Abstract
In recent years, significant progress has been made in the development of quantum algorithms for linear ordinary differential equations as well as linear partial differential equations. There has not been similar progress in the development of quantum algorithms for nonlinear differential equations. In the present work, the focus is on nonlinear partial differential equations arising as governing equations in fluid mechanics. First, the key challenges related to nonlinear equations in the context of quantum computing are discussed. Then, as the main contribution of this work, quantum circuits are presented that represent the nonlinear convection terms in the Navier–Stokes equations. The quantum algorithms introduced use encoding in the computational basis, and employ arithmetic based on the Quantum Fourier Transform. Furthermore, a floating-point type data representation is used instead of the fixed-point representation typically employed in quantum algorithms. A complexity analysis shows that even with the limited number of qubits available on current and near-term quantum computers (<100), nonlinear product terms can be computed with good accuracy. The importance of including sub-normal numbers in the floating-point quantum arithmetic is demonstrated for a representative example problem. Further development steps required to embed the introduced algorithms into larger-scale algorithms are discussed.
Highlights
Quantum computing [1] and quantum communication are research areas that have seen significant developments and progress in recent years, as is apparent from the work covered in this book
The focus is on the development of quantum algorithms for solving nonlinear differential equations, highlighting key challenges that arise from the non-linearity of the equations to be solved
In developing the proposed method, efficient quantum circuits involving floating-point arithmetic were created, in contrast to the more commonly used fixed-point arithmetic employed in a range of quantum algorithms
Summary
Quantum computing [1] and quantum communication are research areas that have seen significant developments and progress in recent years, as is apparent from the work covered in this book. The focus is on the development of quantum algorithms for solving nonlinear differential equations, highlighting key challenges that arise from the non-linearity of the equations to be solved. For this application of quantum computing, progress has so far been relatively limited and in this work, a promising approach to deriving efficient quantum algorithms is proposed. In developing the proposed method, efficient quantum circuits involving floating-point arithmetic were created, in contrast to the more commonly used fixed-point arithmetic employed in a range of quantum algorithms This aspect of the work described here should be useful for a wider audience.
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