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 · #Block (permutation group theory) #Numerical methods for differential equations #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum algorithm #Quantum computer #Quantum error correction #Quantum operation #Quantum phase estimation algorithm #Qubit #Value (mathematics)

paper · doi:10.48550/arxiv.2512.22163

published in Open MIND

openalex publication_date 2025/12/16 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

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

• Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator
• Quantum algorithm for the advection-diffusion equation by direct block encoding of the time-marching operator
• Hamiltonian Simulation for Advection-Diffusion Equation with arbitrary transport field
• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms