed on a quantum computer. The efficacy of the proposed framework is demonstrated on a time series classification task employing a Bag-of-Words pre-processing solution. The advantage of quadratic sample reduction is recovered especially in the regime with large number of samples. This may allow quantum computers to efficiently scale randomized smoothing to more complex tasks beyond the reach of classical methods.

Figures (6)

• Figure 1: 1-Hamming distant state preparation for $\mathbf{x}=(0, 1, 1)$.
• Figure 2: Uniform distant state preparation for $\mathbf{x}=(0, 1, 1)$.
• Figure 3: Binary representation of discretized patterns in time series, visualized through the Bag-of-Words representation. The method operates by splitting in half a sample from the gun-point UCRArchive time series dataset, then dividing into windows of size 15 and bins of two with a word size of 2 (binary).
• Figure 4: Certified accuracy of quantum randomized smoothing with perturbations limited to a Hamming distance of 1 from the original input, across different $\sigma$ values. Each evaluation maintained a consistent number of 10$^9$ shots.
• Figure 5: Certified accuracy of quantum randomized smoothing between QuAdRo and RS in terms of shots number for 1-Hamming distant and uniform distributions. Both tests have been conducted with a $\sigma$ of 0.5 for the Iris dataset.

Theorems & Definitions (3)

• Definition 2.1: Basis Embedding
• Definition 2.2: Amplitude Embedding
• Definition 4.1: $k$-Hamming distant states