Efficient Quantum Hermite Transform
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909827148546048 |
|---|---|
| author | Jain, Siddhartha Iyer, Vishnu Somma, Rolando D. Bao, Ning Jordan, Stephen P. |
| author_facet | Jain, Siddhartha Iyer, Vishnu Somma, Rolando D. Bao, Ning Jordan, Stephen P. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient Quantum Hermite Transform Jain, Siddhartha Iyer, Vishnu Somma, Rolando D. Bao, Ning Jordan, Stephen P. Quantum Physics Computational Complexity 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. |
| title | Efficient Quantum Hermite Transform |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2510.04929 |