Analysis of Quantum Algorithms for Solving Differential Equations with Application to Computational Fluid Dynamics

  • Williams, Chelsea (University of Exeter)
  • Kyriienko, Oleksandr (University of Exeter)
  • Elfving, Vincent (Pasqal)

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Quantum computing utilises the fundamental laws of quantum physics to develop algorithms that have the potential to solve classically intractable problems. These algorithms utilise the quantum mechanical principles of parallelism, superposition and entanglement to provide potentially an exponential speed-up and improved algorithmic scaling in comparison to classical algorithms. The field of computational fluid dynamics (CFD) presents an exciting area for applying quantum computing. CFD requires obtaining accurate solutions to complex problems that are described by systems of differential equations with a large number of variables or high nonlinearity. Quantum computing therefore has the potential to revolutionise the way CFD problems are solved in the future. In my contribution to the Minisymposium, I will present the scaling analysis for several quantum algorithms for solving industrially relevant linear and nonlinear differential equations. The broad approach taken is to convert a differential equation to a linear system of equations of the form Ax=b using classical finite difference methods and linearisation techniques. The quantum algorithms then are used to solve for x by either inverting matrix A or using a quantum-classical hybrid loop to optimise the parameters of a cost function as in classical machine learning models. Additional approaches exist for nonlinear problems that look to utilise various quantum linearisation techniques. These algorithms are analysed in terms of the quantum resources required (the width and depth of the quantum circuit) and the scaling behaviour (as a function of the dimension, sparsity and condition number of A). Finally, I will consider industrially relevant problems in the CFD domain and analyse the resources required for high quality solutions with future fault-tolerant quantum computers.