# How long does quantum annealing take to find the solution to a given problem?

Quantum annealing is an optimization protocol that, thanks to quantum tunneling, allows in given circumstances to maximize/minimize a given function more efficiently than classical optimization algorithms.

A crucial point of quantum annealing is the adiabaticity of the algorithm, which is required for the state to stay in the ground state of the time-dependent Hamiltonian. This is however also a problem, as it means that find a solution can require very long times.

How long do these times have to be for a given Hamiltonian? More precisely, given a problem Hamiltonian $\mathcal H$ of which we want to find the ground state, are there results saying how long would it take a quantum annealer to reach the solution?

• Answers to this question should consider taking noise into account, as it is a critical part of what determines the speed of quantum tunneling. – DanielSank Mar 17 '18 at 21:33
• Is it not related to the spectral properties of $\mathcal{H}$? – Mark S Aug 15 '18 at 2:14