This dissertation presents and prove the viability of a non-standard method for controlling the state of a quantum system by modifying its boundary conditions instead of relying on the action of external fields. The standard approach to quantum control bases on the use of an external field to manipulate the system. Some technological difficulties appear when controlling a quantum system in this way, due to the complications of manipulating a system made of few particles while maintaining the quantum correlations. As a consequence the systems need to be kept at very low temperatures and the interactions have to be performed very fast. The Quantum Control at the Boundary approach is radically different to the standard one. Instead of seeking the control of the quantum system by directly interacting with it through an external field, the control is achieved by manipulating the boundary conditions of the system. The spectrum of a quantum system, for instance an electron moving in a box, depends on the boundary conditions imposed on it. Hence, a modification of such boundary conditions modifies the state of the system allowing for its manipulation and, eventually, its control. This kind of interaction is weaker, which makes one to expect that it may help maintaining the quantum correlations. For showing the viability of the Quantum Control at the Boundary method, a family of boundary control systems on Quantum Circuits (a generalization of quantum grahs) is introduced. Before being able to address the problem of controllability, the problem of existence of solutions for the Schrödinger equation with time-dependent boundary conditions is addressed. The approximate controllability of the systems under study is proven using a controllability result by T. Chambrion et al. (2009) and a stability result which constitutes another original contribution of this dissertation.
The problem of quantum state preparation is one of the main challenges in achieving the quantum advantage. Furthermore, classically, for multi-level problems, our ability to solve the corresponding quantum optimal control problems is rather limited. The ability of the latter to feed into the former may result in significant progress in quantum computing. To address this challenge, we propose a formulation of quantum optimal control that makes use of artificial boundary conditions for the Schrödinger equation in combination with spectral methods. The resulting formulations are well suited for investigating periodic potentials and lend themselves to direct numerical treatment using conventional methods for bounded domains.
Farhi et al. recently proposed a class of quantum algorithms, the quantum approximate optimization algorithm (QAOA), for approximately solving combinatorial optimization problems (E. Farhi et al., arXiv:1411.4028; arXiv:1412.6062; arXiv:1602.07674). A level-
p QAOA circuit consists of
p steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The
2p times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As
p increases, parameter optimization becomes inefficient due to the curse of dimensionality. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here we analytically and numerically study parameter setting for the QAOA applied to MaxCut. For the level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher
p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MaxCut, the ``ring of disagrees,'' or the one-dimensional antiferromagnetic ring, we provide an analysis for an arbitrarily high level. Using a fermionic representation, the evolution of the system under the QAOA translates into quantum control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of the QAOA for any
p. It also greatly simplifies the numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional submanifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.
We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in arXiv:1811.03126 [cs.CC]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In another region of the parameter space which is larger than the previously known FPRASable region, we show that computing the partition function can be reduced to (with or without approximation) counting perfect matchings. Moreover, we give a complete characterization of nonnegatively weighted (not necessarily planar) 4-ary matchgates, which has been open for several years. The key ingredient of our proof is a geometric lemma. We also identify a region of the parameter space where approximating the partition function on planar 4-regular graphs is feasible but on general 4-regular graphs is equivalent to approximately counting perfect matchings. To our best knowledge, these are the first problems of this kind.