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arXiv:2412.05173

Fast Laplace transforms on quantum computers

Julien Zylberman

Published Dec 6, 2024• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2412.05173

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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×NN\times NN×N discrete Laplace transforms on quantum states encoded in ⌈log⁡2(N)⌉\lceil \log_2(N)\rceil⌈log2​(N)⌉-qubits. In many cases, the associated quantum circuits have a depth that scales with NNN as O(log⁡(log⁡(N)))O(\log(\log(N)))O(log(log(N))) and a size that scales as O(log⁡(N))O(\log(N))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.
We introduce a quantum algorithm to perform the Laplace transform on quantum computers. Already, the quantum Fourier transform (QFT) is the cornerstone of many quantum algorithms, but the Laplace transform or its discrete version has not seen any efficient implementation on quantum computers due to its dissipative nature and hence non-unitary dynamics. However, a recent work has shown an efficient implementation for certain cases on quantum computers using the Taylor series. Unlike previous work, our work provides a completely different algorithm for doing Laplace Transform using Quantum Eigenvalue Transformation and Lap-LCHS, very efficiently at points which form an arithmetic progression. Our algorithm can implement N×NN \times NN×N discrete Laplace transform in gate complexity that grows as O((log N)3)O((log\,N)^3)O((logN)3), ignoring the state preparation cost, where N=2nN=2^nN=2n and nnn is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity O(N⋅log N)O(N\cdot log\,N)O(N⋅logN) for the same cases. Also, the circuit width grows as O(log N)O(log\,N)O(logN). Quantum Laplace Transform (QLT) may enable new Quantum algorithms for cases like solving differential equations in the Laplace domain, developing an inverse Laplace transform algorithm on quantum computers, imaginary time evolution in the resolvent domain for calculating ground state energy, and spectral estimation of non-Hermitian matrices.
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The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length NNN, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any ddd-dimensional Hilbert transform in depth O(dlog⁡N)O(d\log N)O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
Current classical simulations of physical phenomena face many limitations, particularly when dealing with complex systems such as turbulent fluids, plasmas or strongly correlated quantum materials. These simulations often struggle with high computational costs, limited scalability across multiple dimensions and time scales, and difficulties in capturing inherently non-linear or chaotic behaviors. Even with access to powerful high performance computing (HPC) resources, accurately resolving fine-scale dynamics or long time evolutions remains a significant challenge. The development of novel numerical methods is therefore essential for both academic research and industrial applications. Quantum computers emerge as novel numerical tools for studying such systems and solving the partial differential equations that model them. Thanks to the superposition and entanglement properties of qubits, digital computations can be performed in novel ways, offering promising theoretical speedups for certain problems. In this context, this thesis contributes to the development and understanding of quantum algorithms, ranging from the design of quantum circuits to the numerical solutions of partial and ordinary differential equations. First, efficient quantum circuits are presented for the implementation of crucial routines that are prevalent in quantum algorithms (multi-controlled operations, quantum state preparation, diagonal operations), and for a novel routine: the quantum Laplace transform. These routines limited the efficiency of quantum algorithms due to generic implementations that required complexities scaling exponentially with the number of qubits. Now, these tasks can be performed with tailored quantum circuits that leverage the structure of the physical problems. In the second part, quantum numerical schemes are introduced to address anisotropic transport equations and non-linear Hamiltonian ODEs. These schemes are composed of three steps (initialization, evolution, and measurement) and are based on high-ordered finite-difference and product formula approximations, also known as Trotterization. Efficient protocols to measure quantities of interest are presented, and a novel numerical analysis based on the vector norm instead of the generic operator norm is introduced, proving that the number of time-steps can be reduced by a factor that scales exponentially with the number of qubits while reaching similar accuracy. Additionally, non-linear Madelung transforms are studied to solve fluid equations on quantum computers. The various results of this thesis are illustrated with classical simulations of the quantum algorithms and simulations on real quantum hardware. This work lays the foundations for the practical simulations of classical physical systems on both current noisy-intermediate-scale quantum devices and future fault-tolerant quantum computers.
Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for non-normal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation (LCHS) method [An, Liu, Lin, Phys. Rev. Lett. 131, 150603, 2023; An, Childs, Lin, arXiv:2312.03916] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, A−kA^{-k}A−k, and the exponential of the matrix inverse, e−A−1e^{-A^{-1}}e−A−1. The latter can be interpreted as the solution of a mass-matrix differential equation of the form Au′(t)=−u(t)A u'(t)=-u(t)Au′(t)=−u(t). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting AAA, reducing the computational complexity.
Abstract. Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, [Formula: see text], and the exponential of the matrix inverse, [Formula: see text]. The latter can be interpreted as the solution of a mass-matrix differential equation of the form [Formula: see text]. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting [Formula: see text], thereby reducing the computational complexity.