Controlled Quantum Amplification

We propose a new framework for turning quantum search algorithms that decide into quantum algorithms for finding a solution. Consider we are given an abstract quantum search algorithm A that can determine whether a target g exists or not. We give a general construction of another operator U that both determines and finds the target, whenever one exists. Our amplification method at most doubles the cost over using A, has little overhead, and works by controlling the evolution of A. This is the first known general framework to the open question of turning abstract quantum search algorithms into quantum algorithms for finding a solution. We next apply the framework to random walks. We develop a new classical algorithm and a new quantum algorithm for finding a unique marked element. Our new random walk finds a unique marked element using H update operations and 1/eps checking operations. Here H is the hitting time, and eps is the probability that the stationary distribution of the walk is in the marked state. Our classical walk is derived via quantum arguments. Our new quantum algorithm finds a unique marked element using H^(1/2) update operations and 1/eps^(1/2) checking operations, up to logarithmic factors. This is the first known quantum algorithm being simultaneously quadratically faster in both parameters. We also show that the framework can simulate Grover\u27s quantum search algorithm, amplitude amplification, Szegedy\u27s quantum walks, and quantum interpolated walks

Efficient Framework for Quantum Walks and Beyond

In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk $P$ and a single marked state, the quantum walk corresponding to $P$ can find the solution using a number of queries that is quadratically smaller than the classical hitting time of $P$. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal