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

2025/12/16 by Gard Olav Helle, Tommaso Benacchio, Helle, Gard Olav +5
Computer Science · Mathematics · #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Numerical methods for differential equations

paper · doi:10.48550/arxiv.2512.22163

Abstract

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.

Citations

• Quantum circuits for partial differential equations in Fourier space
• Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator
• What if? Numerical weather prediction at the crossroads
• Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling
• Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters
• Generalized Quantum Signal Processing
• Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation
• Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices
• Improved quantum algorithms for linear and nonlinear differential equations
• Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis
• Efficient quantum circuits for dense and non-unitary operators
• Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision
• Quantum Computing, Postselection, and Probabilistic Polynomial-Time
• Addition on a Quantum Computer

Related

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