We analyze the operation of a sensor based on atom interferometry, which can achieve supraclassical sensitivity by exploiting quantum correlations in mixed states of many qubits. The interferometer is based on quantum gates which use coherently controlled Rydberg interactions between a single atom (which acts as a control qubit) and an atomic ensemble (which provides register qubits). In principle, our scheme can achieve precision scaling with the size of the ensemble - which can extend to large numbers of atoms - while using only single-qubit operations on the control and bulk operations on the ensemble. We investigate realistic implementation of the interferometer, and our main aim is to develop an approach to quantum metrology that can achieve quantum-enhanced measurement precision by exploiting coherent operations on large impure quantum states. We propose an experiment to demonstrate the enhanced sensitivity of the protocol and to investigate a transition from classical to supraclassical sensitivity which occurs when using highly mixed probe states.