Passarelli, Gianluca (2021) Quantum annealing and advanced optimization strategies of closed and open quantum systems. [Tesi di dottorato]

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Adiabatic quantum computation and quantum annealing are powerful methods designed to solve optimization problems more efficiently than classical computers. The idea is to encode the solution to the optimization problem into the ground state of an Ising Hamiltonian, which can be hard to diagonalize exactly and can involve long-range and multiple-body interactions. The adiabatic theorem of quantum mechanics is exploited to drive a quantum system towards the target ground state. More precisely, the evolution starts from the ground state of a transverse field Hamiltonian, providing the quantum fluctuations needed for quantum tunneling between trial solution states. The Hamiltonian is slowly changed to target the Ising Hamiltonian of interest. If this evolution is infinitely slow, the system is guaranteed to stay in its ground state. Hence, at the end of the dynamics, the state can be measured, yielding the solution to the problem. In real devices, such as in the D-Wave quantum annealers, the evolution lasts a finite amount of time, which gives rise to Landau-Zener diabatic transitions, and occurs in the presence of an environment, inducing thermal excitations outside the ground state. Both these limitations have to be carefully addressed in order to understand the true potential of these devices. The present thesis aims to find strategies to overcome these limitations. In the first part of this work, we address the effects of dissipation. We show that a low-temperature Markovian environment can improve quantum annealing, compared with the closed-system case, supporting other previous results known in the literature as thermally-assisted quantum annealing. In the second part, we combine dissipation with advanced annealing schedules, featuring pauses and iterated or adiabatic reverse annealing, which, in combination with low-temperature environments, can favor relaxation to the ground state and improve quantum annealing compared to the standard algorithm. In general, however, dissipation is detrimental for quantum annealing especially when the annealing time is longer than the typical thermal relaxation and decoherence time scales. For this reason, it is essential to devise shortcuts to adiabaticity so as to reach the adiabatic limit for relatively short times in order to decrease the impact of thermal noise on the performances of QA. To this end, in the last part of this thesis we study the counterdiabatic driving approach to QA. In counterdiabatic driving, a new term is added to the Hamiltonian to suppress Landau-Zener transitions and achieve adiabaticity for any finite sweep rate. Although the counterdiabatic potential is nonlocal and hardly implementable on quantum devices, we can obtain approximate potentials that dramatically enhance the success probability of short-time quantum annealing following a variational formulation.

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