Boundary Integral Algorithms for linear and nonlinear PDEs in Quantum Computation of Fluid Dynamics

  • Bharadwaj, Sachin Satish (New York University)
  • Nadiga, Balu (Los Alamos National Laboratory (LANL))
  • Eidenbenz, Stephan (Los Alamos National Laboratory (LANL))
  • R. Sreenivasan, Katepalli (New York University)

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Given the potential of quantum computing (QC) to achieve large speed-ups on tasks that are difficult from the point of view of classical computing, we are interested in exploring the use of QC for simulating fluid flows. While the field of quantum algorithms has seen frenzied activity over the past couple of decades with new and improved quantum algorithms being continually proposed, a large number of caveats remain to be explicitly addressed before such algorithms can be used to simulate a fluid flow. For example, given the linearity of quantum mechanics, QC speed-up of solving linear systems was proven early on with the HHL (Harrow-Hassidim-Lloyd) algorithm and further significant theoretical improvements have since been made under the rubric of Quantum Linear System Algorithms (QLSA). However, nonlinearity is a common feature of fluid flows. Thus, to be able to leverage QLSA to simulate nonlinear fluid flows and to realize QC speed-up one needs to reformulate fluid flow problems in ways that are fundamentally different from those used in classical computing. In this setting, and with a view to permit end-to-end solutions, we have developed a high performance quantum simulator called QuON; QuON is specifically designed for simulating fluid flows using QC. Using QuOn, the time-dependent Poiseuille and Couette flows are solved as initial examples to demonstrate, for the first time, a high precision, fully gate level implementation of a QLSA method for a CFD simulation. Next, we propose a Quantum Boundary Integral Algorithm (QBIA) formulation that seeks to solve nonlinear partial differential equations by combining the Homotopy Analysis Method (HAM) with QLSA. We expect that our integral formulation is better suited for QC than the usual methods that focus of discretizing the strong differential form of the governing dynamics, both in terms of accuracy and efficiency. We discuss some preliminary efforts to solve a 1D Burgers flow using this approach. Along with this we also aim at evaluating the minimum requirements as well as the potential and obstacles for transitioning from computational fluid dynamics (CFD) to Quantum Computation of Fluid Dynamics (QCFD).