Diabatic Quantum Annealing using XX-Catalysts

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My proposed research will look at a quantum algorithm known as Quantum Annealing. This algorithm is used for solving combinatorial optimisation problems with the aim of reaching a solution faster than is classically achievable.Quantum Annealing works by encoding the optimal solution for such a problem into the ground state of what is called a Hamiltonian - essentially a mathematical object that defines the energy levels of a system. In this way, solving the optimisation problem becomes a matter of getting a quantum system into this ground state so that it can be measured. However, for any non-trivial problem this Hamiltonian will be incredibly complicated and so getting a quantum system into its ground state is not a simple task.In traditional Quantum Annealing, the system is initialised in the ground state of a simple Hamiltonian before slowly transitioning to the Hamiltonian defining the problem. By what is known as the Adiabatic theorem, the system will remain in the ground state throughout this transition provided it happens slowly enough and so the system ends in the ground state of the problem Hamiltonian. However, for some problems, this must occur intractably slowly to avoid transitions away from the ground state - destroying any chance of quantum speedup.My PhD will examine a potential solution to this problem in which transitions to higher energy states are allowed to occur but in such a way that ensures the system has returned to the ground state by the end of the algorithm. This is achieved through the introduction of a third Hamiltonian that facilitates these transitions back into the ground state. This so-called diabatic approach to quantum annealing has the potential to lead to dramatic speedup over its adiabatic counterpart and indeed over classical algorithms.