Fast Laplace transforms on quantum computers

TL;DR

This paper introduces the Quantum Laplace Transform (QLT), a quantum algorithm that efficiently implements discrete Laplace transforms with exponentially fewer operations than classical methods, enabling new quantum algorithms across various fields.

Contribution

The paper presents the first quantum algorithm for discrete Laplace transforms with logarithmic circuit depth and size, significantly improving efficiency over classical algorithms.

Findings

Quantum Laplace Transform (QLT) enables efficient implementation of Laplace transforms on quantum states.

Quantum circuits for QLT have depth $O(\log(\log(N)))$ and size $O(\log(N))$, requiring exponentially fewer operations.

Potential applications include quantum algorithms in physics, engineering, chemistry, machine learning, and finance.

Abstract

While many classical algorithms rely on Laplace transforms, it has remained an open question whether these operations could be implemented efficiently on quantum computers. In this work, we introduce the Quantum Laplace Transform (QLT), which enables the implementation of $N\times N$ discrete Laplace transforms on quantum states encoded in $\lceil \log_2(N)\rceil$-qubits. In many cases, the associated quantum circuits have a depth that scales with $N$ as $O(\log(\log(N)))$ and a size that scales as $O(\log(N))$, requiring exponentially fewer operations and double-exponentially less computational time than their classical counterparts. These efficient scalings open the possibility of developing a new class of quantum algorithms based on Laplace transforms, with potential applications in physics, engineering, chemistry, machine learning, and finance.