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For combinatorial optimization, the quantum approximate optimization algorithm (QAOA) [6] briefly had a better approximation ratio than any known polynomial time classical algorithm (for a certain problem), [7] until a more effective classical algorithm was proposed. [8] The relative speed-up of the quantum algorithm is an open research question.
The quantum approximate optimization algorithm takes inspiration from quantum annealing, performing a discretized approximation of quantum annealing using a quantum circuit. It can be used to solve problems in graph theory. [50] The algorithm makes use of classical optimization of quantum operations to maximize an "objective function."
In quantum computing, the variational quantum eigensolver (VQE) is a quantum algorithm for quantum chemistry, quantum simulations and optimization problems.It is a hybrid algorithm that uses both classical computers and quantum computers to find the ground state of a given physical system.
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"A cross-disciplinary introduction to quantum annealing-based algorithms" [37] presents an introduction to combinatorial optimization problems, the general structure of quantum annealing-based algorithms and two examples of this kind of algorithms for solving instances of the max-SAT (maximum satisfiable problem) and Minimum Multicut problems ...
Qiskit Optimization is an open-source framework that covers the whole range from high-level modeling of optimization problems, with automatic conversion of problems to different required representations, to a suite of easy-to-use quantum optimization algorithms that are ready to run on classical simulators, as well as on real quantum devices ...
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As an illustrative example of how QUBO can be used to encode an optimization problem, we consider the problem of cluster analysis. Here, we are given a set of 20 points in 2D space, described by a matrix D ∈ R 20 × 2 {\displaystyle D\in \mathbb {R} ^{20\times 2}} , where each row contains two cartesian coordinates .