Feynman showed that the path of least action is determined by quantum interference. The interference may be viewed as part of a quantum algorithm for minimising the action. In fact, Lloyd describes the Universe as a giant quantum computer whose purpose is to calculate its own state. Could the direction of time that the universe is apparently following be determined by a quantum algorithm? The answer lies in the violation of time reversal (T) invariance that is being observed in an increasing number of particle accelerator experiments. The violation signifies a fundamental asymmetry between the past and future and calls for a major shift in the way we think about time. Here we show that processes which violate T invariance induce destructive interference between different paths that the universe can take through time. The interference eliminates all paths except for two that represent continuously forwards and continuously backwards time evolution. This suggests that quantum interference from T violation processes gives rise to the phenomenological unidirectional nature of time. A path consisting exclusively of forward steps gives the shortest path to a point which is in the forwards direction. The quantum interference, therefore, underlies a quantum algorithm that determines shortest path through time.