This chapter covers the multiplicative weights update method, a quantum algorithmic primitive for certain continuous optimization problems. This method is a framework for classical algorithms, but it can be made quantum by incorporating the quantum algorithmic primitive of Gibbs sampling and amplitude amplification. The framework can be applied to solve linear programs and related convex problems, or generalized to handle matrix-valued weights and used to solve semidefinite programs.

This chapter covers applications of quantum computing in the area of continuous optimization, including both convex and nonconvex optimization. We discuss quantum algorithms for computing Nash equilibria for zero-sum games and for solving linear, second-order, and semidefinite programs. These algorithms are based on quantum implementations of the multiplicative weights update method or interior point methods. We also discuss general quantum algorithms for convex optimization which can provide a speedup in cases where the objective function is much easier to evaluate than the gradient of the objective function. Finally, we cover quantum algorithms for escaping saddle points and finding local minima in nonconvex optimization problems.

This chapter covers quantum interior point methods, which are quantum algorithmic primitives for application to convex optimization problems, particularly linear, second-order, and semidefinite programs. Interior point methods are a successful classical iterative technique that solve a linear system of equations at each iteration. Quantum interior point methods replace this step with quantum a quantum linear system solver combined with quantum tomography, potentially offering a polynomial speedup.