Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications

• URL: http://arxiv.org/abs/2502.19688v2
• Date: Tue, 11 Mar 2025 02:59:58 GMT
• Title: Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications
• Authors: Rundi Lu, Hao-En Li, Zhengwei Liu, Jin-Peng Liu,
• Abstract summary: We extend the Linear Combination of Hamiltonian Simulation (LCHS) formula to simulate time-evolution operators in infinite-dimensional spaces.<n>We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics.<n>Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics.
• Score: 3.6320083572773507
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We generalize the Linear Combination of Hamiltonian Simulation (LCHS) formula [An, Liu, Lin, Phys. Rev. Lett. 2023] to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. This extension, named Inf-LCHS for short, bridges the gap between finite-dimensional quantum simulations and the broader class of infinite-dimensional quantum dynamics governed by partial differential equations (PDEs). Furthermore, we propose two sampling methods by integrating the infinite-dimensional LCHS with Gaussian quadrature schemes (Inf-LCHS-Gaussian) or Monte Carlo integration schemes (Inf-LCHS-MC). We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics, including linear parabolic PDEs, queueing models (birth-or-death processes), Schr\"odinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

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