Quantum walks and ground state problems

Abstract

Since the appearance of Shor's factoring algorithm in 1994, the search for novel quantum computer algorithms has proved surprisingly difficult. Two design approaches that have yielded some progress are quantum walks and adiabatic computing. The former has been shown to speed up algorithms whose complexity is related to the classical hitting time of a symmetric Markov chain, and there is evidence that the latter speeds up simulated annealing algorithms for computing ground states of classical Hamiltonians. In this thesis, we look into the possibility of obtaining a quantum speedup for the mixing time of a symmetric Markov chain. We prove that by subjecting a quantum walk to a small amount of decoherence (typically the adversary of a quantum computer), it can be forced to mix to the correct stationary distribution, often considerably faster than its classical counterpart. A more general theorem to this effect would imply quantum speedups for a variety of approximation algorithms for #P-complete problems. We conclude with some observations on adiabatic computing—a time-dependent generalization of the quantum walk framework—and the problem of estimating the ground state energy of a quantum Hamiltonian with local spin interactions.