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Nonlinear difference equation1/18/2024 (b) shows the derivative quantum circuit for the product feature map. Several feature maps can be concatenated to represent a multivariable function. The product feature map can be further generalized to several layers, and different functions. For the nonlinear feature encoding the nonlinear function φ ( x ) is used as an angle of rotation. Specifically, the expectation value of the circuit is shown, with thin pink and green blocks depicting the variational ansatz and the cost function measurement respectively. (a) Quantum feature map of a product type, where single qubit rotations (here chosen as R ̂ y) act at each qubit individually and are parametrized by a function of variable x. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We propose a quantum algorithm to solve systems of nonlinear differential equations.
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