Two quantum algorithms for solving the one-dimensional advection–diffusion equation

We discuss the Carleman approach to the quantum simulation of classical fluids, as applied to (i) lattice Boltzmann, (ii) Navier–Stokes, and (iii) Grad formulations of fluid dynamics. Carleman lattice Boltzmann shows excellent convergence properties, but it is plagued by nonlocality which results in an exponential depth of the corresponding circuit with the number of Carleman variables. The Carleman Navier–Stokes offers a dramatic reduction of the number Carleman variables, which might lead to a viable depth, provided locality can be preserved and convergence can be achieved with a moderate number of iterates also at sizeable Reynolds numbers. Finally, it is argued that Carleman Grad might combine the best of Carleman lattice Boltzmann and Carleman Navier–Stokes.

Three Carleman routes to the quantum simulation of classical fluids

Sanavio,

Scatamacchia,

de Falco

et al. 2024

QFlowS: Quantum simulator for fluid flows

Bharadwaj

2024

Quantum computing presents a possible paradigm shift in computing, given its advantages in memory and speed. However, there is a growing need to demonstrate its utility in solving practical problems that are nonlinear, such as in fluid dynamics, which is the subject of this work. To facilitate this objective, it is essential to have a dedicated toolkit that enables the development, testing, and simulation of new quantum algorithms and flow problems, taken together. To this end, we present here a high performance, quantum computational simulation package called Quantum Flow Simulator (QFlowS), designed for computational fluid dynamics simulations. QFlowS is a versatile tool that can create and simulate quantum circuits using an in-built library of fundamental quantum gates and operations. We outline here all its functionalities with illustrations. Algorithms to solve flow problems can be built using the expanding list of the core functionalities of QFlowS with its hybrid quantum–classical type workflow. This is demonstrated here by solving an example, one-dimensional, diffusion flow problem. These simulations serve as a check on the algorithm's correctness as well as an ideal test-bed for making them more efficient and better suited for near-term quantum computers for addressing flow problems.

Quantum algorithm for nonlinear Burgers' equation for high-speed compressible flows

Esmaeilifar,

Ahn,

Myong

2024

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.