Solving Nonlinear Partial Differential Equations via a Hybrid Newton Method Using Quantum Linear System Solver

Abstract

To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.

Figures (18)

• Figure 1: The general workflow of a quantum Newton method. Here, $J_{F, u_i}$ denotes the Jacobian for the function $F$ of the equation $F(u) = 0$. The quantum system provides the iterative step $\Delta u$.
• Figure 1: The calculated normalized solution of the stated Advection Diffusion equation for $N=16$. The solution is obtained via Qiskit simulation of the QLSS with $m=2$.
• Figure 2: Quantum circuit of the implemented QLSS. Here, $Anc_{C,M}$ is the ancilla for the multiplication. This register is also used together with register $Anc_C$ for the comparison. The uncomputation undoes all the operations, except the comparison.
• Figure 2: Relative error of the calculated Advection Diffusion equation solution using an $m=2$ QLSS with different discretizations resulting in $N\times N$ sized linear systems.
• Figure 3: Relative error for various choices of $m$, the number of qubits for the QPE for solving the Advection Diffusion equation, $N=8$. Theoretically the error in the approximations scales with $\varepsilon = 2^{-\frac{m}{2}}$ for $m$ qubits.