A quantum advection-diffusion solver using the quantum singular value transform

We present a quantum algorithm for the simulation of the linear advection-diffusion equation based on block encodings of high order finite-difference operators and the quantum singular value transform. Our complexity analysis shows that the higher order methods significantly reduce the number of gates and qubits required to reach a given accuracy. The theoretical results are supported by numerical simulations of one- and two-dimensional benchmarks.

💡 Research Summary

The paper introduces a quantum algorithm for solving the linear advection‑diffusion equation ∂ₜu + c·∇u = νΔu by leveraging high‑order finite‑difference discretizations, block‑encoding techniques, and the Quantum Singular Value Transform (QSVT). The authors begin by discretizing the spatial domain into a uniform grid and constructing symmetric finite‑difference operators D₂p (for first derivatives) and D(2)₂p (for second derivatives) of arbitrary even order 2p. These operators are expressed as linear combinations of translation unitaries, enabling the use of the Linear Combination of Unitaries (LCU) method to build efficient unitary block‑encodings of the discretized differential operator L = −c D₂p + ν D₂p² (or the alternative L = −c D₂p + ν D(2)₂p).

A key technical contribution is the systematic construction of block‑encodings for any order‑p finite‑difference operator, together with rigorous error bounds showing that the truncation error scales as O(Δx^{2p}) where Δx is the grid spacing. Once a block‑encoding of the Hermitian matrix H = iβ D₂p (with a normalization constant β ensuring ‖H‖ ≤ 1) is obtained, the algorithm applies QSVT to approximate the matrix exponential e^{L T}. The target function is f(x) = e^{−M₁x²} cos(M₂x) + i e^{−M₁x²} sin(M₂x), where M₁ = c T/β and M₂ = ν T/β². By approximating f with a polynomial q of degree d that satisfies |q(x)| ≤ 1 on