Can quantum search accelerate evolutionary algorithms?
Proceedings of the 12th annual conference on Genetic and evolutionary computation
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The last decade witnessed a great surge of fruitful studies in the new paradigm of quantum computing. Remarkable progress has been made in all the areas including quantum algorithms, quantum complexity theory, quantum error-correcting code, quantum information theory, quantum cryptography and physical implementations of quantum computers. However, quantum algorithm design, probably the most important task in quantum computing, did not make as much progress as people expected. This thesis aims at both designing new quantum algorithms and studying why efficient quantum algorithms are so hard to design by proving many lower bounds on quantum query complexity for various problems. We first review (and generalize) known techniques for designing quantum algorithms and for proving quantum lower bounds. Using these techniques, we then study the quantum query complexity for many problems, including some specific problems such as Graph Matching and other natural graph properties, the general tuple search, and local search. We also study the lowest possible quantum query complexity for a class of functions as a whole, such as graph properties, circular functions, and functions invariant of a transitive permutation group, where tight results are given in all these cases. Finally for the quantum lower bounds techniques themselves, we study the main two techniques and show that they are incomparable in power. For the most widely used one, which has a lot of parameters to choose, we show limitations for it in terms of certificate complexity. This implies that we cannot use the method to prove better lower bounds for almost all known open problems in this area.