Grover’s algorithm on a Feynman computer

We present an implementation of Grover’s algorithm in the framework of Feynman’s cursor model of a quantum computer. The cursor degrees of freedom act as a quantum clocking mechanism, and allow Grover’s algorithm to be performed using a single, time-independent Hamiltonian. We examine issues of locality and resource usage in implementing such a Hamiltonian. In the familiar language of Heisenberg spin–spin coupling, the clocking mechanism appears as an excitation of a basically linear chain of spins, with occasional controlled jumps that allow for motion on a planar graph: in this sense our model implements the idea of ‘timing’ a quantum algorithm using a continuous-time random walk. In this context we examine some consequences of the entanglement between the states of the input/output register and the states of the quantum clock.