A Polylogarithmic-Time Quantum Algorithm for the Laplace Transform

TL;DR

The paper tackles efficient quantum computation of the Laplace transform by introducing Quantum Laplace Transform (QLT) built on Laplace-based Linear Combination of Hamiltonian Simulation (Lap-LCHS). By encoding the Laplace variable into the eigenvalues of a diagonal matrix and embedding the input via LCU with a carefully structured SELECT that leverages arithmetic-progression properties, the authors achieve a gate complexity of $O((\log N)^3)$ and circuit width $O(\log N)$, representing a substantial speedup over classical benchmarks for the same discretized transform. Key contributions include a detailed Lap-LCHS-based construction, a rigorously analyzed SELECT operator, and practical implementations and simulations on Qiskit and Pennylane, showing close agreement with classical results. The work highlights QLT as a powerful primitive with potential applications in differential equations in the Laplace domain, inverse Laplace transforms, and spectral problems, while acknowledging state-preparation costs and positioning QLT as a subroutine within larger quantum algorithms. Overall, this establishes a solid theoretical and practical foundation for quantum Laplace-transform techniques and their integration into broader quantum workflows.

Abstract

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 $N \times N$ discrete Laplace transform in gate complexity that grows as $O((log\,N)^3)$, ignoring the state preparation cost, where $N=2^n$ and $n$ is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity $O(N\cdot log\,N)$ for the same cases. Also, the circuit width grows as $O(log\,N)$. 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.

Figures (12)

• Figure 1: Linear Combination of Unitaries (LCU) circuit architecture acting on $\ket{\psi}$. The circuit is divided into three conceptual stages. In the Preparation stage, a Hadamard gate prepares the ancilla register in a coherent superposition that encodes the weighting coefficients of the linear combination. In the Selection stage, controlled unitaries $U_1, U_2, \ldots$ act on the system register conditioned on the ancilla state, implementing the weighted sum of operators through coherent control. In the Unpreparation stage, is the uncomputation step. Then, a measurement is done to post-select the desired outcome.
• Figure 2: QLT implementation follows the standard LCU framework, consisting of the Preparation, Select, and Unpreparation operators. The target register is initialized in an equal superposition using Hadamard gates, a crucial step for our algorithm.
• Figure 3: Explicit one-qubit realization of the QLT circuit. The diagram shows detailed decompositions of the preparation, selection, and un-preparation stages. Single-qubit rotations encode the coefficient structure of the Laplace transform, and controlled-phase as well as controlled-unitary gates implement the Select block.
• Figure 4: Comparison between numerical (classical) lap-LCHS calculation using numpy with analytical Laplace transform calculation for (a) $g(t) = e^{-0.9} sin(t)$ and (b) $g(t) = e^{-0.9t}$.
• Figure 5: Real-weighted LCU implementation using single-qubit rotations and controlled operations.

Theorems & Definitions (5)

• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 4.1