Query suggestion algorithms, which aim to suggest a set of similar but independent queries to users, have been widely studied to simplify user searches. However, in many cases, the users will accomplish their search tasks through a sequence of search behaviors instead of by one single query, which may make the classical query suggestion algorithms fail to satisfy end users in terms of task completion. In this paper, we propose a quantum path integral inspired algorithm for personalized user search behavior prediction, through which we can provide sequential query suggestions to assist the users complete their search tasks step by step. In detail, we consider the sequential search behavior of a user as a trajectory of a particle that moves in a query space. The query space is represented by a graph with each node is a query, which is named as query-path graph. Inspired by the quantum theorems, each edge in query-path graph is represented by both amplitude and phase respectively. Using this graph, we modify the quantum path integral algorithm to predict a user's follow-up trajectory based on her behavioral history in this graph. We empirically show that the proposed algorithm can well predict the user search behavior and outperform classical query suggestion algorithms for user search task completion using the search log of a commercial search engine.