We investigate time complexities of finite difference methods for solving the multiscale transport equation by quantum algorithms.It is found that the time complexity of classical and quantum treatments of the standard explicit scheme scale is (1/ε), where ε is a small scaling parameter.On the other hand, the complexity of the even-odd parity based asymptotic-preserving (AP) schemes do not depend on ε.This indicates that in quantum computing, AP schemes (and probably other multiscale ones) are of great importance for solving multiscale transport and kinetic equations.
We investigate time complexities of finite difference methods for solving the multiscale transport equation by quantum algorithms.It is found that the time complexity of classical and quantum treatments of the standard explicit scheme scale is (1/ε), where ε is a small scaling parameter.On the other hand, the complexity of the even-odd parity based asymptotic-preserving (AP) schemes do not depend on ε.This indicates that in quantum computing, AP schemes (and probably other multiscale ones) are of great importance for solving multiscale transport and kinetic equations.
We investigate time complexities of finite difference methods for solving the multiscale transport equation with quantum algorithms. We find that the time complexities of both the classical treatment and quantum treatment for a standard explicit scheme scale as O(1/ε), where ε is the small scaling parameter, while the complexities for the even-odd parity based Asymptotic-Preserving (AP) scheme do not depend on ε. This indicates that it is still of great importance to use AP (and probably other efficient multiscale) schemes for multiscale problems in quantum computing when solving multiscale transport or kinetic equations.
In this paper, we present two Hamiltonian simulation algorithms for multiscale linear transport equations, combining the Schrödingerization method [S. Jin, N. Liu and Y. Yu, Phys. Rev. Lett, 133(2024), 230602] [S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603] and exponential integrator while incorporating incoming boundary conditions. These two algorithms each have advantages in terms of design easiness and scalability, and the query complexity of both algorithms, outperforms classical algorithms for solving this equation. In terms of the theoretical framework, these are the first quantum Hamiltonian simulation algorithms for multiscale linear transport equation to combine the Schrödingerization method with effective asymptotic-preserving schemes, which are efficient for handling multiscale problems with stiff terms.
We develop a mathematically rigorous framework for simulating \emph{multiscale physical systems} using quantum computational resources, by translating the \emph{language of asymptotic-preserving (AP) schemes} into the formalism of quantum channels and Lindbladian dynamics. For stiff open quantum systems governed by singularly perturbed generators $\cL_\eps = \eps^{-1}\cL_{\mathrm{fast}} + \cL_{\mathrm{slow}}$ with $\eps \to 0$, we prove that layered quantum protocols, which implement fast-scale relaxation via native analog evolution or analytic manifold projection, converge uniformly in the diamond norm to consistent discretizations of the limiting slow dynamics, with explicit error bound $\mathcal{O}(\epsΔt + Δt^2)$ independent of stiffness. We establish precise resource-complexity bounds showing that superlinear gate-count savings Ω(κ⋅(dtot/dslow)c) arise if and only if fast dynamics are resolved via (i) hardware-native analog evolution, or (ii) analytic adiabatic elimination reducing effective Hilbert space dimension. The framework is illustrated through cavity QED in the bad-cavity limit and a quantum-inspired AP discretization of kinetic equations converging to fluid limits, with quantified error propagation in trace and diamond norms. This work provides a principled mathematical bridge between classical multiscale numerical analysis and quantum simulation algorithms.
Abstract. Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems inPDEs, including elliptic, parabolic, and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schrödingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, and analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.
In this paper, we present a quantum implicit-explicit (IMEX) scheme for multiscale ordinary and partial differential equations whose discretization parameters are independent of the scaling parameter ε. A key ingredient of our approach is a continuous-time formulation of classical IMEX schemes, which decouples the evolution time of the quantum algorithm from the physical time of the differential equation and is therefore particularly useful in multiscale settings. Building on this idea, we employ the Schrödingerization framework [Phys. Rev. Lett. 133 (2024), 230602] to implement IMEX schemes on quantum computers. Compared to previous HHL type quantum AP scheme [J. Comput. Phys. 471 (2022), 111641], this new method requires narrower -- an extra logarithmic factor -- auxiliary register numerical examples on linear heat and multiscale telegraph equations demonstrate the independence in ε of the method.
Multiscale modeling of complex chemical systems requires algorithms that operate coherently across electronic, atomistic, mesoscopic, and continuum scales. While quantum algorithms have been proposed for each regime, no systematic framework exists to compose them across scale boundaries. Here, we identify the conditions under which fault-tolerant quantum algorithms might preserve scale-specific quantum advantages. We map quantum phase estimation, Hamiltonian simulation with Gibbs state preparation, quantum random walks, and quantum partial differential equation solvers onto electronic structure, molecular dynamics, mesoscopic kinetics, and continuum reactor physics, respectively. Crucially, these correspondences do not imply unconditional end-to-end quantum advantage; speedups depend heavily on state preparation, memory architectures, matrix conditioning, and classical readout costs. Six unresolved questions define this composition problem, illustrated via a quantum hierarchy for \ce{CO} oxidation over \ce{Pt(111)}. We propose viewing inter-scale transfer as a quantum channel composition problem at the interface of algorithm design and non-equilibrium statistical mechanics, and ask whether information loss at scale boundaries is intrinsic to multiscale modeling or merely a consequence of lossy classical transduction between algorithmic layers. The resulting roadmap suggests that multiscale quantum advantage is governed primarily by the structure of information transfer between algorithmic layers, rather than by performance at individual scales alone.