Identifying jets formed in high-energy particle collisions requires solving optimization problems over potentially large numbers of final-state particles. In this work, we consider the possibility of using quantum computers to speed up jet-clustering algorithms. Focusing on the case of electron-positron collisions, we consider a well-known event shape, called thrust, whose optimum corresponds to the most jet-like separating plane among a set of particles, thereby defining two-hemisphere jets. We show how to formulate thrust both as a quantum annealing problem and as a Grover search problem. A key component of our analysis is the consideration of realistic models for interfacing classical data with a quantum algorithm. With a sequential computing model, we show how to speed up the well-known O(N^3) classical algorithm to an O(N^2) quantum algorithm, including the O(N) overhead of loading classical data from N final-state particles. Along the way, we also identify a way to speed up the classical algorithm to O(N^2 log N) using a sorting strategy inspired by the SISCone jet algorithm, which has no natural quantum counterpart. With a parallel computing model, we achieve O(N log N) scaling in both the classical and quantum cases. Finally, we consider the generalization of these quantum methods to other jet algorithms more closely related to those used for proton-proton collisions at the Large Hadron Collider.