Efficient Quantum Hermite Transform

2025-10-06

AI Summary

This work addresses the efficient quantum implementation of the discrete Hermite transform. We introduce the first logarithmic-complexity quantum Hermite transform primitive: for dimension $d$ and precision $varepsilon$, it requires only $O(log(d/varepsilon))$ quantum gates to map a computational basis state to a quantum state whose amplitudes are proportional to Hermite polynomial evaluations. Our method leverages fast-forwarding of quantum harmonic oscillator time evolution, enabling the first exponential speedup for the Hermite transform. It further supports efficient Gaussian-weighted quantum state preparation and sampling. Applications include strict improvements in query complexity for property testing—e.g., low-degree Hermite coefficient detection—and learning tasks—e.g., Gaussian-domain Goldreich–Levin learning. Notably, this establishes the first provable quantum query advantage under the Gaussian assumption, providing a foundational tool for continuous-variable quantum algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that depends logarithmically in both the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast forward the evolution of the quantum harmonic oscillator, which significantly improves over prior art. We apply this Hermite transform to give examples of provable quantum query advantage in property testing and learning. In particular, we show how to efficiently test the property of being close to a low- degree in the Hermite basis when inputs are sampled from the Gaussian distribution, and how to solve a Gaussian analogue of the Goldreich-Levin learning task efficiently. We also comment on other potential uses of this transform to simulating time dynamics of quantum systems in the continuum.

Problem

• Efficient quantum implementation of discrete Hermite transform
• Testing properties of functions under Gaussian distribution
• Solving Gaussian analogue of Goldreich-Levin learning task

Innovation

• Efficient quantum discrete Hermite transform algorithm
• Logarithmic time dependence on dimension and error
• Fast forwarding quantum harmonic oscillator evolution