Efficient Quantum Hermite Transform

TL;DR

The paper introduces a discrete quantum Hermite transform (QHT) that maps computational-basis states to Hermite-function-based states with polylogarithmic dependence on dimension and error, by leveraging an exponentially fast-forwardable quantum harmonic oscillator (QHO). The approach combines Plancherel–Rotach-based state preparation, fast-forwarded QPE, and fixed-point amplitude amplification to realize the transform in time $O(( ext{log}N+ ext{log}(1/oldsymbol{ au}))^3 ext{log}(1/oldsymbol{ au}))$, with exponential fast-forwarding enabling efficient low-energy subspace operations. The QHT enables Hermite sampling, yielding quantum advantages in Gaussian-based property testing and learning tasks, including Gaussian Goldreich–Levin-type problems and tolerant low-degree testing, while also supporting applications to Hamiltonian simulation and continuum dynamics. This work establishes a foundational quantum primitive beyond the QFT, with potential to unlock exponential-speedups in sampling, learning, and quantum simulation across Gaussian-distributed data and continuum quantum systems. The results provide a concrete algorithmic framework for constructing and utilizing the QHT in a range of quantum-information processing tasks.

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.

Figures (3)

• Figure 1: A visualization of the Quantum Hermite Transform. On the left we have the Plancherel-Rotach functions (visualized without the decaying envelope for simplicity), we prepare states corresponding to these, which is then fed into the fixed-point search algorithm using our fast-forwarding result as a subroutine. This allows us to prepare the states corresponding to the Hermite functions in superposition to arbitrary precision. Both the functions are visualized for $n=10$.
• Figure 2: Visualization of discretization error for the QHO. Here $N = \Theta(M/\log M)$, as specified in \ref{['thm:discrete factoring']}. In the "low-energy" subspace, the fast-forwarding algorithm provably works. Meanwhile, in the "medium-energy" subspace \ref{['fact:somma-discretization-facts']} still holds, although we no longer obtain rigorous guarantees on fast-forwarding.
• Figure 3: The overlap between the Plancherel-Rotach states and the Hermite states for dimension $M=10^5$ and $0 \le n \le 100$. The overlap approaches $2/3$, which corresponds to the probability mass of the Hermite functions in the domain $|x| \le \sqrt{(3/4)(2n+1)}$.

Theorems & Definitions (117)

• Theorem 1: Exponential fast-forwarding of the QHO
• Theorem 2: Efficient QHT, informal
• Theorem 3: Factorization of the QHO evolution QA07
• Theorem 3: Exponential fast-forwarding of the QHO
• Theorem 5
• Theorem 6
• proof : Proof of \ref{['thm:discrete factoring']}
• proof : Proof of \ref{['thm:fastforwarding']}
• Lemma 7
• proof