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Bulletin of the American Physical Society

Dimensionality reduction for adiabatic quantum optimizer in presence of local disorder

Gian Giacomo Guerreschi, Salvatore Mandr `a, Al 'an Aspuru-Guzik

Published Mar 4, 2015• 0 citations
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Adiabatic quantum optimization is a procedure to solve a vast class of optimization problems by slowly changing the Hamiltonian of a quantum system. The evolution time necessary for the algorithm to be successful scales inversely with the minimum energy gap encountered during the dynamics. Unfortunately, the direct calculation of the gap is strongly limited by the exponential growth in the dimensionality of the Hilbert space associated to the quantum system. Although many special-purpose methods have been devised to reduce the effective dimensionality, they are strongly limited to particular classes of problems with evident symmetries. Moreover, little is known about the computational power of adiabatic quantum optimizers in real-world conditions. Here we propose and implement a general purposes reduction method that does not rely on any explicit symmetry and which requires, under certain general conditions, only a polynomial amount of classical resources. Thanks to this method, we are able to analyze the performance of ``nonideal'' quantum adiabatic optimizers to solve the well-known Grover problem, namely the search of target entries in an unsorted database, in the presence of discrete local defects. In this case, we show that adiabatic quantum optimization, even if affected by random noise, is still potentially faster than any classical algorithm.
Non-equilibrium phenomena in quantum physics are ubiquitous in current research because of their extremely rich phenomenology. Among them, adiabatic processes are particularly interesting because they connect two of the cornerstone of modern condensed matter theory: topological phases of matter and quantum information. Indeed with adiabatic periodic driving, it is possible to engineer exotic non-equilibrium phases with tunable topological properties, which are of great interest for both fundamental research and future quantum technologies. Adiabatic processes can also be used to implement optimization processes on quantum hardware and lie at the basis of hybrid quantum-classical algorithms that are extremely promising for near-term quantum devices. In this thesis, I focus on these two aspects of adiabatic dynamics in quantum systems. In the first part, I investigate the robustness of topological phases, arising in periodically driven systems, with respect to the driving protocol and disorder. In particular, I show that quantized transport and Anderson localization coexist in one-dimensional systems displaying Thouless pumping, because of a delocalization-localization transition in the Floquet spectrum. This transition is linked to the topological nature of the adiabatic driving and disappears if the phase is trivial in the clean limit. In the second part, I study one of the most popular hybrid optimization method, the quantum approximate optimization algorithm (QAOA). I show that QAOA can tackle the first-order phase transition arising in the infinite range p-spin model with polynomial resources, in stark contrast with adiabatic quantum computation, which requires exponentially long evolution time to reach similar performances. Finally, I present an approach to QAOA based on reinforcement learning (RL). Interestingly, the RL agent automatically adopts strategies that converge towards optimal adiabatic schedules and that can be easily transferred between systems with different sizes, even in the presence of disorder.
The prospects for realizing a topological quantum computer have brightened since the apparent detection of Majorana fermions at the ends of semiconducting nanowires. These Majorana zero-modes persist in the presence of the strong disorder that may be present in such systems, protected by the mobility gap of localized systems. Recent proposals to manipulate quantum information in a Majorana chain involve adiabatically adjusting gate voltages in one-dimensional nanowire networks. However, as we show, in the adiabatic limit, a disordered system with multiple zero-mode Majorana fermions will undergo a non-local response to a local perturbation via a multi-level Landau-Zener transition. This will disperse the quantum information stored in the target zero-modes amongst other zero-modes that are present in the system, complicating the realization of controlled gates. There is however a solution that may work in practice: we show that there is an optimum window of intermediate time-scales during which the quantum information is approximately preserved under manipulation.