Recent advances in quantum hardware and quantum computing algorithms promise significant breakthroughs in computational capabilities. Quantum computers can achieve exponential improvements in speed vs classical computers by employing principles of quantum mechanics like superposition and entanglement. However, designing quantum algorithms to solve the nonlinear partial differential equations governing fluid dynamics is challenging due to the inherent linearity of quantum mechanics, which requires unitary transformation. In this study, we first address in detail several challenges that arise when trying to deal with nonlinearity using quantum algorithms and then propose a novel pure quantum algorithm for solving a nonlinear Burgers' equation. We employed multiple copies of the state vector to calculate the nonlinear term, which is necessary due to the no-cloning theorem. By reusing qubits from the previous time steps, we significantly reduced the number of qubits required for multi-step simulations, from exponential/quadratic scaling in earlier studies to linear scaling in time in the current study. We also employed various advanced quantum techniques, including block-encoding, quantum Hadamard product, and the linear combination of unitaries, to design a quantum circuit for the proposed quantum algorithm. The quantum circuit was executed on quantum simulators, and the obtained results demonstrated excellent agreement with those from classical simulations.

For understanding many real-world problems involving rarefied hypersonic, micro-, and nanoscale gas flows, the primary method may be the direct simulation Monte Carlo (DSMC). However, its computational cost is prohibitive in comparison with the Navier–Stokes–Fourier (NSF) solvers, eclipsing the advantages it provides, especially for situations where flow is in the near continuum regime or three-dimensional applications. This study presents an alternate computational method that bypasses this issue by taking advantage of data-driven modeling and nonlinear coupled constitutive relations. Instead of using numerical solutions of higher-order constitutive relations in conventional partial differential equation-based methods, we build compact constitutive relations in advance by applying deep neural network algorithms to available DSMC solution data and later combine them with the conventional finite volume method for the physical laws of conservation. The computational accuracy and cost of the methodology thus developed were tested on the shock wave inner structure problem, where high thermal non-equilibrium occurs due to rapid compression, for a range of Mach numbers from 2 to 10. The simulation results obtained with the computing time comparable to that of the NSF solver showed almost perfect agreement between the neural network-based combined finite volume method and DSMC and original DSMC solutions. We also present a topology of DSMC constitutive relations that allows us to study how the DSMC topology deviates from the NSF topology. Finally, several challenging issues that must be overcome to become a robust method for solving practical problems were discussed.