Quantum Speed-Up of Markov Chain Based Algorithms

We develop a generic method for quantizing classical algorithms based on random walks. We show that under certain conditions, the quantum version gives rise to a quadratic speed-up. This is the case, in particular, when the Markov chain is ergodic and its transition matrix is symmetric. This generalizes the celebrated result of L. K. Grover (1996)and a number of more recent results, including the element distinctness result of Ambainis and the result of Ambainis, Kempe and Rivosh that computes properties of quantum walks on the d-dimensional torus. Among the consequences is a faster search for multiple marked items. We show that the quantum escape time, just like its classical version, depends on the spectral properties of the transition matrix with the marked rows and columns deleted.