Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem (according to some criteria) from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. Different optimization techniques are applied in various fields such as mechanics, economics and engineering, and as the complexity and amount of data involved rise, more efficient ways of solving optimization problems are needed. Quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.
Quantum data fitting Data fitting is a process of constructing a mathematical function that best fits a set of data points. The fit's quality is measured by some criteria, usually the distance between the function and the data points.
Quantum least squares fitting One of the most common types of data fitting is solving the least squares problem, minimizing the sum of the squares of differences between the data points and the fitted function. The algorithm is given N {\displaystyle N} input data points ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x N , y N ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),...,(x_{N},y_{N})} and M {\displaystyle M} continuous functions f 1 , f 2 , . . . , f M {\displaystyle f_{1},f_{2},...,f_{M}} . The algorithm finds and gives as output a continuous function f λ → {\displaystyle f_{\vec {\lambda }}} that is a linear combination of f j {\displaystyle f_{j}} :
f λ → ( x ) = ∑ j = 1 M f j ( x ) λ j {\displaystyle f_{\vec {\lambda }}(x)=\sum _{j=1}^{M}f_{j}(x)\lambda _{j}}
In other words, the algorithm finds the complex coefficients λ j {\displaystyle \lambda _{j}} , and thus the vector λ → = ( λ 1 , λ 2 , . . . , λ M ) {\displaystyle {\vec {\lambda }}=(\lambda _{1},\lambda _{2},...,\lambda _{M})} . The algorithm is aimed at minimizing the error, which is given by:
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