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Proceedings of the Quantum Computation for Physical Modeling Workshop Held in North Falmouth, Massachusetts on October 18-19, 2000

Jeffrey Yepez, Bruce M. Boghosian

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

A practical and efficient way to use the power of quantum mechanics (quantum parallelism due to the superposition and entanglement of states) could enormously speed up numerical simulations of interest to computational physicists. The diffusion equation the nonlinear Burgers equations, the turbulent Navier-Stokes equations, and the many-body Schroedinger equation could all be solved much more rapidly if efficient quantum algorithms could be implemented for their solution. The presentations at the conference that are published here review the progress toward these goals.

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A practical and efficient way to use the power of quantum mechanics (quantum parallelism due to the superposition and entanglement of states) could enormously speed up numerical simulations of interest to computational physicists. The diffusion equation the nonlinear Burgers equations, the turbulent Navier-Stokes equations, and the many-body Schroedinger equation could all be solved much more rapidly if efficient quantum algorithms could be implemented for their solution. The presentations at the conference that are published here review the progress toward these goals.
Abstract Computational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.
This paper discusses simulation of quantum computing applications solving important physics and engineering problems. Methodologies are developed for quantum computing solutions of the Navier Stokes Equation, Dirac's Equation, and Schrodinger's Equation in complex domains. Turbulent Navier Stokes solutions were generated in complex geometries including urban domains. A quantum computing algorithm is also developed for the processing of sound which has a greater than classical efficiency in compression when run on a quantum computer. The algorithm also has interesting and useful properties on a classical computer, but without the superior speed and storage properties realized on a quantum computer.
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.
Fault-tolerant quantum computing is a promising technology to solve linear partial differential equations that are classically demanding to integrate. It is still challenging to solve non-linear equations in fluid dynamics, such as the Burgers equation, using quantum computers. We propose a novel quantum algorithm to solve the Burgers equation. With the Cole-Hopf transformation that maps the fluid velocity field uuu to a new field ψψψ, we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state ∣ψ⟩\vertψ\rangle∣ψ⟩ that encodes the solution ψψψ. We also propose an efficient way to extract stochastic properties of uuu, namely the multi-point functions of uuu, from the quantum state of ∣ψ⟩\vertψ\rangle∣ψ⟩. Our algorithm offers an exponential advantage over the classical finite difference method in terms of the number of spatial grids when a perturbativity condition in the information-extracting step is met.
Abstract Efficient simulation of a quantum system's dynamics is expected to be an important application area for quantum computers as existing classical computers cannot do this. However, quantum systems are not unique in being hard to simulate. For example, classical nonlinear continuum systems and fields are governed by nonlinear partial differential equations whose solution is also hard for classical computers. Solving such equations is essential for many economically important industries/applications such as the aerospace industry, weather‐forecasting, fiber‐optics communication, and plasma magneto‐hydrodynamics. This raises the question: can a quantum computer speed‐up solving these equations? In this Review, a new quantum algorithm is described for solving nonlinear partial differential equations for which the answer is yes. First, a new quantum algorithm is discussed for solving the Navier‐Stokes nonlinear partial differential equations which govern the flow of a viscous fluid. Its construction, verification, and computational cost are described, and it is shown that a significant quantum speed‐up is possible. Its generalization to a quantum algorithm for solving nonlinear partial differential equations is described. The Review closes with a discussion of next steps forward. These new quantum algorithms open up a large new application area for quantum computing with substantial economic impact, including the trillion‐dollar aerospace industry.