Quantum time-marching algorithms for solving linear transport problems including boundary conditions

TL;DR

This work develops a complete quantum time-marching approach for solving multidimensional diffusion/linear transport problems with arbitrary boundary conditions on fault-tolerant quantum computers. It combines amplitude encoding, Hamiltonian simulation, block encoding, and linear combination of unitaries to implement the diffusive time-step while enforcing boundaries via the method of images or direct unitary decompositions, achieving intrinsic optimal success probabilities. The authors demonstrate 2D heat-equation simulations with Neumann, Dirichlet, and mixed boundaries, validating accuracy against classical finite-difference methods and showing linear-time scaling with a polynomial quantum speedup over classical solvers. The framework is poised for CFD-scale applications on quantum hardware and can be extended to higher dimensions and non-homogeneous boundary conditions using tensor-network techniques.

Abstract

This article presents the first complete application of a quantum time-marching algorithm for simulating multidimensional linear transport phenomena with arbitrary boundaries, whereby the success probabilities are problem intrinsic. The method adapts the linear combination of unitaries algorithm to block encode the diffusive dynamics, while arbitrary boundary conditions are enforced by the method of images only at the cost of one additional qubit per spatial dimension. As an alternative to the non-periodic reflection, the direct encoding of Neumann conditions by the unitary decomposition of the discrete time-marching operator is proposed. All presented algorithms indicate optimal success probabilities while maintaining linear time complexity, thereby securing the practical applicability of the quantum algorithm on fault-tolerant quantum computers. The proposed time-marching method is demonstrated through state-vector simulations of the heat equation in combination with Neumann, Dirichlet, and mixed boundary conditions.

Figures (13)

• Figure 1: Three-qubit example for the implementation of the shift operators $S_0$ and $S_0^\dagger$.
• Figure 2: Generic circuit for the temporal evolution of $\ket{\psi^0}$ given by the Hamiltonian simulation of Eqn. \ref{['eq:schroedinger']}.
• Figure 3: Exemplary quantum circuit implementations of the block encoding strategies for $U_{\text{M}}$ (top) and $\hat{U}_{\text{M}}$ (bottom).
• Figure 4: Quantum circuit to implement one explicit time step of the one-dimensional periodic heat equation, Eqn. \ref{['eq:heat_equation']}.
• Figure 5: LCU circuit for the two-dimensional periodic heat equation.